Fractional Kirchhoff-Hardy Problem: Breaking of Resonance and General Existence Results

Abstract

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient: ${\displaystyle M\left({\int \int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}\frac{{|u\left(x\right)-u\left(y\right)|}^q}{{|x-y|}^{N+qs}}dxdy\right){(-\Delta )}^{s}u=\lambda \frac{u}{{\left|x\right|}^{2s}}+f(x,u,\nabla u)\mathrm{in}\Omega ,}$ where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain containing the origin, $s\in (0,1)$, $q\in (1,2]$ with $N>2s$, $\lambda >0$, and f is a measurable non-negative function satisfying suitable hypotheses. The main objective is to establish the existence of positive solutions for the largest possible class of nonlinearities f without imposing restrictions on $\lambda$. Two main cases areconsidered: $\left(I\right)-f(x,u,\nabla u)=u^p+\mu ,\mathrm{and}\left(II\right)-f(x,u,\nabla u)={|\nabla u|}^p+\mu g.$ Existence is proved under suitable hypotheses on $q,p$ and the data $g,\mu$. The results are new, including for the local case $s=1$.

1. Introduction

The present work is concerned with the existence of positive solutions for a class of fractional equation involving a Kirchhoff term and singular potential. More precisely, we consider the problem

$$\left\{\begin{array}{cccc}\hfill {\displaystyle M\left({\int \int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}\frac{{|u\left(x\right)-u\left(y\right)|}^q}{{|x-y|}^{N+qs}}dxdy\right){(-\Delta )}^{s}u}& =& \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+f(x,u,\nabla u)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& >& 0\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(1)

where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain containing the origin, $s\in (0,1)$, $q\in (1,2]$ with $N>2s$, $\lambda >0$ and f is a measurable nonnegative function with suitable hypotheses. M is a positive function defined on ${\mathbb{R}}^{+}$.

Here, for $0<s<1$, the fractional Laplacian ${(-\Delta )}^s$ is defined by

$${(-\Delta )}^{s}u\left(x\right):=a_{N,s}\mathrm{P}.\mathrm{V}.{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{u\left(x\right)-u\left(y\right)}{{|x-y|}^{N+2s}}}dy,$$

(2)

where

$$a_{N,s}:=2^{2s-1}{\pi }^{-{\displaystyle \frac{N}{2}}}{\displaystyle \frac{\Gamma \left(\frac{N+2s}{2}\right)}{|\Gamma (-s\left)\right|}}$$

is the normalization constant such that the identity

$${(-\Delta )}^{s}u={\mathcal{F}}^{-1}{\left(\right|\xi |}^{2s}\mathcal{F}u),\xi \in {\mathbb{R}}^N$$

holds for all $u\in \mathcal{S}\left({\mathbb{R}}^N\right)$, where $\mathcal{F}u$ denotes the Fourier transform of u and $\mathcal{S}\left({\mathbb{R}}^N\right)$ is the Schwartz class of tempered functions.

The problem (1) is related to the following Hardy inequality, proven in [1,2] (see [3]),

$${\displaystyle \frac{a_{N,s}}{2}}\int {\int }_{D_{\Omega }}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+2s}}}dxdy\ge {\Lambda }_{N,s}{\int }_{\Omega }{\displaystyle \frac{u^2}{{\left|x\right|}^{2s}}}dx,u\in {\mathcal{C}}_0^{\infty }\left({\mathbb{R}}^N\right),$$

(3)

where

$$D_{\Omega }:={\mathbb{R}}^N\times {\mathbb{R}}^N\setminus \left(\mathcal{C}\Omega \times \mathcal{C}\Omega \right)$$

(4)

and

$${\Lambda }_{N,s}=2^{2s}{\displaystyle \frac{{\Gamma }^2\left(\frac{N+2s}{4}\right)}{{\Gamma }^2\left(\frac{N-2s}{4}\right)}}.$$

(5)

Notice that ${\Lambda }_{N,s}$ is optimal and not attained.

Hence, if we consider the bilinear form

$$Q(u,u)={\displaystyle \frac{a_{N,s}}{2}}\int {\int }_{D_{\Omega }}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+2s}}}dxdy-\lambda {\int }_{\Omega }{\displaystyle \frac{u^2}{{\left|x\right|}^{2s}}}dx,u\in {\mathcal{C}}_0^{\infty }\left({\mathbb{R}}^N\right),$$

(6)

then Q is coercive in a suitable fractional Sobolev space if and only if $\lambda \le {\Lambda }_{N,s}$, see [1,2]. This restriction on the value of $\lambda$ does not only affect the study of problems with a variational structure, but also influences problems with general data. In such cases, the condition appears as a necessary requirement and has a significant impact on the regularity of the solution, even when solutions are understood in weaker senses.

Notice that local and non-local problems related to the Hardy potential were widely studied in the last years. Before establishing the main result of our paper, let us begin with some previous results.

In the case where $M\equiv 1$ and $f(x,u,\nabla u)=f(x,u)$, then the previous problem takes the form

$${(-\Delta )}^{s}u=\lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+f(x,u)\mathrm{in}\Omega ,u\ge 0 \mathrm{in} \Omega .$$

(7)

It is well known that, in order for problem (7) to admit a solution, it is necessary that the condition $\lambda \le {\Lambda }_{N,s}$ holds, together with additional assumptions depending on the nonlinear nature of f.

Setting $w_{\alpha }={\left|x\right|}^{-{\displaystyle \frac{N-2s}{2}}+\alpha }$ where $\alpha$ is obtained by the equality

$$\lambda =\lambda \left(\alpha \right)=\lambda (-\alpha )={\displaystyle \frac{2^{2s}\Gamma \left(\frac{N+2s+2\alpha }{4}\right)\Gamma \left(\frac{N+2s-2\alpha }{4}\right)}{\Gamma \left(\frac{N-2s+2\alpha }{4}\right)\Gamma \left(\frac{N-2s-2\alpha }{4}\right)}}.$$

(8)

It follows that $w_{\alpha }$ solves the equation

$${(-\Delta )}^{s}w=\lambda {\displaystyle \frac{w}{{\left|x\right|}^{2s}}} \mathrm{in} ({\mathbb{R}}^N\setminus \left\{0\right\}).$$

(9)

From [4], we know that if $u\ge 0$ in ${\mathbb{R}}^N$ satisfies

$${(-\Delta )}^{s}u\ge \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}} \mathrm{in} B_r\left(0\right),$$

then $u\left(x\right)\ge {C\left|x\right|}^{-{\gamma }_{\lambda }}$ in $B_{\rho }\left(0\right)$ with ${\gamma }_{\lambda }={\displaystyle \frac{N-2s}{2}}-\alpha \left(\lambda \right)$.

Now, always in the case $M=1$, for $\lambda \in (0,{\Lambda }_{N,s}]$ fixed and taking into consideration the singular behavior near the origin of the Hardy potential, it holds that additional conditions on f are needed in order to get the existence of a solution. More precisely:

• If f does not depend on u ($f(x,u):=f\left(x\right))$, then problem (1) (with $M=1$) has a solution if and only if ${\displaystyle {\int }_{\Omega }{\left|x\right|}^{-{\gamma }_{\lambda }}f\left(x\right)dx<\infty }$ where ${\gamma }_{\lambda }$ is given as above.
• If $f(x,u)=u_{+}^p$, then existence of solutions holds if and only if $p<p_{+}\left(\lambda \right)={\displaystyle \frac{{\gamma }_{\lambda }+2s}{{\gamma }_{\lambda }}}$. We refer, for instance, to [4,5] for more details.
• In the case where $f(x,u,\nabla u)={|\nabla u|}^q+\mu g$, then the authors in [6] proved that the existence of a solution holds if and only if $\lambda \le {\Lambda }_N$ and $q<q_{+}\left(\lambda \right)={\displaystyle \frac{{\gamma }_{\lambda }+2s}{{\gamma }_{\lambda }+1}}<2$.

Where M is not identically constant, then for the local case $s=1$ and without the Hardy potential, the problem can be expressed in the general form

$$\left\{\begin{array}{ccc}\hfill {\displaystyle -M\left({\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\right)\Delta u}& =& g(x,u)\mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\mathrm{on}\partial \Omega .\hfill \end{array}\right.$$

(10)

Problem (10) is well known as a Kirchhoff-type problem, see [7], and is widely studied in the literature. The first motivation was the study of the stationary solution to the classical equation

$$\rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\left({\displaystyle \frac{p_0}{h}}+{\displaystyle \frac{E}{2L}}{\int }_0^L{\left|{\displaystyle \frac{\partial u}{\partial x}}\right|}^{2}dx\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0.$$

This is an extension of the classical d’Alembert’s wave equation by considering the effects of the changes in the length of the string during the vibrations. In this equation, L is the length of the string, h is the area of the cross-section, E is the Young modulus of the material, $\rho$ is the mass density and $p_0$ is the initial tension. The stationary multidimensional Kirchhoff equation given by (10) has received much attention after the work of Lions [8], where a functional setting was proposed to the problem.

The Kirchhoff function M is assumed Lipschitz-continuous, but not always monotone, even if the model proposed by Kirchhoff in 1883, $M\left(t\right)=a+b{t}^{\beta }$ where $a,b\ge 0$ and $\beta >0$, is monotone; see discussions in [9,10]. If $M\left(0\right)=0$, the Kirchhoff problem is called degenerate; we may refer to [11] and references therein for more details on this case.

In [12], using variational arguments, the authors proved the existence of a positive solution of (10) if M is a positive non-increasing function and g satisfies the Ambrosetti–Rabinowitz condition. The case where M is a non-decreasing function was considered in [13]. In [14], the author collected some related problems to (10) and unified the presentation of the results. In particular, for g such that $g(x,u)u\le a{\left|u\right|}^2+b\left|u\right|$ ($a,b>0$), he showed both existence and uniqueness of solutions. In the case where $g(x,u)$ is asymptotically linear at infinity with respect to u, some existence results are proved in [15]. The singular case was considered in [16].

The case of a general potential, which may be singular or degenerate, has also been addressed in [17,18]. In these works, the potential enters as an absorption term, which facilitates the derivation of precise estimates and ensures the establishment of suitable a priori bounds.

It is important to emphasize that a key feature of these previous studies is the presence of a variational structure, which enables the authors to exploit variational methods in the construction of solutions. Moreover, when perturbations by a linear term are considered, the derivation of the corresponding estimates relies critically on coercivity assumptions and suitable regularity conditions on the data. See also [19,20,21,22] where some partial existence results are obtained under the coercivity of the main operator.

The non-local case, $s\in (0,1)$, was considered recently by several authors, including the critical and singular cases. We refer to [9,23,24] and references therein for further details.

The novelty of the present paper lies in uncovering a nontrivial interaction between the Kirchhoff term and the Hardy potential, which allows us to establish the existence of nonnegative solutions for a broad class of data f, without imposing any restriction on the parameter $\lambda$.

In contrast to the purely linear problem involving the Hardy potential, whose solvability is governed by a spectral condition requiring $\lambda$ to remain below the optimal Hardy constant, our analysis shows that the Kirchhoff structure fundamentally alters this scenario. In the linear framework, existence is typically constrained by resonance phenomena and may also require additional assumptions on the datum f. Here, the nonlocal nature of the Kirchhoff coefficient changes the balance of the equation and prevents the spectral obstruction associated with the Hardy operator. In this sense, the Kirchhoff term breaks the resonance mechanism characteristic of the linear problem and allows solvability for every $\lambda >0$, thereby significantly enlarging the admissible range of parameters and data.

Kirchhoff-type equations with gradient terms arise naturally in nonlinear models where the diffusion coefficient depends on the global energy of the solution, while lower-order contributions involve the gradient explicitly.

From a physical standpoint, these models can be used to describe vibrating strings, membranes, or beams in which the tension is not constant but depends on the total deformation, or equivalently on the global energy, of the structure.

This combination produces a strongly nonlocal and nonhomogeneous structure, which significantly complicates the analytical framework. In particular, the presence of a gradient term generally destroys the standard variational setting and may lead to a loss of compactness, making classical existence techniques inapplicable.

One of the main contributions of this work is the identification of a delicate interaction between the nonlocal Kirchhoff coefficient, the Hardy potential, and the gradient term. This interplay compensates for the inherent lack of compactness and allows us to establish robust existence results.

Our approach shows that the Kirchhoff structure modifies the energy balance of the equation in a way that compensates for the difficulties introduced by the Hardy potential and the gradient term, allowing solvability under weaker assumptions on the data.

Notice that the restriction $s>{\displaystyle \frac{1}{2}}$ is natural taking into consideration the regularity result obtained in [25], see Theorem 5 and Remark 7 below.

The result covers also the local case $s=1$.

Through this paper we will assume that M is a continuous function such that:

(M1)

$M:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is increasing,

(M2)

$\exists m_0>0;M\left(t\right)\ge m_0\forall t\in {\mathbb{R}}^{+}$,

(M3)

$\underset{t\to \infty }{\lim }M\left(t\right)=+\infty$.

We begin by analyzing the case where f does not depend on u, namely $f(x,u):=f\left(x\right)$. Then, for $q=2$, and hence the problem has a variational structure, we are able to prove the next result.

Theorem 1.

Assume that $f\in L^2(\Omega )$ is such that $f\gneqq 0$, then for all $\lambda >0$, problem (1), with $q=2$, has a unique positive solution, $\overline{u}\in H_0^s(\Omega )$.

We mention here the strong regularity enjoyed by the solution. In the case where $M\equiv 1$, such a regularity is false, including for $f\in L^{\infty }(\Omega )$, where $\lambda$ is closed to the Hardy constant ${\Lambda }_{N,s}$.

In the general case $q<2$, we get the existence of a solution for a large class of the datum f. More precisely, we have

Theorem 2.

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded domain such that $0\in \Omega$ and $1<q<{\displaystyle \frac{N}{N-s}}$. Suppose that $f\in L^1(\Omega )$ is such that ${\displaystyle {\int }_{\Omega }f{\left|x\right|}^{-\theta }dx<c}$ for some positive constant θ. Then, for all $\lambda >0$, problem (1) has a minimal solution $\overline{u}$ such that $\overline{u}\in W_0^{s,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-s}}$.

As a consequence, it holds that problem (1) has a solution for all $\lambda >0$ and for all $f\in L^{\sigma }(\Omega )$ with $\sigma >1$.

In the second part of the paper, we treat the case where $f(x,u)=u_{+}^p+\mu g$. In this case, if $1<p<2_s^{*}-1$, then under suitable hypotheses on $\mu$ and g, we will show the existence of a solution without any restriction on $\lambda$. Our approach will not use any comparison principle, and then we allow a more general class of datum g under the suitable integrability condition.

To conclude our results, we will consider the case where the nonlinear term depends also on $|\nabla u|$. In the local case, this class of problem appears when considering a function energy in the form

$$J\left(u\right)={\displaystyle \frac{1}{2}}\widehat{M}({\int }_{\Omega }{k\left(u\right)|\nabla u|}^{2}dx)-{\int }_{\Omega }H(x,u)dx,$$

with $\widehat{M}\left(t\right)={\int }_0^{t}M\left(\tau \right)d\tau$. Notice that $J^{\prime }$ is given by

$$⟨J^{\prime }\left(u\right),v⟩=M({\int }_{\Omega }{k\left(u\right)|\nabla u|}^{2}dx\left)\right(k\left(u\right)(-\Delta u)+k^{\prime }{\left(u\right)|\nabla u|}^2)-{\int }_{\Omega }{H}^{\prime }(x,u)vdx.$$

Hence, the interest to consider the case where f depends also on the gradient of u.

This paper is organized as follows. In Section 2, we provide some results related to fractional Sobolev spaces and some functional inequalities. The sense for which solutions are defined is also presented, namely energy solutions and weak solutions.

The case where f depends only on x is treated in Section 3. We will consider both cases, variational and non-variational structure. We also prove a useful compactness result that will be used by the sequel.

In Section 4, we treat the case where $f(x,u)=u^p+\mu g$. According to some condition on p, we are able to get the existence for the largest possible class of the datum g. The main difficulty is the fact that we cannot use the monotony argument to show the existence. Finally, in Section 5, we consider the case where f also depends on the gradient of u. Further results and open problems are given in the last section.

2. Preliminaries and Functional Setting

Let $\Omega \subset {\mathbb{R}}^N$, for $s\in (0,1)$ and $1\le p<+\infty$. The fractional Sobolev spaces $W^{s,p}(\Omega )$ are defined by

$$W^{s,p}(\Omega )\equiv \left\{\varphi \in L^p(\Omega ):{\int }_{\Omega }{\int }_{\Omega }{\displaystyle \frac{{|\varphi \left(x\right)-\varphi \left(y\right)|}^p}{{|x-y|}^{N+ps}}}dxdy<+\infty \right\}.$$

$W^{s,p}(\Omega )$ is a Banach space endowed with the following norm

$${\displaystyle {\parallel \varphi \parallel }_{W^{s,p}(\Omega )}={\left({\int }_{\Omega }{\left|\varphi \left(x\right)\right|}^{p}dx\right)}^{\frac{1}{p}}+{\left({\int }_{\Omega }{\int }_{\Omega }\frac{{|\varphi \left(x\right)-\varphi \left(y\right)|}^p}{{|x-y|}^{N+ps}}dxdy\right)}^{\frac{1}{p}}.}$$

We define the space $W_0^{s,p}(\Omega )$ as

$$W_0^{s,p}(\Omega ):=\left\{u\in W^{s,p}\left({\mathbb{R}}^N\right):u=0 \mathrm{in} {\mathbb{R}}^N\setminus \Omega \right\}.$$

If $\Omega$ is a bounded regular domain, then using Poincaré type inequality, we can endow $W_0^{s,p}(\Omega )$ with the equivalent norm

$${\parallel u\parallel }_{W_0^{s,p}(\Omega )}:={\left({\int \int }_{D_{\Omega }}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{N+sp}}}dxdy\right)}^{1/p},$$

where $D_{\Omega }$ is defined in (4).

The next Sobolev inequality is proved in [26].

Theorem 3 (Fractional Sobolev inequality).

Assume that $0<s<1,p>1$ in order to get $ps<N$. Then, there exists a positive constant $S\equiv S(N,s,p)$ such that for all $v\in {\mathbf{C}}_0^{\infty }\left({\mathbb{R}}^N\right)$,

$${\displaystyle {\int \int }_{{\mathbb{R}}^{2N}}\frac{{|v\left(x\right)-v\left(y\right)|}^p}{{|x-y|}^{N+ps}}dxdy\ge S{\left({\int }_{{\mathbb{R}}^N}{\left|v\left(x\right)\right|}^{p_s^{*}}dx\right)}^{\frac{p}{p_s^{*}}},}$$

where $p_s^{*}={\displaystyle \frac{pN}{N-ps}}$.

In the particular case $p=2$, $H^s(\Omega ):=W^{s,2}(\Omega )$ turns out to be a Hilbert space. If $\Omega ={\mathbb{R}}^N$, the Fourier transform provides yet an alternative definition; more precisely, the Plancherel identity leads to

$${\int }_{{\mathbb{R}}^N}{\left|\xi \right|}^{2s}{\left|\mathcal{F}\left(u\right)\left(\xi \right)\right|}^{2}d\xi ={\displaystyle \frac{a_{N,s}}{2}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+2s}}}dxdy.$$

The operator ${(-\Delta )}^s$ defined for $u\in \mathcal{S}\left({\mathbb{R}}^N\right)$ by

$${(-\Delta )}^{s}u:=a_{N,s}\mathrm{P}.\mathrm{V}.{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{u\left(x\right)-u\left(y\right)}{{|x-y|}^{N+2s}}}dy,s\in (0,1),$$

can be extended by density from $\mathcal{S}\left({\mathbb{R}}^N\right)$ to $H^s\left({\mathbb{R}}^N\right)$. In this way, the associated scalar product can be reformulated as

$$\begin{array}{cc}\hfill {⟨u,v⟩}_{H^s\left({\mathbb{R}}^N\right)}& :=⟨{(-\Delta )}^{s}u,v⟩+(u,v)\hfill \\ \hfill & :=\mathrm{P}.\mathrm{V}.{\int \int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}{\displaystyle \frac{\left(u\right(x)-u(y\left)\right)\left(v\right(x)-v(y\left)\right)}{{|x-y|}^{N+2s}}}dxdy+{\int }_{{\mathbb{R}}^N}uvdx.\hfill \end{array}$$

In the case of bounded domain, if we denote by $H^{-s}(\Omega ):={\left[H_0^s(\Omega )\right]}^{*}$ the dual space of $H_0^s(\Omega )$, then

$${(-\Delta )}^s:H_0^s(\Omega )\to H^{-s}(\Omega ),$$

is a continuous operator.

We state, by the next definition, the sense in which the solution is defined.

Definition 1.

Assume that $h\in L^1(\Omega )$ and consider the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^{s}u& =& h\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(11)

we say that $u\in L^1(\Omega )$ is a weak solution to (11) if $u=0$ in ${\mathbb{R}}^N\setminus \Omega$ and for all $\psi \in {\mathbb{X}}_s$, we have

$${\int }_{\Omega }u\left({(-\Delta )}^s\psi \right)dx={\int }_{\Omega }h\psi dx.$$

where

$${\mathbb{X}}_s\equiv \left\{\psi \in \mathcal{C}\left({\mathbb{R}}^N\right)|\mathit{supp}\left(\psi \right)\subset \overline{\Omega },{(-\Delta )}^s\psi \left(x\right)\mathit{pointwise}\mathit{defined}\mathit{and}\left|{(-\Delta )}^s\psi \left(x\right)\right|<C\mathit{in}\Omega \right\}.$$

Remark 1.

With respect to the above definition, it is appropriate in view of the fact that if $\psi \in {\mathcal{C}}_0^{\infty }(\Omega )$, then ${(-\Delta )}^s\psi \in \mathcal{S}\left({\mathbb{R}}^N\right)$. This justifies the choice of the corresponding space of test functions. Such a framework is consistent with the nonlocal nature of the operator and allows for a proper interpretation of the Dirichlet condition outside Ω. Moreover, as shown in [6,25], the solution is unique and can be constructed through an iteration scheme, which coincides with the notion of SOLA (Solutions Obtained as Limits of Approximations) or entropy solutions introduced for the classical Laplacian.

The next existence result is proved in [27,28]; see also [29].

Theorem 4.

Assume that $h\in L^1(\Omega )$, then problem (11) has a unique weak solution u obtained as the limit of the sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}$, where $u_n$ is the unique solution of the approximating problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^s{u}_n& =& h_n\left(x\right)\hfill & \mathrm{in} \Omega ,\hfill \\ \hfill u_n& =& 0\hfill & \mathrm{in} {\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(12)

with $h_n=T_n\left(h\right)$ and $T_n\left(m\right)=max(-n,min(n,m))$. Moreover, $T_k\left(u_n\right)\to T_k\left(u\right)\mathit{strongly}\mathit{in}{H}_0^s(\Omega )$ for all $k>0$. In addition, we have $u\in L^{\theta }(\Omega )$ for all $\theta \in [1,{\displaystyle \frac{N}{N-2s}})$ and $|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|\in L^r(\Omega )$ for all $r\in [1,{\displaystyle \frac{N}{N-s}})$. Furthermore, $u_n\to u strongly in W_0^{s,q_1}(\Omega ) for all q_1<{\displaystyle \frac{N}{N-s}}$.

Subsequently, we get a compactness result, which will be systematically used in the sequel.

Theorem 5.

Assume that $1<q<{\displaystyle \frac{N}{N-s}}$ is fixed and consider the operator $D:L^1(\Omega )\to W_0^{s,q}(\Omega )$ defined by $D\left(h\right)=u$, where u is the unique solution to problem (11). Then, D is a compact operator; moreover, there exists a positive constant $C\equiv C(\Omega ,s,N,q)$, such that

$${\left|\right|u\left|\right|}_{W_0^{s,q}(\Omega )}\le {C\left|\right|h\left|\right|}_{L^1(\Omega )}.$$

(13)

In addition, if $s>{\displaystyle \frac{1}{2}}$, then $u\in W_0^{1,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-2s+1}}$ and

$${\left|\right|u\left|\right|}_{W_0^{1,\sigma }(\Omega )}\le \widehat{C}{\left|\right|h\left|\right|}_{L^1(\Omega )}.$$

(14)

For $\sigma <{\displaystyle \frac{N}{N-2s+1}}$ fixed, setting $\widehat{D}:L^1(\Omega )\to W_0^{1,\sigma }(\Omega )$ defined by $\widehat{D}\left(h\right)=u$, then $\widehat{D}$ is compact.

Remark 2.

The proof of the preceding compactness result follows exactly from Propositions 2.3 and 2.4 in [25], where interpolation identities and the properties of the inverse fractional Laplacian are employed. For the convenience of the reader, we briefly recall here the main estimates leading to the corresponding Sobolev regularity. According to the previous theorem, if u denotes the unique solution to problem (11), then it admits the representation formula

$$u\left(x\right)={\int }_{\Omega }{\mathcal{G}}_s(x,y)h\left(y\right)dy,$$

where ${\mathcal{G}}_s:{\mathbb{R}}^{2N}\to \mathbb{R}$ denotes the Green kernel associated with ${(-\Delta )}^s$ in Ω.

From [30] (Theorem 1.1) and [31] (Corollary 3.3), it follows that for a.e. $x,y\in \Omega$,

$${\mathcal{G}}_s(x,y)\le {\displaystyle \frac{C}{{|x-y|}^{N-2s}}},$$

(15)

and

$$|{\nabla }_x{\mathcal{G}}_s(x,y)|\le Cmax\left\{{\displaystyle \frac{1}{{|x-y|}^{N-2s+1}}},{\displaystyle \frac{1}{{|x-y|}^{N-s}}}{\displaystyle \frac{1}{{\delta }^{1-s}\left(x\right)}}\right\},$$

(16)

where $\delta \left(x\right)=\mathrm{dist}(x,\partial \Omega )$. Consequently,

$$\left|u\left(x\right)\right|\le C_0{\int }_{\Omega }{\displaystyle \frac{\left|h\right(y\left)\right|}{{|x-y|}^{N-2s}}}dy,$$

and

$$|\nabla u\left(x\right)|\le C_1{\int }_{\Omega }{\displaystyle \frac{\left|h\right(y\left)\right|}{{|x-y|}^{N-2s+1}}}dy+{\displaystyle \frac{C_2}{{\delta }^{1-s}\left(x\right)}}{\int }_{\Omega }{\displaystyle \frac{\left|h\right(y\left)\right|}{{|x-y|}^{N-s}}}dy.$$

By the classical estimates for singular integrals (see, e.g., [32] (Theorem I, Section 1.2, Chapter V)), we deduce

$${\parallel u\parallel }_{L^{\sigma }(\Omega )}\le C{\parallel h\parallel }_{L^1(\Omega )}\mathit{for}\mathit{all}\sigma <{\displaystyle \frac{N}{N-2s}},$$

and

$${\parallel \nabla u\parallel }_{L^{\theta }(\Omega )}\le C{\parallel h\parallel }_{L^1(\Omega )}\mathit{for}\mathit{all}\theta <{\displaystyle \frac{N}{N-2s+1}}.$$

Observe that

$${\left({\displaystyle \frac{N}{N-2s}}\right)}^{*}={\displaystyle \frac{N}{N-2s+1}},$$

which is consistent with the corresponding Sobolev conjugation.

In what follows, C denotes any positive constant that does not depend on the solution and that can change from one line to another.

3. Existence Results in the Case $\mathit{f}(\mathit{x},\mathit{u}):=\mathit{f}\left(\mathit{x}\right)$

For the sake of simplicity, we set ${\displaystyle {\left|\right|u\left|\right|}_{s,q}^q={\int \int }_{D_{\Omega }}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^q}{{|x-y|}^{N+qs}}}dxdy.}$

Along this section, we will assume that the function M satisfies the conditions $\left(M1\right),\left(M2\right)$ stated in the introduction.

Recall that we are considering the problem

$$\left\{\begin{array}{cccc}\hfill {M\left(\right|\left|u\right||}_{s,2}^2{)(-\Delta )}^{s}u& =& \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+f\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& >& 0\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(17)

Our first existence result consists of the following.

Theorem 6.

Assume that $f\in L^2(\Omega )$ is such that $f\gneqq 0$, then for all $\lambda >0$, problem (17) has a unique positive solution $\overline{u}$ in $H_0^s(\Omega )$.

Proof. 

Let $f\in L^2(\Omega )$ and fix $\lambda >0$; we will proceed by approximations. Let us consider the problem

$$\left\{\begin{array}{cccc}\hfill M\left(\right||u_n{\left|\right|}_{s,2}^2{)(-\Delta )}^s{u}_n& =& \lambda {\displaystyle \frac{u_n^{+}}{(1+\frac{1}{n}{u}_n^{+}){\left(\right|x|}^{2s}+\frac{1}{n})}}+f\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u_n& >& 0\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u_n& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(18)

Define $K_n\left(\sigma \right)={\displaystyle \frac{{\sigma }^{+}}{1+\frac{1}{n}{\sigma }^{+}}}$, then it is clear that the solutions of problem (18) are critical points of the functional $J_n$ given by

$$J_n\left(v\right)={\displaystyle \frac{1}{2}}\widehat{M}\left({∥v∥}_s^2\right)-\lambda {\int }_{\Omega }{\displaystyle \frac{{\widehat{K}}_n\left(v^{+}\right)}{{\left|x\right|}^{2s}+\frac{1}{n}}}dx-{\int }_{\Omega }fvdx,$$

where

$$\widehat{M}\left(t\right)={\int }_0^{t}M\left(\tau \right)d\tau ,{\widehat{K}}_n\left(t\right)={\int }_0^t{K}_n\left(\tau \right)d\tau .$$

Taking into account the behavior of M, and using Hölder’s inequality together with the fractional Sobolev embedding, we deduce that

$$J_n\left(v\right)\ge c_1{\parallel v\parallel }_s^2-{\parallel f\parallel }_{L^2(\Omega )}{\parallel v\parallel }_s-C,$$

for some positive constants $c_1$ and C. Hence, $J_n$ is coercive and bounded from below.

We now prove that $J_n$ is weakly sequentially lower semicontinuous. Let $v_k\rightharpoonup v$ weakly in $H_0^s(\Omega )$. Since the norm is weakly lower semicontinuous and $\widehat{M}$ is continuous and nondecreasing, we have $\widehat{M}\left({\parallel v\parallel }_s^2\right)\le {lim\; inf}_{k\to \infty }\widehat{M}(\parallel v_k{\parallel }_s^2).$ Moreover, as $f\in L^2(\Omega )$ and $H_0^s(\Omega )\hookrightarrow L^2(\Omega )$ continuously, it follows that ${\int }_{\Omega }f{v}_{k}dx\to {\int }_{\Omega }fvdx.$

Since $v_k\rightharpoonup v$ in $H_0^s(\Omega )$, by the compact fractional Sobolev embedding we have, up to a subsequence, $v_k\to v\mathrm{strongly}\mathrm{in}{L}^r(\Omega ),\mathrm{for}\mathrm{all}1\le r<2_s^{*}.$ In particular, $v_k^{+}\to v^{+}$ strongly in $L^r(\Omega )$. For fixed n, the function $K_n$ is globally Lipschitz; hence, by the dominated convergence theorem, ${\int }_{\Omega }{\displaystyle \frac{{\widehat{K}}_n\left(v_k^{+}\right)}{{\left|x\right|}^{2s}+\frac{1}{n}}}dx\to {\int }_{\Omega }{\displaystyle \frac{{\widehat{K}}_n\left(v^{+}\right)}{{\left|x\right|}^{2s}+\frac{1}{n}}}dx.$ Collecting the previous estimates, we conclude that $J_n$ is weakly sequentially lower semicontinuous. Define

$$D_n=\underset{H_0^s(\Omega )\setminus \left\{0\right\}}{inf}{J}_n\left(u\right),$$

then we reach that $D_n$ is achieved, and then we get the existence of $u_n\in H_0^s(\Omega )$ that solves (18). We claim that the sequence ${\left\{u_n\right\}}_n$ is bounded in $H_0^s(\Omega )$. We argue by contradiction. Assume that $\left|\right|u_n{\left|\right|}_{s,2}\to \infty$ as $n\to \infty$. Taking $u_n$ as a test function in (18) and using Hardy and Hölder inequalities, it follows that

$$\begin{array}{c}{\displaystyle M\left(\right||u_n{\left|\right|}_{s,2}^2\left)\right||u_n{\left|\right|}_{s,2}^2\le \lambda {\int }_{\Omega }{\displaystyle \frac{{\left|u_n\right|}^2}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }f{u}_{n}dx}\hfill \\ \le {\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}{∥u_n∥}_{s,2}^2+{∥f∥}_{L^2(\Omega )}\left|\right|u_n{\left|\right|}_{s,2}.\hfill \end{array}$$

(19)

Thus

$${\left|\right|u\left|\right|}_{s,2}^2\left(M\left(\left|\right|u_n{\left|\right|}_{s,2}^2\right)-{\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}\right)\le 2^{-1}\left(\left|\right|u_n{\left|\right|}_{s,2}^2+{∥f∥}_{L^2(\Omega )}^2\right),$$

(20)

from which we have

$$2\left|\right|u_n{\left|\right|}_{s,2}^2\left(M\left(\right||u_n{\left|\right|}_{s,2}^2)-{\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}-1\right)\le {∥f∥}_{L^2(\Omega )}^2.$$

(21)

By $\left(M2\right)$, we deduce that

$$M\left(\right||u_n{\left|\right|}_{s,2}^2)-{\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}-1\to \infty \mathrm{as} n\to \infty .$$

Hence, we reach a contradiction with (21). Thus, we conclude that ${\left\{u_n\right\}}_n$ is bounded in $H_0^s(\Omega )$ and the claim follows.

Therefore, we get the existence of $\overline{u}\in H_0^s(\Omega )$ such that, up to a sub-sequence, $u_n\rightharpoonup \overline{u}$ weakly in $H_0^s(\Omega )$, and $u_n⟶\overline{u}$ strongly in $L^q(\Omega )$ for all $1\le q<2_s^{*}$.

Once more, we claim that there exists a positive constant $\rho$ such that up to a sub-sequence denoted by $u_n$, we have

$${\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}<{\Lambda }_{N,s}-\rho \mathrm{for} \mathrm{all} n.$$

(22)

By virtue of $\left(M1\right)$ and since ${\left|\right|u\left|\right|}_{s,2}\le C$, we obtain

$$\underset{n\to \infty }{lim\; inf}M\left(\right||u_n{\left|\right|}_{s,2}^2):=\mu >0.$$

Therefore, we get the existence of a subsequence denoted by ${\left\{u_n\right\}}_n$ such that

$$M\left(\right|\left|u_n\right|{|}_{s,2}^2)\to \mu \mathrm{as} n\to \infty .$$

Obviously, $u_n\rightharpoonup \overline{u}$ weakly in $H_0^s(\Omega )$, thus by passing to the limit in the problem of $u_n$, it follows that $\overline{u}$ solves

$$\mu {(-\Delta )}^s\overline{u}=\lambda {\displaystyle \frac{\overline{u}}{{\left|x\right|}^{2s}}}+f.$$

Thus $\overline{u}\in H_0^s(\Omega )$ satisfies $\overline{u}\ge 0$ in $\Omega$ and

$${(-\Delta )}^s\overline{u}={\displaystyle \frac{\lambda }{\mu }}{\displaystyle \frac{\overline{u}}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{f\left(x\right)}{\mu }}\mathrm{in}\Omega .$$

From the result of [4], we conclude that ${\displaystyle \frac{\lambda }{\mu }}\le {\Lambda }_{N,s}$.

Let us show that ${\displaystyle \frac{\lambda }{\mu }}<{\Lambda }_{N,s}-\rho$ for some positive constant $\rho >0$. By contradiction, if ${\displaystyle \frac{\lambda }{\mu }}={\Lambda }_{N,s}$, then $\overline{u}$ is solution of the equation

$${(-\Delta )}^s\overline{u}={\Lambda }_{N,s}{\displaystyle \frac{\overline{u}}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{f\left(x\right)}{\mu }}\mathrm{in}\Omega ,$$

Using again the result of [4], one can see that $\overline{u}\left(x\right)\ge C{\left|x\right|}^{-{\displaystyle \frac{N-2s}{2}}}$ in a small ball $B_{\eta }\left(0\right)$. Thus

$${\int }_{B_{\eta }\left(0\right)}{\overline{u}}^{2_s^{*}}dx\ge C{\int }_{B_{\eta }\left(0\right)}{\left|x\right|}^{-N}dx=\infty ,$$

which is a contradiction with the fact that $\overline{u}\in H_0^s(\Omega )$. Hence, we get the existence of a positive constant $\rho$ such that

$${\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}<{\Lambda }_{N,s}-\rho .$$

(23)

To conclude, we prove the strong convergence of the sequence ${\left\{u_n\right\}}_n$ in $H_0^s(\Omega )$.

Using $u_n-\overline{u}$ as a test function, we obtain that

$$\begin{array}{ccc}{\displaystyle {\int }_{\Omega }{(-\Delta )}^s{u}_n(u_n-\overline{u})dx}\hfill & =\hfill & {\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}{\int }_{\Omega }\frac{u_n(u_n-\overline{u})}{(1+\frac{1}{n}{u}_n){\left(\right|x|}^{2s}+\frac{1}{n})}dx}\hfill \\ & +\hfill & {\displaystyle \frac{1}{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}{\int }_{\Omega }f\left(x\right)(u_n-\overline{u})dx.}\hfill \end{array}$$

(24)

Since the term ${\displaystyle \frac{1}{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}$ is bounded away from zero and $u_n-\overline{u}\to 0$ strongly in $L^2(\Omega )$, it holds that

$${\displaystyle \frac{1}{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}{\int }_{\Omega }f\left(x\right)(u_n-\overline{u})dx\to 0 \mathrm{as} n\to \infty .$$

We deal now with the term ${\displaystyle {\int }_{\Omega }{\displaystyle \frac{u_n(u_n-\overline{u})}{(1+\frac{1}{n}{u}_n)({\left|x\right|}^{2s}+\frac{1}{n})}}dx}$.

We have

$$\begin{array}{cc}& {\displaystyle {\int }_{\Omega }\frac{u_n(u_n-\overline{u})}{(1+\frac{1}{n}{u}_n)({\left|x\right|}^{2s}+\frac{1}{n})}dx={\int }_{\Omega }\frac{{(u_n-\overline{u})}^{2}dx}{(1+\frac{1}{n}{u}_n)({\left|x\right|}^{2s}+\frac{1}{n})}+{\int }_{\Omega }\frac{\overline{u}(u_n-\overline{u})dx}{(1+\frac{1}{n}{u}_n)({\left|x\right|}^{2s}+\frac{1}{n})}dx}\\ & {\displaystyle ={\int }_{\Omega }\frac{{(u_n-\overline{u})}^{2}dx}{(1+\frac{1}{n}{u}_n)({\left|x\right|}^{2s}+\frac{1}{n})}+{\int }_{\Omega }\frac{u_n-\overline{u}}{{\left(\left|x\right|+\frac{1}{n}\right)}^s}\frac{\overline{u}dx}{(1+\frac{1}{n}{u}_n){(\left|x\right|+\frac{1}{n})}^s}.}\end{array}$$

Taking into consideration that ${\displaystyle \frac{\overline{u}}{(1+\frac{1}{n}{u}_n){(\left|x\right|+n^{-1})}^s}}\to {\displaystyle \frac{\overline{u}}{{\left|x\right|}^s}}$ strongly in $L^2(\Omega )$ and that ${\displaystyle \frac{u_n-\overline{u}}{{\left(\left|x\right|+n^{-1}\right)}^s}}\rightharpoonup 0$ weakly in $L^2\left(\Omega \right)$ (as $n\to \infty$), we obtain

$${\displaystyle {\int }_{\Omega }{\displaystyle \frac{u_n-\overline{u}}{{\left(\left|x\right|+n^{-1}\right)}^s}}{\displaystyle \frac{\overline{u}}{{\left(\left|x\right|+n^{-1}\right)}^s}}dx\to 0 \mathrm{as} n\to \infty .}$$

Consequently, and using the Hardy inequality, we get

$${\int }_{\Omega }{\displaystyle \frac{u_n(u_n-\overline{u})}{(1+\frac{1}{n}{u}_n)({\left|x\right|}^{2s}+\frac{1}{n})}}dx={\int }_{\Omega }{\displaystyle \frac{{\left(u_n-\overline{u}\right)}^2}{(1+\frac{1}{n}{u}_n){(\left|x\right|+n^{-1})}^{2s}}}dx+o\left(1\right)\le {\displaystyle \frac{1}{{\Lambda }_{N,s}}}{∥u_n-\overline{u}∥}_{s,2}^2+o\left(1\right).$$

(25)

Now, by the weak convergence, it holds that

$${\displaystyle {\int }_{\Omega }{(-\Delta )}^s{u}_n(u_n-\overline{u})dx={∥u_n-\overline{u}∥}_{s,2}^2+o\left(1\right).}$$

Therefore, by estimates (24) and (25), it follows that

$${∥u_n-\overline{u}∥}_{s,2}^2\le {\displaystyle \frac{\lambda }{{\Lambda }_{N,s}M({\left|\right|u_n\left|\right|}_{s,2}^2)}}{∥u_n-\overline{u}∥}_{s,2}^2+o\left(1\right).$$

Recall that by (23), we have ${\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}<{\Lambda }_{N,s}-\rho$ for all n. Thus

$${\displaystyle {∥u_n-\overline{u}∥}_{s,2}^2<\frac{{\Lambda }_{N,s}-\rho }{{\Lambda }_{N,s}}{∥u_n-\overline{u}∥}_{s,2}^2+o\left(1\right).}$$

Since ${\displaystyle 0<{\displaystyle \frac{{\Lambda }_{N,s}-\rho }{{\Lambda }_{N,s}}}<1}$, then ${∥u_n-\overline{u}∥}_{s,2}^2⟶0$ as $n\to \infty$ and then $u_n⟶\overline{u}$ strongly in $H_0^s(\Omega )$.

As a consequence, it follows that $M\left(\right||u_n{\left|\right|}_{s,2})\to M({\left|\right|u\left|\right|}_{s,2})$. Passing to the limit in problem (18), we obtain that $\overline{u}$ solves (17).

It still remains to prove the uniqueness part.

Assume that $u,v$ are two positive solutions to problem (18). To show the uniqueness, we just have to prove that ${M\left(\right|\left|u\right||}_{s,2}^2{)=M(\left|\right|v\left|\right|}_{s,2}^2)$. Assume by contradiction that ${M\left(\right|\left|u\right||}_{s,2}^2{)<M(\left|\right|v\left|\right|}_{s,2}^2)$. Setting $u_1={M\left(\right|\left|u\right||}_{s,2}^2)u$ and $v_1={M\left(\right|\left|v\right||}_{s,2}^2)v$, it holds that

$${(-\Delta )}^s{v}_1={\displaystyle \frac{\lambda }{M\left(\right|\left|v\right|{|}_{s,2}^2)}}{\displaystyle \frac{v_1}{{\left|x\right|}^{2s}}}+f\left(x\right),$$

and

$${(-\Delta )}^s{u}_1={\displaystyle \frac{\lambda }{M\left(\right|\left|u\right|{|}_{s,2}^2)}}{\displaystyle \frac{u_1}{{\left|x\right|}^{2s}}}+f\left(x\right).$$

Since ${\displaystyle \frac{\lambda }{M\left(\right|\left|v\right|{|}_{s,2}^2)}}<{\displaystyle \frac{\lambda }{M\left(\right|\left|u\right|{|}_{s,2}^2)}}$ and $u_1=v_1=0$ in ${\mathbb{R}}^N\setminus \Omega$, then $v_1<u_1$ in $\Omega$. Thus

$${M\left(\right|\left|v\right||}_{s,2}^2{)v<M(\left|\right|u\left|\right|}_{s,2}^2)u \mathrm{in} \Omega .$$

Recall that by hypothesis, ${M\left(\right|\left|u\right||}_{s,2}^2{)<M(\left|\right|v\left|\right|}_{s,2}^2)$, thus $v<u$ in $\Omega$. Going back to the equation of v and using v as a test function, it follows that

$${M\left(\right|\left|v\right||}_{s,2}^2{\left)\right|\left|v\right||}_{s,2}^2=\lambda {\int }_{\Omega }{\displaystyle \frac{v^2}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }fvdx\le \lambda {\int }_{\Omega }{\displaystyle \frac{u^2}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }fudx={M\left(\right|\left|u\right||}_{s,2}^2{\left)\right|\left|u\right||}_{s,2}^2$$

Since $M\left(t^2\right)t^2$ is a strict monotone function, we conclude that ${\left|\right|v\left|\right|}_{s,2}^2{<\left|\right|u\left|\right|}_{s,2}^2$ and then ${M\left(\right|\left|v\right||}_{s,2}^2{)\le M(\left|\right|u\left|\right|}_{s,2}^2)$, a contradiction with the main hypothesis; hence the uniqueness result follows. □

Remark 3.

It is not difficult to show that the above existence and the uniqueness result holds also for all $f\in L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$.

As a consequence of the previous construction, we have the next regularity result.

Proposition 1.

Assume that $f\in L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$, then for all $\sigma <2_s^{*}$, there exists a positive constant $C:=C(N,\Omega ,\lambda ,\sigma )$, such that if u is the solution to problem (17), we have

$${\left|\right|u\left|\right|}_{L^{\sigma }(\Omega )}\le {C\left|\right|f\left|\right|}_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )}.$$

Proof. 

Since u solves (17), then ${\displaystyle \frac{\lambda }{M\left(\right|\left|u\right|{|}_{s,2}^2)}}\le {\Lambda }_{N,s}$ and

$${(-\Delta )}^{s}u={\displaystyle \frac{\lambda }{M\left(\right|\left|u\right|{|}_{s,2}^2)}}{\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{f\left(x\right)}{M\left(\right|\left|u\right|{|}_{s,2}^2)}}\le {\Lambda }_{N,s}{\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{{\Lambda }_{N,s}}{\lambda }}f\left(x\right).$$

Setting w to be the unique solution to the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^{s}w& =& {\Lambda }_{N,s}{\displaystyle \frac{w}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{{\Lambda }_{N,s}}{\lambda }}f\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill w& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

then $u\le w$ in $\Omega$. Notice that from [4] we obtain that, for all $\sigma <2_s^{*}$,

$${\left|\right|w\left|\right|}_{L^{\sigma }(\Omega )}\le C\left|\right|{\displaystyle \frac{{\Lambda }_{N,s}}{\lambda }}{f\left|\right|}_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )}\le {C(\Omega ,\lambda ,s,N)\left|\right|f\left|\right|}_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )}.$$

Hence we conclude. □

Remark 4.

1. 

Using u as a test function in (17) and taking into consideration that ${\displaystyle \frac{\lambda }{M\left(\right|\left|u\right|{|}_{s,2}^2)}}\le {\Lambda }_{N,s}$, we obtain that

$${\displaystyle \frac{a_{N,s}}{2}}\int {\int }_{D_{\Omega }}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+2s}}}dxdy-{\Lambda }_{N,s}{\int }_{\Omega }{\displaystyle \frac{u^2}{{\left|x\right|}^{2s}}}dx\le {\displaystyle \frac{{\Lambda }_{N,s}}{\lambda }}{\int }_{\Omega }fudx.$$

Now, if $f\in L^{\sigma }(\Omega )$ for some $\sigma >{\displaystyle \frac{2N}{N+2s}}$, then, using the improved Hardy inequality proved in [33], it holds that, for all $q<2$, we have

$$\left({\int }_{\Omega }{\int }_{\Omega }{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^2}{{|x-y|}^{N+qs}}}dxdy\right)\le {C\left|\right|f\left|\right|}_{L^{\sigma }(\Omega )}.$$

2. 

In the case $f\in L^{\sigma }(\Omega )$ for some $\sigma \ge {\displaystyle \frac{2N}{N+2s}}$, then as in the proof of the affirmation (22), we can show that ${\displaystyle \frac{\lambda }{M\left(\right|\left|u\right|{|}_{s,2}^2)}}\le {\Lambda }_{N,s}-\rho$ with $\rho >0$. Again using u as a test function in (17), we reach that

$${M\left(\right|\left|u\right||}_{s,2}^2{\left)\right|\left|u\right||}_{s,2}^2\le {\left|\right|u\left|\right|}_{s,2}^2+{\int }_{\Omega }fudx.$$

Using Hölder and Sobolev inequalities and taking into consideration that $M\left(t\right)\to \infty$ as $t\to \infty$, it follows that

$${\left|\right|u\left|\right|}_{s,2}\le C_1+C_2{\left|\right|f\left|\right|}_{L^{\sigma }(\Omega )},$$

where $C_1,C_2$ are independent of u and f.

To complete this part, we have the next compactness result.

Proposition 2.

Let ${\left\{f_n\right\}}_n\subset L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$ be such that $f_n\ge 0$ and $\left|\right|f_n{\left|\right|}_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )}\le C$ for all n. Define $u_n$, the unique nonnegative solution to problem (17) with $f:=f_n$. Then, there exists a $u\in H_0^s(\Omega )$, such that, to a subsequence, $u_n\to u$ strongly in $W_0^{s,q}(\Omega )$ for all $q<2$.

Proof. 

Let $u_n$ be a solution to problem (17), then $u_n$ solves the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^s{u}_n& =& {\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}{\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{f_n}{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}\hfill & \mathrm{in}\Omega \hfill \\ \hfill u_n& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

Thus, ${\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}\le {\Lambda }_{N,s}$ and then ${\displaystyle \frac{1}{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}\le {\displaystyle \frac{{\Lambda }_{N,s}}{\lambda }}=\widehat{C}$.

Define $w_n$ to be the unique solution to the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^s{w}_n& =& {\Lambda }_{N,s}{\displaystyle \frac{w_n}{{\left|x\right|}^{2s}}}+\widehat{C}{f}_n\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill w_n& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

then $u_n\le w_n$ and for all $\sigma <2_s^{*}$,

$$\left|\right|w_n{\left|\right|}_{L^{\sigma }(\Omega )}\le C\left|\right|f_n{\left|\right|}_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )}.$$

Hence $\left|\right|u_n{\left|\right|}_{L^{\sigma }(\Omega )}\le \overline{C}$ for all n. If $M\left(\right||u_n{\left|\right|}_{s,2}^2)\le 2{\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}$, then $\left|\right|u_n{\left|\right|}_{s,2}^2\le C$. Now, if $M\left(\right||u_n{\left|\right|}_{s,2}^2)\ge 2{\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}$, then using $u_n$ as a test function in the problem solved by $u_n$, it follows that

$$M\left(\right||u_n{\left|\right|}_{s,2}^2\left)\right||u_n{\left|\right|}_{s,2}^2\le {\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}\left|\right|u_n{\left|\right|}_{s,2}^2+{\int }_{\Omega }{f}_n{u}_{n}dx.$$

Thus

$$\left(M\right(\left|\right|u_n{\left|\right|}_{s,2}^2)-{\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}\left)\right||u_n{\left|\right|}_{s,2}^2\le {\int }_{\Omega }{f}_n{u}_{n}dx.$$

Hence, using Hölder and Sobolev inequalities, we conclude that

$$\left|\right|u_n{\left|\right|}_{s,2}^2\le C \mathrm{for} \mathrm{all} n.$$

Hence, ${\left\{u_n\right\}}_n$ is bounded in $H_0^s(\Omega )$. Therefore, we get the existence of $u\in H_0^s(\Omega )$ such that, up to a sub-sequence, $u_n\rightharpoonup u$ weakly in $H_0^s(\Omega )$, $u_n\to u$ strongly in $L^{\sigma }(\Omega )$ for all $\sigma <2_s^{*}$ and $\left(M\right(\left|\right|u_n{\left|\right|}_{s,2}^2):={\alpha }_n\to {\alpha }_0\ge {\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}$.

Setting $h_n\left(x\right):={\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}{\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{f_n}{M\left(\right|\left|u_n\right|{|}_{s,2}^2)}}$, it holds that $\left|\right|h_n{\left|\right|}_{L^1(\Omega )}\le C$. Therefore, by the compactness result in [29], we reach that, up to a sub-sequence, $u_n\to u$ strongly in $W_0^{s,q}(\Omega )$ for all $q<{\displaystyle \frac{N}{N-s}}$. Thus, by interpolation, it holds that $u_n\to u$ strongly in $W_0^{s,q}(\Omega )$ for all $q<2$, and the result follows. □

Some Results Related to Non Variational Problems

In this subsection, we will assume that $q<2$. Consider the problem

$$\left\{\begin{array}{cccc}\hfill M\left({∥u∥}_{s,q}^q\right){(-\Delta )}^{s}u& =& f\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(26)

It is clear that problem (26) cannot be viewed and treated in a variational setting. We begin by proving the next proposition.

Proposition 3.

Assume that $f\in L^2(\Omega )$ and $1<q<2$, then problem (26) has a unique solution $u_q$ in $H_0^s(\Omega )$.

Proof. 

To show the existence of a solution, we will use Schauder’s Fixed Point Theorem. Let $E=W_0^{s,q}(\Omega )$ and consider the operator $T:E⟶E$ defined by $T\left(v\right)=u$, where u is the unique solution for the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^{s}u& =& {\displaystyle \frac{f}{M\left(\right|\left|v\right|{|}_{s,q}^q)}}\hfill & \mathrm{in}\Omega \hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

Taking into consideration the properties of M and classical results on regularity of the fractional Laplacian, we conclude that T is well defined and

$${\left|\right|u\left|\right|}_{H_0^s(\Omega )}\le {\displaystyle \frac{c_0}{M\left(\right|\left|v\right|{|}_{s,q}^q)}}{\parallel f\parallel }_{L^2(\Omega )}\le {\displaystyle \frac{c_0}{m_0}}{\parallel f\parallel }_{L^2(\Omega )}.$$

(27)

Hence,

$${\left|\right|u\left|\right|}_E\le {\displaystyle \frac{c_1}{m_0}}{∥f∥}_{L^2(\Omega )}$$

for some positive constant $c_1$ depending only on $\Omega ,N$ and s. Choosing $r=m_0^{-1}{c}_1{\left|\right|f\left|\right|}_{L^2(\Omega )}$, it holds that $T\left(B_r\left(0\right)\right)\subset B_r\left(0\right)$, where $B_r\left(0\right)$ is the ball centered at zero with radius $r>0$ of E.

We begin by proving that the operator T is continuous. Let ${\left\{v_n\right\}}_n\subset E$ be a sequence such that $v_n\to v$ strongly in E as $n\to \infty$. Then, in particular, ${\parallel v_n\parallel }_{s,q}^q\to {\parallel v\parallel }_{s,q}^q$. By the continuity of the function M, it follows that $M\left({\parallel v_n\parallel }_{s,q}^q\right)\to M\left({\parallel v\parallel }_{s,q}^q\right)$. Recall that $M\left(t\right)\ge m_0>0$ for all $t\ge 0$. Define $g_n={\displaystyle \frac{f}{M\left({\parallel v_n\parallel }_{s,q}^q\right)}}$, then $g_n\to g={\displaystyle \frac{f}{M\left({\parallel v\parallel }_{s,q}^q\right)}}$ strongly in $L^2(\Omega )$. Since $L^2(\Omega )\subset H^{-s}(\Omega )$, the dual space of $H_0^s(\Omega )$, we also have $g_n\to g$ in $H^{-s}(\Omega )$. Therefore, by the continuity of the inverse fractional Laplacian

$${(-\Delta )}^{-s}:H^{-s}(\Omega )\to H_0^s(\Omega ),$$

we conclude that $T\left(v_n\right)\to T\left(v\right)$ in $H_0^s(\Omega )$. Finally, since $H_0^s(\Omega )\subset E$, the continuity of T in E follows.

We prove now that T is compact.

Let ${\left\{v_n\right\}}_n$ be a bounded sequence in E. Define $u_n=T\left(v_n\right)$, then from (27), it holds that ${\left\{u_n\right\}}_n$ is bounded in $H_0^s(\Omega )$ and a priori in E. Hence, up to a sub-sequence, we get the existence of $\widehat{u}\in H_0^s(\Omega )$ such that $u_n\rightharpoonup \widehat{u}$ weakly in $H_0^s(\Omega )$.

Setting $g_n={\displaystyle \frac{f}{M\left({∥v_n∥}_{s,q}^q\right)}}$, then $\left|\right|g_n{\left|\right|}_{L^1(\Omega )}\le C$ for all n. Hence, from the compactness result in Theorem 5, it holds that $u_n\to \widehat{u}$ strongly in $W_0^{s,p}(\Omega )$ for all $p<{\displaystyle \frac{N}{N-s}}$. Taking into consideration that the sequence ${\left\{u_n\right\}}_n$ is bounded in $H_0^s(\Omega )$ and using Vitali’s Lemma, we conclude that $u_n\to \widehat{u}$ strongly in $W_0^{s,\sigma }(\Omega )$ for all $\sigma <2$, in particular $u_n\to \widehat{u}$ strongly in E. Hence, T is compact.

Thus, by Schauder’s Fixed Point Theorem, we get the existence of $\tilde{u}\in E$ such that $T\left(\tilde{u}\right)=\tilde{u}$, and then $\tilde{u}$ solves problem (26). It is not difficult to show that $\tilde{u}\in H_0^s(\Omega )$.

We now prove the uniqueness part. Suppose that $u_1$ and $u_2$ are two solutions to problem (26), then

$$M\left({∥u_1∥}_{s,q}^q\right){\left(-\Delta \right)}^s{u}_1=f=M\left({∥u_2∥}_{s,q}^q\right){\left(-\Delta \right)}^s{u}_2.$$

Thus

$${(-\Delta )}^s\left(M\left(\right||u_1{\left|\right|}_{s,q}^q)u_1-M\left(\right||u_2{\left|\right|}_{s,q}^q)u_2\right)=0.$$

Since $M\left(\right||u_i{\left|\right|}_{s,q}^q)u_i\in H_0^s(\Omega )$ for $i=1,2$, then

$$M\left({∥u_1∥}_{s,q}^q\right)u_1=M\left({∥u_2∥}_{s,q}^q\right)u_2.$$

(28)

Therefore, we get $M\left(\right||u_1{\left|\right|}_{s,q}^q\left)\right||u_1{\left|\right|}_{s,q}=M\left(\right||u_2{\left|\right|}_{s,q}^q\left)\right||u_2{\left|\right|}_{s,q}$. Recall that $q>1$, then by $\left(M1\right)$ and $\left(M2\right)$, the function $t⟼M\left(t^q\right)t$ is increasing and then $\left|\right|u_1{\left|\right|}_{s,q}=\left|\right|u_2{\left|\right|}_{s,q}$. Going back to (28), we obtain $u_1=u_2$. □

In the case where $1<q<{\displaystyle \frac{N}{N-s}}$, we are able to show that problem (26) has a unique solution for all $f\in L^1(\Omega )$.

Theorem 7.

Assume that $f\in L^1(\Omega )$ and $1<q<{\displaystyle \frac{N}{N-s}}$, then problem (26) has a unique solution $\overline{u}$ in $W_0^{s,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-s}}$. In particular, $\overline{u}\in W_0^{s,q}(\Omega )$. Moreover, defining $T:L^1(\Omega )\to W_0^{s,q}(\Omega )$ with $u=T\left(f\right)$ being the unique solution to problem (26), then T is a compact operator.

Proof. 

We set $f_n=T_n\left(f\right)$, then $f_n\in L^{\infty }(\Omega )$. By Theorem 3, it follows that the problem

$$\left\{\begin{array}{cccc}\hfill M\left({∥u_n∥}_{s,q}^q\right){(-\Delta )}^s{u}_n& =& f_n\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u_n& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(29)

has a unique solution $u_n$. We claim that $M\left({∥u_n∥}_{s,q}^q\right)<C$ for all $n\ge 1$.

We argue by contradiction. Assume that $M\left({∥u_n∥}_{s,q}^q\right)⟶+\infty$ as $n\to \infty$. Then, by the properties of M, we reach that $\underset{n\to \infty }{lim}{∥u_n∥}_{s,q}=\infty$.

Define $g_n\equiv {\displaystyle \frac{f_n}{M\left({∥u_n∥}_{s,q}^q\right)}}$, then $\left|g_n\right|\le {\displaystyle \frac{|f_n|}{m_0}}$ and hence $\left|\right|g_n{\left|\right|}_{L^1(\Omega )}\le C$. Thus, using Theorem 5, it holds that $\left|\right|u_n{\left|\right|}_{W_0^{s,\sigma }(\Omega )}\le {\displaystyle \frac{C}{m_0}}\left|\right|f_n{\left|\right|}_{L^1(\Omega )}\le C$ for all n and for all $\sigma <{\displaystyle \frac{N}{N-s}}$. Choosing $\sigma =q$, we get $\left|\right|u_n{\left|\right|}_{W_0^{s,q}(\Omega )}\le C$, a contradiction with the main hypothesis. Thus, $M\left({∥u_n∥}_{s,q}^q\right)<C$ for all n, and then the claim follows.

Consequently ${∥u_n∥}_{s,q}\le C$, therefore there exists $\overline{u}\in W_0^{s,q}(\Omega )$ such that, up to a sub-sequence, $u_n\rightharpoonup \overline{u}$ weakly in $W_0^{s,q}(\Omega )$.

Let us show that $u_n⟶\overline{u}$ strongly in $W_0^{s,q}(\Omega )$.

Recall that

$${(-\Delta )}^s{u}_n={\displaystyle \frac{f_n}{M\left({∥u_n∥}_{s,q}^q\right)}}.$$

By the boundedness of the sequence ${\left\{{\displaystyle \frac{f_n}{M\left({∥u_n∥}_{s,q}^q\right)}}\right\}}_n$ in $L^1(\Omega )$ and the compactness result in Theorem 5, we conclude that $u_n⟶\overline{u}$ strongly in $W_0^{s,\sigma }(\Omega )$, for all $\sigma <{\displaystyle \frac{N}{N-s}}$, and in particular, $u_n⟶\overline{u}$ strongly in $W_0^{s,q}(\Omega )$

Hence, $M\left({∥u_n∥}_{s,q}^q\right)⟶M\left({∥\overline{u}∥}_{s,q}^q\right)$. It is clear that $\overline{u}$ solves problem (26). The uniqueness follows as in the proof of Theorem 3.

Define now $T:L^1(\Omega )\to W_0^{s,q}(\Omega )$, where $u=T\left(f\right)$ is the unique solution to problem (26). It is clear that T is well defined. To show that T is compact, we consider a sequence ${\left\{f_n\right\}}_n\subset L^1(\Omega )$ and define $u_n=T\left(f_n\right)$. Then,

$${(-\Delta )}^s{u}_n={\displaystyle \frac{f_n}{M\left({∥u_n∥}_{s,q}^q\right)}}.$$

Now, repeating the same computations as above and using the compactness result in Theorem 5, we deduce that, up to a sub-sequence, $u_n⟶\overline{u}$ strongly in $W_0^{s,\sigma }(\Omega )$, for all $\sigma <{\displaystyle \frac{N}{N-s}}$, and in particular, $u_n⟶\overline{u}$ strongly in $W_0^{s,q}(\Omega )$. Hence, the result follows. □

We are now able to state the main result of this subsection.

Theorem 8.

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded domain such that $0\in \Omega$ and $1<q<{\displaystyle \frac{N}{N-s}}$. Suppose that $f\in L^1(\Omega )$ is such that ${\displaystyle {\int }_{\Omega }f{\left|x\right|}^{-\theta }dx<c}$ for some $\theta >0$. Then, for all $\lambda >0$, the problem

$$\left\{\begin{array}{cccc}\hfill M\left({∥u∥}_{s,q}^q\right){(-\Delta )}^{s}u& =& \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+f\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(30)

has a solution $\overline{u}$ such that $\overline{u}\in W_0^{s,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-s}}$.

Proof. 

Let $\lambda >0$ be fixed and define $f_n=T_n\left(f\right)$. Consider the approximating problem

$$\left\{\begin{array}{cccc}\hfill M\left(\right||u_n{\left|\right|}_{s,q}^q{)(-\Delta )}^s{u}_n& =& \lambda {\displaystyle \frac{T_n\left(u_n\right)}{{\left|x\right|}^{2s}+\frac{1}{n}}}+f_n\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u_n& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(31)

Using a direct variation of Theorem 5, we can show that problem (31) has a solution $u_n\in H_0^s(\Omega )$.

Let’s begin by proving that $M\left({\left|\right|u_n\left|\right|}_{s,q}^q\right)<m_1$. We argue by contradiction, if $M\left(\right|\left|u_n\right|{|}_{s,q}^q)⟶+\infty$, then ${\left|\right|u\left|\right|}_{s,q}^q⟶+\infty$. Observe that

$${(-\Delta )}^s{u}_n={\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,q}^q)}}{\displaystyle \frac{T_n\left(u_n\right)}{{\left|x\right|}^{2s}+\frac{1}{n}}}+{\displaystyle \frac{f_n}{M\left(\right|\left|u_n\right|{|}_{s,q}^q)}}$$

(32)

For simplicity of typing, we set

$$g_n\equiv {\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,q}^q)}}{\displaystyle \frac{T_n\left(u_n\right)}{{\left|x\right|}^{2s}+\frac{1}{n}}}+{\displaystyle \frac{f_n}{M\left(\right|\left|u_n\right|{|}_{s,q}^q)}}.$$

(33)

We claim that ${\parallel {\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\parallel }_{L^1(\Omega )}<C$ for some positive constant C.

To prove the claim, we define $\phi$ as the unique solution to the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^s\phi & =& \beta {\displaystyle \frac{\phi }{{\left|x\right|}^{2s}}}+1\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill \phi & =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(34)

where $0<\beta <{\Lambda }_{N,s}$ is chosen later. Notice that $\phi \in L^{\infty }(\Omega \setminus B_r\left(0\right))$ and $\phi \approx {\left|x\right|}^{-{\gamma }_{\beta }}$ in $B_r\left(0\right)$ where

$${\gamma }_{\beta }:={\displaystyle \frac{N-2s}{2}}-\alpha ,$$

(35)

and $\alpha$ is given by (8) where $\lambda$ is substituted by $\beta$. It is not difficult to show that

${\gamma }_{\beta }\to 0$ as $\beta \to 0$.

Using $u_n$ as a test function in (34) and integrating over $\Omega$, it holds that

$$\beta {\int }_{\Omega }{\displaystyle \frac{\phi u_n}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }{u}_{n}dx={\displaystyle \frac{\lambda }{M\left({∥u_n∥}_{s,q}^q\right)}}{\int }_{\Omega }{\displaystyle \frac{u_n\phi }{{\left|x\right|}^{2s}}}dx+{\displaystyle \frac{1}{M\left({∥u_n∥}_{s,q}^q\right)}}{\int }_{\Omega }{f}_n\phi dx.$$

Then,

$$(\beta -{\displaystyle \frac{\lambda }{M\left({∥u_n∥}_{s,q}^q\right)}}){\int }_{\Omega }{\displaystyle \frac{\phi u_n}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }{u}_{n}dx\le {\displaystyle \frac{1}{m_0}}{\int }_{\Omega }{f}_n\phi dx\le {\displaystyle \frac{1}{m_0}}{\int }_{\Omega }{f}_n{\left|x\right|}^{-{\gamma }_{\beta }}dx.$$

Now, since $M\left(\right|\left|u_n\right|{|}_{s,q}^q)⟶+\infty$, we can choose $n_0$ large enough such that $\beta -{\displaystyle \frac{\lambda }{M\left(\right|\left|u_n\right|{|}_{s,q}^q)}}>{\beta }_0>0$ if $n\ge n_0$. Hence, for $n\ge n_0$,

$${\beta }_0{\int }_{\Omega }{\displaystyle \frac{\phi u_n}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }{u}_{n}dx\le {\displaystyle \frac{1}{m_0}}{\int }_{\Omega }f{\left|x\right|}^{-{\theta }_{\epsilon }}dx.$$

Recall that ${\displaystyle {\int }_{\Omega }f{\left|x\right|}^{-\theta }dx<c}$, then now, we fix $\beta$ small enough such that ${\theta }_{\epsilon }\le \theta$. Thus, ${\displaystyle {\int }_{\Omega }f{\left|x\right|}^{-{\theta }_{\beta }}<\infty }$. Thus,

$${\displaystyle {\beta }_0{\int }_{\Omega }\frac{\phi u_n}{{\left|x\right|}^{2s}}dx+{\int }_{\Omega }{u}_{n}dx<m_0^{-1}c,}$$

(36)

from which we deduce that ${\parallel {\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\parallel }_{L^1(\Omega )}<m_0^{-1}c$, and the claim follows.

Now, going back to (32) and using the fact that the sequence ${\left\{g_n\right\}}_n$ is bounded in $L^1(\Omega )$, we reach that the sequence ${\left\{u_n\right\}}_n$ is bounded in $W_0^{s,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-s}}$, in particular for $\sigma =q$. Thus, ${∥u_n∥}_{s,q}<C_1$ and then $M\left({∥u_n∥}_{s,q}^q\right)<\overline{C}$, a contradiction with the main hypothesis. Thus $\left\{M\right(\left|\right|u_n{\left|\right|}_{s,q}^q{\left)\right\}}_n$ is bounded in ${\mathbb{R}}^{+}$.

Hence, we get the existence of $\overline{u}\in W_0^{s,q}(\Omega )$ such that, up to a sub-sequence, $u_n\rightharpoonup \overline{u}$ weakly in $W_0^{s,q}(\Omega )$, $u_n\to \overline{u}$ strongly in $L^{\sigma }(\Omega )$ for all $\sigma <q_s^{*}={\displaystyle \frac{qN}{N-qs}}$ and $u_n\to \overline{u}a.e \mathrm{in} \Omega$.

Setting ${\alpha }_n:=M\left({∥u_n∥}_{s,q}^q\right)$, then we can assume that ${\alpha }_n⟶\overline{\alpha }\ge m_0.$

Let us show that ${\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\to {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}$ strongly in $L^1(\Omega )$. It is clear that ${\displaystyle {\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\to \frac{\overline{u}}{{\left|x\right|}^{2s}}}$ a.e. in $\Omega .$ Using the fact that $u_n\to \overline{u}$ strongly in $L^{\sigma }(\Omega )$ for all $\sigma <q_s^{*}$, we reach that

$${\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\to {\displaystyle \frac{\overline{u}}{{\left|x\right|}^{2s}}} \mathrm{strongly} \mathrm{in} L^1(\Omega \setminus B_R\left(0\right) \mathrm{for} B_R\left(0\right)\subset \subset \Omega .$$

Hence, to conclude, we have just to show that

$${\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\to {\displaystyle \frac{\overline{u}}{{\left|x\right|}^{2s}}} \mathrm{strongly} \mathrm{in} L^1\left(B_R\left(0\right)\right) \mathrm{for} \mathrm{some} B_R\left(0\right)\subset \subset \Omega .$$

Recall that from (36), we have ${\displaystyle {\int }_{\Omega }{\displaystyle \frac{\phi u_n}{{\left|x\right|}^{2s}}}dx<C}$ for all n. By the fact that $\phi \ge C_{\beta }{\left|x\right|}^{-{\theta }_{ϵ}}$ in $B_R\left(0\right)$, it holds that

$$C_{\beta }{R}^{-{\theta }_{\beta }}{\int }_{B_R\left(0\right)}{\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\le C \mathrm{for} \mathrm{all} n.$$

Fix $R>0$ to be chosen later and consider $E\subset B_R\left(0\right)$ a measurable set. Then,

$$\begin{array}{ccc}\hfill {\int }_E{\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}dx& =& {\int }_{E\cap B_R\left(0\right)}{\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}dx+{\int }_{E\setminus B_R\left(0\right)}{\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}dx\hfill \\ & \le & C_{\beta }{R}^{{\theta }_{\beta }}.\hfill \end{array}$$

Let $ϵ>0$, then we can choose R small enough such that

$$C_{\beta }{R}^{{\theta }_{\beta }}\le {\displaystyle \frac{ϵ}{2}}.$$

Combining the above estimates, it holds that for $n\ge n_0$ and if $\left|E\right|\le \delta$, then

$${\displaystyle {\int }_E\frac{u_n}{{\left|x\right|}^{2s}}dx\le ϵ.}$$

Hence, using Vitali’s lemma, we reach that ${\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}\to {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}$ strongly in $L^1(\Omega )$.

Recalling the definition of $g_n$ in (33) and using the compactness result in Theorem 5, we conclude, up to a subsequence, $u_n\to \overline{u}$ strongly in $W_0^{\sigma ,q}(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-s}}$. Thus,

$$M\left({∥u_n∥}_{s,q}^q\right)\to M\left({∥\overline{u}∥}_{s,q}^q\right) \mathrm{as} n\to \infty .$$

Therefore, we conclude that $\overline{u}$ satisfies

$${(-\Delta )}^s\overline{u}={\displaystyle \frac{\lambda }{M\left({∥\overline{u}∥}_{s,q}^q\right)}}.{\displaystyle \frac{\overline{u}}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{f}{M\left({∥\overline{u}∥}_{s,q}^q\right)}},$$

and then $\overline{u}$ is the solution of Problem $\left(P_{s,q}^{\lambda }\right)$. □

Remark 5.

If $f\in L^{\sigma }(\Omega )$ for $\sigma >1$, then using Hölder inequality we can prove that the condition stated in Theorem 8 holds for f for some $\theta >0$. Thus, problem (30) has a solution for all $f\in L^{\sigma }(\Omega )$ with $\sigma >1$ and for all $\lambda >0$.

To end this section, we present the next regularity result that will be used later.

Theorem 9.

Assume that f is a nonnegative function such that ${\displaystyle {\int }_{\Omega }f{\left|x\right|}^{-\theta }dx<c}$ for some positive constant θ. Let u be the solution to problem (30) obtained in Theorem 8, then for all $\sigma <{\displaystyle \frac{N}{N-s}}$, we have

$${\displaystyle {\left|\right|u\left|\right|}_{W_0^{s,\sigma }(\Omega )}\le C{\int }_{\Omega }f{\left|x\right|}^{-\theta }dx,}$$

where $C=C(\lambda ,M,\theta ,\Omega ,N,s)$ and it is independent of u.

In addition, if $s\in ({\displaystyle \frac{1}{2}},1)$, then for all $\sigma <{\displaystyle \frac{N}{N-(2s-1)}}$, we have

$${\displaystyle {\left|\right|u\left|\right|}_{W_0^{1,\sigma }(\Omega )}\le C{\int }_{\Omega }f{\left|x\right|}^{-\theta }dx.}$$

Proof. 

Assume that f is nonnegative function with ${\int }_{\Omega }f{\left|x\right|}^{-\theta }dx<\infty$. Let u be the solution to problem (30) obtained in Theorem 8, then $u\in W_0^{s,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-s}}$ and u solves

$${(-\Delta )}^{s}u=\widehat{\lambda }{\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{1}{M\left(\right|\left|u\right|{|}_{s,q}^q)}}f\left(x\right),$$

(37)

with $\widehat{\lambda }={\displaystyle \frac{\lambda }{M\left(\right|\left|u\right|{|}_{s,q}^q)}}$. Then we deduce that f satisfies ${\int }_{\Omega }f{\left|x\right|}^{-{\gamma }_{\widehat{\lambda }}}dx$ where ${\gamma }_{\widehat{\lambda }}:={\displaystyle \frac{N-2s}{2}}-\widehat{\alpha }$ and $\widehat{\alpha }$ is given by (8) with $\lambda$ is substituted by $\widehat{\lambda }$. According to the integral condition on f, we conclude that ${\gamma }_{\widehat{\lambda }}\le \theta$.

Define $\widehat{\phi }$ as the unique solution to the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^s\widehat{\phi }& =& \widehat{\lambda }{\displaystyle \frac{\widehat{\phi }}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{1}{{\left|x\right|}^{2s}}}\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill \widehat{\phi }& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(38)

then $\widehat{\phi }\le {C\left|x\right|}^{-{\gamma }_{\widehat{\lambda }}}\le C{\left|x\right|}^{-\theta }$. Using $\widehat{\phi }$ as a test function in (37), we obtain that

$${\int }_{\Omega }{\displaystyle \frac{u}{{\left|x\right|}^{2s}}}dx={\displaystyle \frac{1}{M\left(\right|\left|u\right|{|}_{s,q}^q)}}{\int }_{\Omega }f\widehat{\phi }dx\le C{\displaystyle \frac{{\Lambda }_{N,s}}{\lambda }}{\int }_{\Omega }f{\left|x\right|}^{-\theta }dx.$$

Setting $g=\lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+f$, it follows that

$${\left|\right|g\left|\right|}_{L^1(\Omega )}\le C{\Lambda }_{N,s}{\int }_{\Omega }f{\left|x\right|}^{-\theta }dx.$$

Going back to (30) and using estimate (13) in Theorem 5, we obtain that

$${\displaystyle {\left|\right|M\left(\right|\left|u\right||}_{s,q}^q{\left)u\right||}_{W_0^{s,\sigma }(\Omega )}\le {C\left|\right|g\left|\right|}_{L^1(\Omega )}\le C{\int }_{\Omega }f{\left|x\right|}^{-\theta }dx.}$$

Since ${M\left(\right|\left|u\right||}_{s,q}^q)\ge {\displaystyle \frac{\lambda }{{\Lambda }_{N,s}}}$, then

$${\displaystyle {\left|\right|u\left|\right|}_{W_0^{s,\sigma }(\Omega )}\le \frac{{\Lambda }_{N,s}}{\lambda }C{\int }_{\Omega }f{\left|x\right|}^{-\theta }dx}$$

and the result follows in this case.

If $s>{\displaystyle \frac{1}{2}}$, then we use estimate (14) in Theorem 5 to get the desired estimate. □

4. Existence Results in the Case $\mathit{f}(\mathit{x},\mathit{u}):={\mathit{u}}^{\mathit{p}}+\mathit{\mu }\mathit{g}\left(\mathit{x}\right)$

Let consider now the nonlinear problem

$$\left\{\begin{array}{cccc}\hfill M\left({∥u∥}_{s,2}^2\right){(-\Delta )}^{s}u& =& \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+u^p+\mu g\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& >& 0\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(39)

In the case where $M\equiv 1$, then, as in the local case, for $\lambda <{\Lambda }_{N,s}$ fixed, it is possible to show the existence of a critical value $p_{*}>2_s^{*}-1$ such that problem (39) has a non-negative super-solution if and only if $p<p_{*}$. This can be obtained using suitable radial computation, using the behavior of any super-solution near the origin and a suitable test function; see [5,34].

The goal of this section is to show, under a suitable hypothesis on g and p, the existence of a solution to problem (39) for all $\lambda >0$. The first existence result is the following.

Theorem 10.

Let $g\in L^{\sigma }(\Omega )$ with $\sigma \ge {\displaystyle \frac{2N}{N+2s}}$ and suppose that $p<2_s^{*}-1$, then there exists ${\mu }^{*}>0$ such that for all $\lambda >0$ and for all $\mu <{\mu }^{*}$, problem (39) has a solution, $u\in H_0^s(\Omega )$.

Proof. 

In the case $M=1$, the existence of solution is proved under the condition $\lambda \le {\Lambda }_{N,s}$, by construing a suitable radial super-solution using the comparison principle. However, this argument is not applicable in our case and we have to use a different approach in order to show the existence of a solution.

The main idea is to use a suitable fixed-point argument.

Fix $p<r<2_s^{*}$ such that $p{\displaystyle \frac{2N}{N+2s}}<r<2_s^{*}$; we can show the existence of ${\mu }^{*}>0$ such that if $\mu <{\mu }^{*}$, then the algebraic equation

$$C(l+{\mu }^{*}{\parallel g\parallel }_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )})=l^{{\displaystyle \frac{1}{p}}},$$

where C is a universal constant that will be chosen later.

Define the set

$$E:=\left\{\phi \in W_0^{s,1}{(\Omega ):\left|\right|\phi \left|\right|}_{L^r(\Omega )}\le l^{{\displaystyle \frac{1}{p}}}\right\},$$

then E is a closed convex set of $W_0^{s,1}(\Omega )$.

Now, we define $T:E\to W_0^{s,1}(\Omega )$ by $T\left(\phi \right)=u$, where u is the weak solution to

$$\left\{\begin{array}{cccc}\hfill {M\left(\right|\left|u\right||}_{s,2}^2{)(-\Delta )}^{s}u& =& \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+{\phi }_{+}^p+\mu g\left(x\right),\hfill & \mathrm{in} \Omega ,\hfill \\ \hfill u& =0,& \mathrm{in} {\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(40)

It is clear that if $u=T\left(u\right)$, then u solves problem (39). Hence, to prove Theorem 10, we shall show that T has a fixed point belonging to $W_0^{s,2}(\Omega )\cap {\mathcal{C}}^{0,\alpha }(\Omega )$ for some $\alpha >0$.

The proof will be given in several steps.

Step 1: T is well defined.

Since $\phi \in E$, then ${\phi }_{+}^p+\mu g\in L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$. Thus, using Theorem 6 and Remark 3, we get the existence and uniqueness of $u\in H_0^s(\Omega )$ such that u solves problem (40). Hence, $u\in W_0^{s,1}(\Omega )$, and then T is well defined.

Step 2: $T\left(E\right)\subset E$.

Let $\phi \in E$. We define $u=T\left(\phi \right)$. By the regularity result in Proposition 1 and using Hölder inequality, it follows that

$$\begin{array}{cc}\hfill {\left|\right|u\left|\right|}_{L^r(\Omega )}& \le C{\parallel {\phi }_{+}^p+\mu g\parallel }_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )}\hfill \\ \hfill & \le C({\parallel \phi \parallel }_{L^r(\Omega )}^{{\displaystyle \frac{p}{r}}}+{\mu }^{*}\parallel f{\parallel }_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )})\hfill \\ \hfill & \le C(l+{\mu }^{*}{\parallel f\parallel }_{L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )})\le l^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(41)

Since we have proved previously that $u\in W_0^{s,1}(\Omega )$, then using the definition of l, we conclude that $u\in E$ and so, $T\left(E\right)\subset E$.

Step 3: T is continuous.

Let ${\left\{{\phi }_n\right\}}_n\subset E$ be such that ${\phi }_n\to \phi$ in $W_0^{s,1}(\Omega )$. Consider $u_n=T\left({\phi }_n\right)$ and $u=T\left(\phi \right)$. Define

$$h_n\left(x\right):={\left({\phi }_n\right)}_{+}^p+\lambda h\left(x\right),$$

then, to get the desired result, we have just to show that $h_n\to h$ strongly in $L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$, where

$$h\left(x\right):={\left(\phi \right)}_{+}^p+\lambda g\left(x\right).$$

Using Sobolev inequality in $W_0^{s,1}(\Omega )$ and since ${\phi }_n\to \phi$ strongly in $W_0^{s,1}(\Omega )$, then ${\phi }_n\to \phi$ strongly in $L^{{\displaystyle \frac{N}{N-1}}}(\Omega )$. Since ${\left\{{\phi }_n\right\}}_n$ is bounded in $L^r(\Omega )$, then using an interpolation argument, it holds that ${\phi }_n\to \phi$ strongly in $L^{\sigma }(\Omega )$ for all $\sigma <r$, and in particular in $L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$. Hence, $u_n\to u$ strongly in $H_0^s(\Omega )$, and then the continuity of T follows.

Step 4: T is compact.

Let $\left\{{\phi }_n\right\}\subset E$ be a bounded sequence in $W_0^{s,1}(\Omega )$ and define $u_n=T\left({\phi }_n\right)$. We will prove that, up to a sub-sequence, $u_n\to u$ in $W_0^{s,1}(\Omega )$ for some $u\in W_0^{s,1}(\Omega )$.

Recall that ${\left\{{\phi }_n\right\}}_n\subset E$, then as in the proof of Step 3, we reach that ${\left\{{\phi }_n\right\}}_n$ is a bounded sequence in $L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$. Now, setting

$$h_n\left(x\right):={\left({\phi }_n\right)}_{+}^p+\mu g\left(x\right),$$

we have that $\left\{h_n\right\}$ is a bounded sequence in $L^{{\displaystyle \frac{2N}{N+2s}}}(\Omega )$. The result then follows from the compactness result in Proposition 5.

As a conclusion, E is a closed convex set of $W_0^{s,1}(\Omega )$ and, by the previous steps, we can apply the Schauder fixed-point theorem to get the existence of $u\in E$ such that $T\left(u\right)=u$. Thus, we conclude that problem (39) has a weak solution for all $0<\mu \le {\mu }^{*}$. It is clear that $u\in H_0^s(\Omega )$. □

Remark 6.

The condition $p<2_s^{*}$ is classical in this setting in order to ensure a variational structure, in particular when the datum belongs to the dual space of the fractional Sobolev space.

However, in the case $M\equiv 1$, it is possible to consider positive supersolutions for $p\ge 2_s^{*}-1$ under suitable assumptions on the data (for instance, in the radial case by restricting the analysis to radial solutions). In this situation, the solutions may lie outside the standard energy space $H_0^s(\Omega )$.

On the other hand, when $M\not\equiv 1$, this approach seems to be more delicate. Indeed, in order to evaluate the term $M\left({\parallel u\parallel }_{s,2}^2\right)$, it is necessary that the solution belongs a priori to $H_0^s(\Omega )$, which is incompatible with the supercritical exponent.

5. Existence Results in the Case $\mathit{f}(\mathit{x},\nabla \mathit{u}):={|\nabla \mathit{u}|}^{\mathit{p}}+\mathit{\mu }\mathit{g}\left(\mathit{x}\right)$

In this section, we assume that $f(x,\nabla u):={|\nabla u|}^p+\mu g\left(x\right)$, then we consider the problem

$$\left\{\begin{array}{cccc}\hfill M\left({∥u∥}_{s,q}^q\right){(-\Delta )}^{s}u& =& {\displaystyle \lambda \frac{u}{{\left|x\right|}^{2s}}+{|\nabla u|}^p+\mu g}\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& >& 0\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(42)

The main existence result in this section is the following.

Theorem 11.

Let $s\in ({\displaystyle \frac{1}{2}},1)$, $1<q<{\displaystyle \frac{N}{N-s}}$ and $\lambda >0$. Consider $g\in L^1(\Omega )$ to be a nonnegative function such that ${\int }_{\Omega }g{\left|x\right|}^{-\theta }dx<\infty$, for some $\theta >0$. Suppose that $1<p<{\displaystyle \frac{N}{N-(2s-1)}}$. Then, there exists ${\mu }^{*}:={\rho }^{*}(N,p,s,g,\lambda ,\Omega )>0$ such that if $\mu <{\mu }^{*}$, problem (42) has a solution $u\in W_0^{1,\sigma }(\Omega )$, for all $1<\sigma <{\displaystyle \frac{N}{N-(2s-1)}}$. Moreover, ${\int }_{{\mathbb{R}}^N}{|\nabla u|}^{\sigma }{\left|x\right|}^{-\mu \left(\lambda \right)}dx<\infty$.

The assumption on q and p are optimal in the sense that if $q\ge {\displaystyle \frac{N}{N-s}}$ or $p\ge {\displaystyle \frac{N}{N-(2s-1)}}$, then there exists $g\in L^1(\Omega )$ such that ${\int }_{\Omega }g{\left|x\right|}^{-\theta }dx<\infty$ and that problem (42) has a non-positive solution.

Remark 7.

The condition $s>{\displaystyle \frac{1}{2}}$ is natural in this context, due to the consideration of general datum g. Indeed, if we consider the problem

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^{s}w& =& h\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

(43)

it was shown in [25] that the local gradient of w exists and is locally integrable if and only if $s>{\displaystyle \frac{1}{2}}$. Therefore, the restriction $s>{\displaystyle \frac{1}{2}}$ arises naturally in this setting. Another way to see this is via the properties of Green kernel ${\mathcal{G}}_s$ associated with ${(-\Delta )}^s$ in Ω given in (15) and (16). From (16), one has $|{\nabla }_x{\mathcal{G}}_s(x,y)|\le {\displaystyle \frac{C}{{|x-y|}^{N-(2s-1)}}}$, which shows that $\left|{\nabla }_x{\mathcal{G}}_s\right(x,y\left)\right|$ is locally integrable if $s>{\displaystyle \frac{1}{2}}$. This provides an alternative, kernel-based justification for the condition. We refer also to [6] for additional comments.

Proof of Theorem 11.

Fix $1<q<{\displaystyle \frac{N}{N-s}}$ and let $g\in L^1(\Omega )$ be a nonnegative function with ${\int }_{\Omega }g{\left|x\right|}^{-\theta }dx<\infty$ for some $\theta >0$.

Now, let $1<p<{\displaystyle \frac{N}{N-(2s-1)}}$ and consider $r>1$ be such that $1<max\{p,q\}<r<{\displaystyle \frac{N}{N-s}}$. Then, there exists $\widehat{\theta }<\theta$ such that ${\int }_{\Omega }g{\left|x\right|}^{-\widehat{\theta }}dx<\infty$ and $\widehat{\theta }{\displaystyle \frac{r}{r-p}}<N$. Fixed $r,\widehat{\theta }$ as above, we get the existence of ${\mu }^{*}>0$ such that for some $l>0$, we have

$$C_0(l+{\mu }^{*}{\left|\right|g\left|\right|}_{L^1{\left(\right|x|}^{-\widehat{\theta }}dx,\Omega )})=l^{{\displaystyle \frac{1}{p}}},$$

where $C_0$ is a positive constant depending only on $\Omega ,\lambda ,\widehat{\theta }$, and the regularity constant in Theorem 5, estimate (14).

Let $\mu <{\mu }^{*}$ be fixed and consider the set

$$E=\{v\in W_0^{1,1}(\Omega ):v\in W_0^{1,r}(\Omega ) \mathrm{and} ||\nabla v|{|}_{L^r(\Omega )}\le l^{{\displaystyle \frac{1}{p}}}\}.$$

(44)

It is not difficult to show that E is a closed convex set of $W_0^{1,1}(\Omega )$. Consider the operator

$$\begin{array}{ccc}\hfill T:E& \to & W_0^{1,1}(\Omega )\hfill \\ \hfill v& \to & T\left(v\right)=u,\hfill \end{array}$$

where u is the unique solution to problem

$$\left\{\begin{array}{cccc}\hfill M\left({∥u∥}_{s,q}^q\right){(-\Delta )}^{s}u& =& \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+{|\nabla v|}^p+\mu g\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \\ \hfill u& >& 0\hfill & \mathrm{in}\Omega .\hfill \end{array}\right.$$

(45)

Setting $h\left(x\right)={|\nabla v|}^p+\mu g$, then taking into consideration the definition of E, it holds that $h\in L^1{\left(\right|x|}^{-\widehat{\theta }}dx,\Omega )$. Hence, the existence and the uniqueness of u follow using the result of Theorem 8. Moreover, since $s>{\displaystyle \frac{1}{2}}$, using Theorem 9, we deduce that $u\in W_0^{1,\sigma }(\Omega )$ for all $1<\sigma <{\displaystyle \frac{N}{N-(2s-1)}}$ with

$${\displaystyle {\left|\right|u\left|\right|}_{W_0^{1,\sigma }(\Omega )}\le C{\int }_{\Omega }h{\left|x\right|}^{-\widehat{\theta }}dx.}$$

Thus, T is well defined.

We claim that $T\left(E\right)\subset E$. Let $1<\sigma <{\displaystyle \frac{N}{N-(2s-1)}}$; using Hölder inequality, it holds that

$$\begin{array}{ccc}\hfill {\left|\right|u\left|\right|}_{W_0^{1,\sigma }(\Omega )}& \le & {\displaystyle C{\int }_{\Omega }h{\left|x\right|}^{-\widehat{\theta }}dx\le C\left(\mu {\int }_{\Omega }{g\left|x\right|}^{-\widehat{\theta }}dx+{\int }_{\Omega }{|\nabla v|}^p{\left|x\right|}^{-\widehat{\theta }}dx\right)}\hfill \\ & \le & C\left(\mu {\int }_{\Omega }{g\left|x\right|}^{-\widehat{\theta }}dx+({\int }_{\Omega }{|\nabla v|}^r{dx)}^{{\displaystyle \frac{p}{r}}}({\int }_{\Omega }{\left|x\right|}^{-\widehat{\theta }{\displaystyle \frac{r}{r-p}}}{dx)}^{{\displaystyle \frac{r-p}{r}}}\right).\hfill \end{array}$$

Using the fact that $\widehat{\theta }{\displaystyle \frac{r}{r-p}}<N$, it follows that

$${\left|\right|u\left|\right|}_{W_0^{1,\sigma }(\Omega )}\le C\left(\mu {\int }_{\Omega }{g\left|x\right|}^{-\widehat{\theta }}dx+({\int }_{\Omega }{|\nabla v|}^r{dx)}^{{\displaystyle \frac{p}{r}}}\right)\le C\left(\mu {\int }_{\Omega }g{\left|x\right|}^{-\widehat{\theta }}dx+l\right)\le l^{{\displaystyle \frac{1}{p}}}.$$

Choosing $\sigma =r$, it holds that $u\in E$.

To show the continuity of T, we consider a sequence ${\left\{v_n\right\}}_n\subset E$ such that $v_n\to v$ strongly in $W_0^{1,1}(\Omega )$. It is clear that $\left|\right|v_n{\left|\right|}_{W_0^{1,r}(\Omega )}\le C$ and $v\in E$. Using the interpolation inequality, we deduce that $v_n\to v$ strongly in $W_0^{1,\sigma }(\Omega )$ for all $\sigma <r$. Since $\widehat{\theta }{\displaystyle \frac{r}{r-p}}<N$, using the above computations, it follows that ${\int }_{\Omega }{|\nabla v_n|}^p{\left|x\right|}^{-\widehat{\theta }}dx\le C$ for all n and

$${\int }_{\Omega }|\nabla (v_n-v){|}^p{\left|x\right|}^{-\widehat{\theta }}dx\to 0 \mathrm{as} n\to \infty .$$

Setting $h_n={|\nabla v_n|}^p+\mu g$, $h={|\nabla v|}^p+\mu g$, we obtain that $h_n\to h$ strongly in $L^1{\left(\right|x|}^{-\widehat{\theta }}dx,\Omega )$. Let $u_n=T\left(v_n\right),u=T\left(v\right)$, then using Theorem 9, we conclude that $v_n\to v$ strongly in $W_0^{1,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-(2s-1)}}$. Thus, T is continuous. The compactness of T follows in the same way. Therefore, using the Schauder fixed-point theorem, we get the existence of $u\in E$ such that u solves (42).

We show the optimality of the conditions imposed on p and q for the existence of solutions. We begin with the following well-known fact. Consider

$$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^{s}w& =& h\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill w& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$$

(46)

then there exists $C>0$ such that ${\left|\right|w\left|\right|}_{L^{\theta }(\Omega )}\le {C\left|\right|h\left|\right|}_{L^1(\Omega )}$. The regularity result is optimal in the sense that for all $\sigma \ge {\displaystyle \frac{N}{N-2s}}$, there exists $h_{\sigma }\in L^1(\Omega )$ such that the corresponding solution w of (46) satisfies $w\notin L^{\sigma }(\Omega )$. See [6,25].

We now prove that the conditions imposed on p and q are optimal for the existence of a solution to problem (42) when h is arbitrary in $L^1(\Omega )$.

Assume first that for some $p\ge {\displaystyle \frac{N}{N-2s+1}}$, problem (42) admits a solution u for every $h\in L^1(\Omega )$. Define $v=M\left({\parallel u\parallel }_{s,q}^q\right)u$, then v satisfies

$${(-\Delta )}^{s}v\ge h\mathrm{in}\Omega ,$$

and therefore, by the comparison principle $w\ge v=Cu$. If $p\ge {\displaystyle \frac{N}{N-2s+1}}$, using the fact that ${|\nabla v|}^p=C^p{|\nabla u|}^p\in L^1(\Omega )$ and by the Sobolev embedding, we get

$${\int }_{\Omega }{w}^{\theta }\le {\int }_{\Omega }{v}^{\theta }dx=C^p{\int }_{\Omega }{u}^{\theta }dx<\infty \mathrm{for}\mathrm{all}\theta \le {\left({\displaystyle \frac{N}{N-2s+1}}\right)}^{*}={\displaystyle \frac{N}{N-2s}}.$$

Hence, we reach a contradiction with the previous optimal regularity of w. This proves the optimality with respect to p.

Assume now that $q\ge {\displaystyle \frac{N}{N-s}}$. Since $M\left({\parallel u\parallel }_{s,q}^q\right)<\infty$, then ${\parallel u\parallel }_{s,q}<\infty$. By the fractional Sobolev inequality, it holds that $u\in L^{\theta }(\Omega )$ for all $\theta \le {\displaystyle \frac{qN}{N-qs}}$. If $q\ge {\displaystyle \frac{N}{N-s}}$, then ${\displaystyle \frac{qN}{N-qs}}\ge {\displaystyle \frac{N}{N-2s}}$. We reach a contradiction repeating the same argument as above.

Therefore, the conditions imposed on p and q are optimal for the existence of solutions. □

6. Further Results and Open Problems

• Consider the case where $f(x,u):=u^p+g$ and $0<p<1$, then for $q<{\displaystyle \frac{N}{N-s}}$ fixed, we will prove that, for all $\lambda >0$ and for all $g\in L^{\sigma }(\Omega )$ with $\sigma >1$, the problem

  $$\left\{\begin{array}{cccc}\hfill M\left({∥u∥}_{s,q}^q\right){(-\Delta )}^{s}u& =& \lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+u^p+g\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& >& 0\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

  (47)

  has a solution $u\in W_0^{s,\overline{q}}(\Omega )$ for all $\overline{q}<{\displaystyle \frac{N}{N-s}}$. To show that, we follow by approximation. Consider $u_n$, the solution of the problem

  $$\left\{\begin{array}{cccc}\hfill M\left({∥u_n∥}_{s,q}^q\right){(-\Delta )}^s{u}_n& =& \lambda {\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{u_n^p}{1+\frac{1}{n}{u}_n^p}}+T_n\left(g\right)\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u_n& >& 0\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u_n& =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

  (48)
  Notice that the existence of $u_n$ follows using the same arguments as in the proof of Theorem 8. To get our main existence result, we have just to show that $M\left(\right|\left|u_n\right|{|}_{s,q}^q)\le C$ for all n. By contradiction, suppose that $M\left(\right|\left|u_n\right|{|}_{s,q}^q)⟶+\infty$. Since $g\in L^{\sigma }(\Omega )$ with $\sigma >1$, we can fix $\beta >0$ small enough such that ${\int }_{\Omega }g\left(x\right){\left|x\right|}^{-\mu \left(\beta \right)}dx<\infty$. Now consider ${\phi }_{\beta }$, the unique solution to the problem

  $$\left\{\begin{array}{cccc}\hfill {(-\Delta )}^s\phi & =& \beta {\displaystyle \frac{\phi }{{\left|x\right|}^{2s}}}+1\hfill & \mathrm{in}\Omega ,\hfill \\ \hfill \phi & =& 0\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Omega .\hfill \end{array}\right.$$

  (49)
  It is clear that ${\phi }_{\beta }\backsimeq {\left|x\right|}^{-\mu \left(\beta \right)}$ near the origin. Using ${\phi }_{\beta }$ as a test function in (48), it follows that

  $$(\beta -{\displaystyle \frac{\lambda }{M\left({∥u_n∥}_{s,q}^q\right)}}){\int }_{\Omega }{\displaystyle \frac{{\phi }_{\beta }{u}_n}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }{u}_{n}dx\le {\int }_{\Omega }{u}_n^q{\phi }_{\beta }+{\displaystyle \frac{1}{m_0}}{\int }_{\Omega }g{\phi }_{\beta }dx.$$
  Now, using Young inequality and taking into consideration that $q<1$, we deduce that

  $$(\beta -ϵ-{\displaystyle \frac{\lambda }{M\left({∥u_n∥}_{s,q}^q\right)}}){\int }_{\Omega }{\displaystyle \frac{{\phi }_{\beta }{u}_n}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }{u}_{n}dx\le {\displaystyle \frac{1}{m_0}}{\int }_{\Omega }g{\left|x\right|}^{-\mu \left(\beta \right)}dx+C\left(ϵ\right).$$
  For $\beta$ fixed such that ${\int }_{\Omega }g{\left|x\right|}^{-\mu \left(\beta \right)}dx<\infty$, since $M\left(\right|\left|u_n\right|{|}_{s,q}^q)⟶+\infty$, then we can fix $ϵ>0$ small enough such that $(\beta -ϵ-{\displaystyle \frac{\lambda }{M\left({∥u_n∥}_{s,q}^q\right)}})>{\beta }_1>0$. Hence, we deduce that

  $${\int }_{\Omega }{\displaystyle \frac{\phi u_n}{{\left|x\right|}^{2s}}}dx+{\int }_{\Omega }{u}_{n}dx\le C \mathrm{for} \mathrm{all} n.$$
  Setting

  $$h_n\left(x\right)={\displaystyle \frac{1}{M\left({∥u_n∥}_{s,q}^q\right)}}\lambda {\displaystyle \frac{u_n}{{\left|x\right|}^{2s}}}+{\displaystyle \frac{u_n^p}{1+\frac{1}{n}{u}_n^p}}+T_n\left(g\right),$$

  we obtain that $\left|\right|h_n{\left|\right|}_{L^1(\Omega )}\le C$. Thus, we get that ${\left\{u_n\right\}}_n$ is bounded in $W_0^{s,\sigma }(\Omega )$ for all $\sigma <{\displaystyle \frac{N}{N-s}}$, in particular for $\sigma =q$. Thus, ${∥u_n∥}_{s,q}<C_1$ and then $M\left({∥u_n∥}_{s,q}^q\right)<\overline{C}$, a contradiction. Thus, $\left\{M\right(\left|\right|u_n{\left|\right|}_{s,q}^q{\left)\right\}}_n$ is bounded. Now, the rest of the proof follows using closely the same argument as in the proof of Theorem 8.
• Consider now the case $f(x,u)=u^p$ with $p>1$. As it was explained in the introduction, in the case where $M=1$, the existence of a non-negative solution holds if $p<{\displaystyle \frac{{\gamma }_{\lambda }+2s}{{\gamma }_{\lambda }}}$.
  In our case, if u solves

  $$M\left({∥u∥}_{s,q}^q\right){(-\Delta )}^{s}u=\lambda {\displaystyle \frac{u}{{\left|x\right|}^{2s}}}+u^p+g,$$

  (50)

  then setting $\widehat{\lambda }={\displaystyle \frac{\lambda }{M\left({∥u∥}_{s,q}^q\right)}}$, we obtain that $p<{\widehat{p}}_{\widehat{\lambda }}={\displaystyle \frac{{\gamma }_{\widehat{\lambda }}+2s}{{\gamma }_{\widehat{\lambda }}}}$. Taking into consideration that ${\widehat{p}}_{\widehat{\lambda }}\to \infty$ as $\widehat{\lambda }\to 0$, then it seems to be natural to ask if the existence of a solution to problem (50) holds for all $p>1$ under the suitable regularity assumption on g.
• Consider the case where f depends on the gradient. It is well known that in the local case, without the Kirchhoff term and for $f(x,u,\nabla u)={|\nabla u|}^p$, the existence of a solution is obtained under strong limitation on the value of p (in any case $p<2$). It will be interesting to analyze the situation under the presence of the Kirchhoff term. Notice that the comparison principle does not hold in this case, and then radial computations do not allow any conclusion related to the existence of a critical exponent for the existence.

7. Discussion

This paper deals with a fractional Kirchhoff problem involving a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient. The main novelty consists of proving the existence of positive solutions without any restriction on the Hardy parameter $\lambda >0$, a condition usually required in related works.

For nonlinearities depending only on $(x,u)$, existence is obtained for a large class of measurable data, including source terms and combined power-type reactions. When gradient terms are present, the condition $s>{\displaystyle \frac{1}{2}}$ naturally appears as a threshold ensuring sufficient regularity in the fractional framework.

The results also include the local case $s=1$, providing a unified treatment of local and non-local Kirchhoff problems with Hardy potentials. Several issues remain open, notably uniqueness, multiplicity, sign-changing solutions, and critical growth cases, which will be the subject of future investigations.

8. Conclusions and Perspectives

In this work, we have studied Kirchhoff-type problems involving a Hardy-type singular potential acting as a reaction term. We proved existence results for arbitrary values of $\lambda$ and for general data under minimal integrability assumptions ($f\in L^{\theta }(\Omega ),\theta >1$), allowing the treatment of irregular or even measure-valued sources. From a physical viewpoint, such models arise in the description of vibrating strings, membranes, or beams whose tension depends on the total displacement, while Hardy-type singular potentials may represent localized defects, impurities, or highly concentrated forces.

Kirchhoff-type problems with gradient-dependent terms remain relatively underexplored. Such nonlinearities arise naturally when replacing the Laplacian by an operator of the form $-\mathrm{div}\left(a\right(u,\nabla u\left)\right)$, increasing the analytical complexity and allowing the treatment of irregular or even measure-valued data. These models are relevant for media with nonlocal tension effects and localized defects. From a theoretical standpoint, numerous challenging questions remain open and call for deeper and more refined analytical investigation. Several open questions naturally emerge:

• Existence results for general gradient powers, including the case without Hardy potentials.
• Identification of the appropriate functional setting ensuring uniqueness of solutions.
• Analysis of multiplicity phenomena, as in, the absence of the Kirchhoff term where singular solutions appear, concentrating in a singular set with respect to the potential capacities.
• Possible critical exponents arising from the interaction between gradient effects and Hardy-type singularities.
• Under a positivity constraint, it is natural to expect the absence of oscillatory behavior, since positive solutions typically exclude sign changes and therefore prevent oscillations. However, when one considers sign-changing solutions, the situation becomes substantially different. In the hyperbolic Kirchhoff framework, oscillatory solutions may naturally arise, particularly due to the intrinsic wave-like character of the equation. In this context, it is of considerable interest to investigate whether such oscillatory behavior persists in the presence of a Hardy-type singular potential and gradient-dependent nonlinearities, and how the interaction of these effects influences the qualitative structure of sign-changing solutions.