Existence of Solution for a Singular Problem with a General Non-Local Integrated Differential Operator

Abstract

This work examines a singular elliptic problem with a fractional and a non-local integrodifferential operator. The question of whether solutions exist is transformed into the existence of critical points of the associated functional energy, to be more specific. The existence of a critical point is then demonstrated by combining the variational method with some monotonicity arguments. After this, due to the singular non-linearity, we manually demonstrate that this critical point is a weak solution for such a problem.

1. Introduction

In this work, we demonstrate the existence and multiplicity of solutions to the following singular elliptic problem involving a non-local integrodifferential operator

$$\left\{\begin{array}{c}{\mathcal{L}}_A^{\kappa }u+{\left|u\right|}^{r\left(x\right)-2}u=\mu \left(x\right)u^{-m\left(x\right)}+\lambda f(x,u)\mathrm{i}\mathrm{n} \Lambda ,\hfill \\ \\ u=0,\mathrm{o}\mathrm{n} \partial \Lambda ,\hfill \end{array}\right.$$

(1)

where the bounded domain $\Lambda \subset {\mathbb{R}}^N$, $N\ge 2$ has a Lipschitz boundary $\partial \Lambda$, $m\in (0,1)$, $\lambda \in \mathbb{R}$ is the parameter to be specified, and ${\mathcal{L}}_A^{\kappa }$ is the fractional non-local Laplacian operator described as follows:

$${\mathcal{L}}_A^{\kappa }u\left(x\right)=p.v.{\int }_{{\mathbb{R}}^N}A(u\left(x\right)-u\left(y\right))\kappa (x,y)dy,$$

$p.v.$ stands for the Cauchy principal value; $A:\mathbb{R}\times {\mathbb{R}}^N\times {\mathbb{R}}^N\to \mathbb{R}$ and $\kappa :{\mathbb{R}}^N\times {\mathbb{R}}^N\to {\mathbb{R}}_{+}$ are measurable functions.

The non-local integrodifferential operator ${\mathcal{L}}_A^{\kappa }$ is more general than the fractional $p(x,.)$-Laplacian operator ${(-\Delta )}_{p(x,.)}^s$. In particular, when $A(t,x,y)={\left|t\right|}^{p(x,y)-2}t$ and $\kappa (x,y)={\displaystyle \frac{1}{{|x-y|}^{N+sp\left(x\right)}}},$ the operator ${\mathcal{L}}_A^{\kappa }$ is reduced to the fractional $p(x,.)$-Laplacian operator ${(-\Delta )}_{p(x,.)}^s$, which is defined by

$${(-\Delta )}_{p(x,.)}^{s}u\left(x\right)=p.v.{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^{p(x,y)-2}(u\left(x\right)-u\left(y\right))}{{|x-y|}^{N+sp(x,y)}}}dy,$$

A mathematical operator ${(-\Delta )}_{p(x,.)}^s$ expands the classical Laplacian operator to take into account spatially variable exponents and fractional-order derivatives. The origins of this operator can be found in fractional calculus, where it is applied to simulate non-local behavior and irregularity, such as stationary thermo-rheological viscous flows of non-Newtonian fluids, elastic mechanics, electrorheological fluids (see [1]), and image processing (see [2]). The $p(x,.)$-Laplacian operator is a flexible tool in the analysis of complex systems and processes because it recognizes geographical variability and fractional differentiation, in contrast to the normal Laplacian, which assumes constant exponent values and integer-order derivatives. Many authors have recently studied elliptic problems involving the fractional $p(x,·)$-Laplacian; we cite, for instance, the papers [3,4,5,6,7,8,9].

The use of fractional powers of the Laplace operator in elliptic and singular equations has been the subject of numerous works; see, for example, the papers [10,11,12,13,14] and the references therein. In particular, Ratan et al. [13] proved some existing results for the problem

$$\left\{\begin{array}{cc}{\mathcal{L}}_{\Phi }^{\mathbf{K}}u=\lambda {\left|u\left(x\right)\right|}^{q-2}u\left(x\right)\hfill & \mathrm{in}\Lambda ,\hfill \\ \\ u=0,\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Lambda ,\hfill \end{array}\right.$$

with $K:{\mathbb{R}}^N\times {\mathbb{R}}^N⟶(0,\infty [$ is a function that satisfies some suitable hypotheses.

After this, Chammem et al. [15] used the Nehari manifold method to study the problem

$$\left\{\begin{array}{cc}{L}_{\mathbf{K}}^{p(x,.)}\psi =\lambda {\displaystyle \frac{a\left(x\right)}{{\left|\psi \right|}^{\gamma }}}+g(x,\psi )\hfill & \mathrm{in}\Lambda ,\hfill \\ \\ \psi =0,\hfill & \mathrm{in}{\mathbb{R}}^N\setminus \Lambda ,\hfill \end{array}\right.$$

(2)

where $p\in C(\mathbb{R}\times \mathbb{R},(1,+\infty \left)\right),\gamma \in C(\mathbb{R},(0,1\left)\right)$, $\lambda >0$, and the operator $L_{\mathbf{K}}^{p(x,.)}$ is given by

$$L_{\mathbf{K}}^{p(x,.)}u\left(x\right)={\int }_{{\mathbb{R}}^N}{|u\left(x\right)-u\left(y\right)|}^{p(x,y)-2}(u\left(x\right)-u\left(y\right))K(x,y)dy.$$

Under supplementary conditions on the non-linearities a and g, the authors prove that problem (2) admits a non-trivial solution.

We note that the operator ${\mathcal{L}}_A^{\kappa }$ is a natural generalization of many integrodifferential operators, including the fractional Laplacian ${(-\Delta )}^s$, the fractional $p(x,.)$-Laplacian, and the integrodifferential operator $L_{\mathbf{K}}^{p(x,.)}$. Thus, our study is a natural generalization of other works in the literature, such as [15,16]. The novelty of this work is the presence of the general operator ${\mathcal{L}}_A^{\kappa }$ with a singular non-linearity and variable exponents.

The structure of this manuscript is as follows. In Section 2, we introduce essential preliminary insights into variable exponent Lebesgue and fractional Sobolev spaces. In Section 3 and Section 4, we give demonstrations of the existence of solutions of our problem (1); in the first case, the parameter $\lambda$ is non-negative (Theorem (2)), and, in the second case, the parameter $\lambda$ is negative (Theorem 3).

2. Preliminaries

In this part, we present the definitions of the fractional Sobolev and variable exponent Lebesgue spaces, along with their characteristics. To learn more about these spaces, interested readers can consult the works [17,18,19]. Hereafter, $\Lambda$ denotes a bounded domain in ${\mathbb{R}}^N,$ with $N\ge 2$, and $C_{+}(\Lambda )$ denotes the set defined as

$$C_{+}(\Lambda )=\left\{r\in C(\Lambda ),r\left(\xi \right)>1,\forall \xi \in \Lambda \right\}.$$

We introduce the variable exponent Lebesgue space as

$$L^{r\left(\xi \right)}(\Lambda )=\{\psi \in M(\Lambda ),{\int }_{\Lambda }{\left|\psi \left(\xi \right)\right|}^{r\left(\xi \right)}d\xi <\infty \},$$

where $M(\Lambda )$ denotes the set of all measurable functions $\psi :\Lambda \to \mathbb{R}$. It is well known that the space $\left(L^{r\left(\xi \right)}(\Lambda ),{| · |}_{r\left(\xi \right)}\right)$ is a Banach space. Moreover, $\left(L^{r\left(\xi \right)}(\Lambda ),{| · |}_{r\left(\xi \right)}\right)$ becomes separable and reflexive provided that $1<r^{-}\le r^{+}<\infty$, where

$${\left|\psi \right|}_{r\left(\xi \right)}=inf\left\{\mu >0:{\int }_{\Lambda }{\left|{\displaystyle \frac{\psi \left(\xi \right)}{\mu }}\right|}^{r\left(\xi \right)}d\xi \le 1\right\},$$

and, for a given function $\chi \in C_{+}(\Lambda )$, ${\chi }^{-}$ and ${\chi }^{-}$ are given by

$${\chi }^{-}=\underset{\xi \in \overline{\Lambda }}{inf}\chi \left(\xi \right)\mathrm{and}{\chi }^{+}=\underset{\xi \in \overline{\Lambda }}{sup}\chi \left(\xi \right).$$

The space $L^{r\left(\xi \right)}(\Lambda )$ has the same properties as in the classical Lebesgue space; for example, the Hölder inequality holds. Precisely, we have the following.

Proposition 1 

([4]). For any $\psi \in L^{r\left(\xi \right)}(\Lambda )$ and $\varphi \in L^{r^{\prime }\left(\xi \right)}(\Lambda )$, where ${\displaystyle \frac{1}{r\left(\xi \right)}}+{\displaystyle \frac{1}{r^{\prime }\left(\xi \right)}}=1,$ we have

$$|{\int }_{\Lambda }\psi \varphi d\xi |\le ({\displaystyle \frac{1}{r^{-}}}+{\displaystyle \frac{1}{{(r\prime )}^{-}}}){\left|\psi \right|}_{r\left(\xi \right)}{\left|\varphi \right|}_{r\prime \left(\xi \right)}.$$

The modular function is given by

$${\rho }_{r\left(\xi \right)}\left(\psi \right)={\int }_{\Lambda }{\left|\psi \left(\xi \right)\right|}^{r\left(\xi \right)}d\xi .$$

The norm ${\left|\psi \right|}_{r\left(\xi \right)}$ and the modular function ${\rho }_{r\left(\xi \right)}$ have the following relations.

Proposition 2 

([4]). For all $\psi \in L^{r\left(\xi \right)}(\Lambda )$, we have

(1) 

${\left|\psi \right|}_{r\left(\xi \right)}<1(resp=1,>1)\iff {\rho }_{r\left(\xi \right)}\left(\psi \right)<1(resp=1,>1)$;

(2) 

${\left|\psi \right|}_{r\left(\xi \right)}>1\Rightarrow {\left|\psi \right|}_{r\left(\xi \right)}^{r^{-}}\le {\rho }_{r\left(\xi \right)}\left(\psi \right)\le {\left|\psi \right|}_{r\left(\xi \right)}^{r^{+}}$;

(3) 

${\left|\psi \right|}_{r\left(\xi \right)}<1\Rightarrow {\left|\psi \right|}_{r\left(\xi \right)}^{r^{+}}\le {\rho }_{r\left(\xi \right)}\left(\psi \right)\le {\left|\psi \right|}_{r\left(\xi \right)}^{r^{-}}.$

Next, we present an important proposition, which is used in the lower bound of the functional energy.

Proposition 3 

([4]). Let p and r be measurable functions such that $p\in L^{\infty }\left({\mathbb{R}}^N\right)$ and $1\le p\left(\xi \right),r\left(\xi \right)\le \infty$ for all $\xi \in {\mathbb{R}}^N$. Let $\psi \in L^{r\left(\xi \right)}\left({\mathbb{R}}^N\right),\psi \ne 0$. Then,

(1) 

${\left|\psi \right|}_{p\left(\xi \right)r\left(\xi \right)}\le 1\Rightarrow {\left|\psi \right|}_{p\left(\xi \right)r\left(\xi \right)}^{r^{+}}\le {\left|\right|\psi |}^{p\left(\xi \right)}{|}_{r\left(\xi \right)}\le {\left|\psi \right|}_{p\left(\xi \right)r\left(\xi \right)}^{r^{-}}$;

(2) 

${\left|\psi \right|}_{p\left(\xi \right)r\left(\xi \right)}\ge 1\Rightarrow {\left|\psi \right|}_{p\left(\xi \right)r\left(\xi \right)}^{r^{-}}\le {\left|\right|\psi |}^{p\left(\xi \right)}{|}_{r\left(\xi \right)}\le {\left|\psi \right|}_{p\left(\xi \right)r\left(\xi \right)}^{r^{+}}$.

In the rest of this work, we suppose that $s\in (0,1)$, $r\in C\left(\overline{\Lambda }\right)$ and $p\in C(\overline{\Lambda }\times \overline{\Lambda })$ are two continuous functions, where p is symmetric,

$$1<p^{-}=\underset{\overline{\Lambda }\times \overline{\Lambda }}{inf}p(\xi ,\eta )\le p(\xi ,\eta )\le p^{+}=\underset{\overline{\Lambda }\times \overline{\Lambda }}{sup}p(\xi ,\eta )<+\infty ,$$

(3)

and

$$1<r^{-}=\underset{\overline{\Lambda }}{inf}r\left(\xi \right)\le r\left(\xi \right)\le r^{+}=\underset{\overline{\Lambda }}{sup}r\left(\xi \right)<+\infty .$$

(4)

Under hypotheses (3) and (4), the fractional Sobolev space is defined by

$$W^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )=\{\psi \in L^{r\left(\xi \right)}(\Lambda ),{\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\psi \left(\xi \right)-\psi \left(\eta \right)|}^{p(\xi ,\eta )}}{{\mu }^{p(\xi ,\eta )}{|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta <\infty ,$$

for some $\mu >0\}$.

Let

$${\left[\psi \right]}_{s,p(\xi ,\eta )}=inf\{\mu >0:{\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\psi \left(\xi \right)-\psi \left(\eta \right)|}^{p(\xi ,\eta )}}{{\mu }^{p(\xi ,\eta )}{|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta \le 1\},$$

$\left(W^{s,r\left(\xi \right),p(\xi ,\eta )}{(\Lambda ),\Vert \psi \Vert }_{W^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )}\right)$ is a Banach, separable, and reflexive space with the norm

$${\Vert \psi \Vert }_{W^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )}={\left|\psi \right|}_{r\left(\xi \right)}+{\left[\psi \right]}_{s,p(\xi ,\eta )}.$$

Let $W_0^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )=\overline{C_0^{\infty }(\Lambda )}$ in $W^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )$; then, $W_0^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )$ is a Banach and reflexive space with the norm

$$\left|\right|\psi \left|\right|={\left[\psi \right]}_{s,p(\xi ,\eta )}.$$

In the next part of this paper, we denote $X=W^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )$, $E=W_0^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )$.

Lemma 1 

([4]).

(1) 

If $1\le {\left[\psi \right]}_{s,p(\xi ,\eta )}<\infty$, we have

$${\left[\psi \right]}_{s,p(\xi ,\eta )}^{p^{-}}\le {\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\psi \left(\xi \right)-\psi \left(\eta \right)|}^{p(\xi ,\eta )}}{{|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta \le {\left[\psi \right]}_{s,p(\xi ,\eta )}^{p^{+}}.$$

(2) 

If ${\left[\psi \right]}_{s,p(\xi ,\eta )}\le 1$, we have

$${\left[\psi \right]}_{s,p(\xi ,\eta )}^{p^{+}}\le {\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\psi \left(\xi \right)-\psi \left(\eta \right)|}^{p(\xi ,\eta )}}{{|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta \le {\left[\psi \right]}_{s,p(\xi ,\eta )}^{p^{-}}.$$

For $\psi \in X$, we define

$$\rho \left(\psi \right)={\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\psi \left(\xi \right)-\psi \left(\eta \right)|}^{p(\xi ,\eta )}}{{|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta +{\int }_{\Lambda }{\left|\psi \left(\xi \right)\right|}^{r\left(\xi \right)}d\xi$$

and

$${\Vert \psi \Vert }_{\rho }=inf\{\mu >0:\rho \left({\displaystyle \frac{\psi }{\mu }}\right)\le 1\}.$$

Then, $\Vert ·{ \Vert }_{\rho }$ is a norm; moreover, $\Vert ·{ \Vert }_{\rho }$ and $\Vert ·{ \Vert }_{W^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )}$ are equivalent.

We finish this section by presenting the following embedding result.

Theorem 1 

([20,21]). Let $t:\overline{\Lambda }\to (1,\infty )$ be a continuous function such that, for all $\xi \in \overline{\Lambda }$, we have

$$p^{*}\left(\xi \right)>t\left(\xi \right)\ge t^{-}>1,andt\left(\xi \right)\ge p(\xi ,\xi ),$$

where

$$p^{*}\left(\xi \right)={\displaystyle \frac{Np(\xi ,\xi )}{N-sp(\xi ,\xi )}}.$$

If, in addition, for any $\eta \in \overline{\Lambda }$, we have $sp(\xi ,\eta )<N$, then the embedding from E into $L^{t\left(\xi \right)}(\Lambda )$ is continuous and compact. Moreover, there exists $k>0$ such that, for any $\psi \in E$, we have

$${\left|\psi \right|}_{L^{t\left(\xi \right)}}\le {k\left|\right|\psi \left|\right|}_E.$$

3. Existence Result for $\mathit{\lambda }\ge \mathbf{0}$

In this section, we present and prove the first existence result concerning the problem (1) in the case when $\lambda \ge 0$. To this aim, we assume the following hypotheses.

(A₁) 

$A_{(x,y)}:t⟼A(t,x,y)$ is an increasing and odd function, such that the function ${\widehat{A}}_{(x,y)}$, defined by

$${\widehat{A}}_{(x,y)}\left(t\right)={\int }_0^{\left|t\right|}{A}_{(x,y)}\left(\xi \right)d\xi ,$$

is strictly convex for any $(x,y)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$.

(A₂) 

For any $(x,\xi ,\eta )\in \mathbb{R}\times {\mathbb{R}}^N\times {\mathbb{R}}^N$, there exist $c_A>0$ and $C_A>0$, such that

$$A_{(\xi ,\eta )}\left(x\right)\le C_A{\left|x\right|}^{p(\xi ,\eta )-2}x,\mathrm{and}{A}_{(\xi ,\eta )}\left(x\right)x\ge c_A{\left|x\right|}^{p(\xi ,\eta )}.$$

(A₃) 

There exist $0<a_1<a_2$, such that the kernel $\kappa :{\mathbb{R}}^N\times {\mathbb{R}}^N\to {\mathbb{R}}_{+}$ satisfies the following inequalities:

$${\displaystyle \frac{a_1}{{|\xi -\eta |}^{N+sp(\xi ,\eta )}}}\le \kappa (\xi ,\eta )\le {\displaystyle \frac{a_2}{{|\xi -\eta |}^{N+sp(\xi ,\eta )}}},\forall (\xi ,\eta )\in {\mathbb{R}}^N\times {\mathbb{R}}^{N}with\xi \ne \eta .$$

(H₁) 

The function $\mu$ is positive in $L^{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}(\Lambda )$, where t and r satisfy

$$1<t\left(\xi \right)<p^{*}\left(\xi \right),\mathrm{and}p(\xi ,\xi )\le r\left(\xi \right)<p^{*}\left(\xi \right),\forall \xi \in {\mathbb{R}}^N.$$

(H₂) 

There exist $c>0$ and $\sigma \in L^{S\left(\xi \right)}(\Lambda )$, such that, for any $(\xi ,\eta )\in {\mathbb{R}}^N\times \mathbb{R}$, we have

$$f(\xi ,\eta )\le {c\sigma \left(\xi \right)\left|\eta \right|}^{v\left(\xi \right)-2}\eta ,$$

where S and v are continuous functions on $\overline{\Lambda }$ such that

$$1<v\left(\xi \right)<p(\xi ,\eta )<{\displaystyle \frac{N}{s}}<S\left(\xi \right).$$

(H₃) 

There exists ${\Lambda }_1\subset \subset \Lambda$ with $|{\Lambda }_1|>0$, such that, for each $\xi \in {\Lambda }_1$, we have $f(\xi ,\eta )\ge 0$.

The first main result of this work is the following theorem.

Theorem 2. 

Assume that hypotheses $\left(A_1\right)$–$\left(A_3\right)$ and $\left(H_1\right)$–$\left(H_3\right)$ hold. Then, for any $\lambda \ge 0$, the problem (1) admits a non-trivial weak solution.

We note that a function $\psi$ in E is said to be a weak solution of the problem (1) if, for each $\varphi \in E$, we have

$$\begin{array}{c}\hfill {\int }_{\Lambda \times \Lambda }{A}_{(\xi ,\eta )}(\psi \left(\xi \right)-\psi \left(\eta \right))(\varphi \left(\xi \right)-\varphi \left(\eta \right))\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\left|\psi \left(\xi \right)\right|}^{r\left(\xi \right)-2}\psi \left(\xi \right)\varphi \left(\xi \right)d\xi \\ \hfill -{\int }_{\Lambda }\mu \left(\xi \right){\left|\psi \right|}^{-m\left(\xi \right)}\varphi \left(\xi \right)d\xi -\lambda {\int }_{\Lambda }f(\xi ,\psi \left(\xi \right))\varphi \left(\xi \right)d\xi =0.\end{array}$$

Associated with problem (1), we define the functional ${\mathcal{I}}_{\lambda }:E\to \mathbb{R}$ as

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\psi \right)& =& \mathcal{J}\left(\psi \right)-{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}{\left|\psi \right|}^{1-m\left(\xi \right)}d\xi -\lambda {\int }_{\Lambda }F(\xi ,\psi \left(\xi \right))d\xi ,\hfill \end{array}$$

where $F(\xi ,t)={\int }_0^{t}f(\xi ,s)ds$, and

$$\mathcal{J}\left(\psi \right)={\int }_{\Lambda \times \Lambda }{\widehat{A}}_{(\xi ,\eta )}(\psi \left(\xi \right)-\psi \left(\eta \right))\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\displaystyle \frac{{\left|\psi \left(\xi \right)\right|}^{r\left(\xi \right)}}{r\left(\xi \right)}}d\xi .$$

It is noted that the above functional is used to prove the existence of weak solutions, so we need to establish some properties related to the mountain pass geometry, which are summarized in the next three lemmas.

Lemma 2. 

Suppose that hypotheses $\left(A_1\right)$–$\left(A_3\right)$ and $\left(H_1\right)$–$\left(H_3\right)$ hold. Then, for any $\lambda \ge 0$, the functional ${\mathcal{I}}_{\lambda }$ is coercive in E.

Proof. 

Let $\varphi \in E$ with $\left|\right|\varphi \left|\right|>1$; then, from hypotheses $\left(A_1\right)$–$\left(A_3\right)$, Lemma 1, and Proposition 2, we have

$$\begin{array}{ccc}\hfill \mathcal{J}\left(\varphi \right)& \ge & {\displaystyle \frac{a_1{c}_A}{p^{+}}}{\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\varphi \left(\xi \right)-\varphi \left(\eta \right)|}^{p(\xi ,\eta )}}{{|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta \hfill \\ & & +{\displaystyle \frac{1}{r^{+}}}{\int }_{\Lambda }{\left|\varphi \left(\xi \right)\right|}^{r\left(\xi \right)}d\xi \hfill \\ & \ge & min({\displaystyle \frac{a_1{c}_A}{p^{+}}},{\displaystyle \frac{1}{r^{+}}}){\left|\right|\varphi \left|\right|}^{p^{-}}.\hfill \end{array}$$

(5)

Now, from Proposition 1, we have

$${\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}{\varphi }^{1-m\left(\xi \right)}d\xi \le {\displaystyle \frac{1}{1-m^{+}}}{\int }_{\Lambda }\mu \left(\xi \right){\varphi }^{1-m\left(\xi \right)}d\xi \le {\displaystyle \frac{1}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}{\left|\right|\varphi |}^{1-m\left(\xi \right)}{|}_{{\displaystyle \frac{t\left(\xi \right)}{1-m\left(\xi \right)}}}.$$

Then, using hypotheses $\left(H_1\right)$–$\left(H_2\right)$, Proposition 2, and Theorem 1, we obtain

$$\begin{array}{ccc}\hfill {\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m(\xi )}}{\varphi }^{1-m(\xi )}d\xi & \le & {\displaystyle \frac{1}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}{\left(\right|\varphi |}_{t\left(\xi \right)}^{1-m^{+}}+{\left|\varphi \right|}_{t\left(\xi \right)}^{1-m^{-}})\hfill \\ & \le & {\displaystyle \frac{c}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}{\left(\right|\left|\varphi \right||}^{1-m^{+}}+{\left|\right|\varphi \left|\right|}^{1-m^{-}}).\hfill \end{array}$$

(6)

On the other hand, from hypothesis $\left(H_3\right)$, Proposition 1, and Lemma 1, we have

$${\int }_{\Lambda }F(\xi ,\varphi \left(\xi \right))d\xi \le c{\int }_{\Lambda }{\sigma \left(\xi \right)\left|\varphi \left(\xi \right)\right|}^{v\left(\xi \right)}d\xi \le {c\left|\sigma \right|}_{S\left(\xi \right)}{\left|\right|\varphi |}^{v\left(\xi \right)}{|}_{S\prime \left(\xi \right)}\le {c\left|\sigma \right|}_{S\left(\xi \right)}{\left(\right|\varphi |}_{S\prime \left(\xi \right)v\left(\xi \right)}^{v^{+}}+{\left|\varphi \right|}_{S\prime \left(\xi \right)v\left(\xi \right)}^{v^{-}}).$$

So, from $\left(H_3\right)$ and Theorem 1, we obtain

$${\int }_{\Lambda }F(\xi ,\varphi \left(\xi \right))d\xi \le {c\left|\sigma \right|}_{S\left(\xi \right)}{\left(\right|\left|\varphi \right||}^{v^{+}}+{\left|\right|\varphi \left|\right|}^{v^{-}}).$$

(7)

Finally, by combining Equations (5)–(7) with the fact that $\lambda \ge 0$, we deduce that

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\varphi \right)& \ge & min({\displaystyle \frac{a_1{c}_A}{p^{+}}},{\displaystyle \frac{1}{r^{+}}}){\left|\right|\varphi \left|\right|}^{p^{-}}-{\displaystyle \frac{c}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}{\left(\right|\left|\varphi \right||}^{1-m^{+}}+{\left|\right|\varphi \left|\right|}^{1-m^{-}})\hfill \\ & -& {c\lambda \left|\sigma \right|}_{S\left(\xi \right)}{\left(\right|\left|\varphi \right||}^{v^{+}}+{\left|\right|\varphi \left|\right|}^{v^{-}}).\hfill \end{array}$$

Since $1-m^{-}<v^{+}<p^{-},$ then, from the last inequality, we can deduce that ${\mathcal{I}}_{\lambda }\left(\varphi \right)\to \infty$ as $\left|\right|\varphi \left|\right|\to \infty$. Hence, ${\mathcal{I}}_{\lambda }$ is coercive on E. □

Lemma 3. 

Suppose that the hypotheses $\left(A_1\right)-\left(A_3\right)$ and $\left(H_1\right)-\left(H_3\right)$ hold. Then, for each $\lambda \ge 0$, there exists $\varphi \in E$ such that $\varphi \ge 0,\varphi \ne 0$ and ${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0$ provided that $t\in (0,1)$ is small enough.

Proof. 

Let $0<t<1$, $\varphi \in C_0^{\infty }(\Lambda )$ such that $supp\left(\varphi \right)\subset {\Lambda }_1\subset \subset \Lambda ,\varphi =1$ in a subset $\Lambda \prime \subset supp\left(\varphi \right)$, $0\le \varphi \le 1$ in $\Lambda$.

From hypotheses $\left(A_1\right)-\left(A_3\right)$, $\left(H_3\right)$ and Lemma 1, we obtain

$$\begin{array}{ccc}\hfill \mathcal{J}\left(t\varphi \right)& \le & a_2{C}_A{\int }_{\Lambda \times \Lambda }{\displaystyle \frac{t^{p(\xi ,\eta )}{|\varphi \left(\xi \right)-\varphi \left(\eta \right)|}^{p(\xi ,\eta )}}{{p(\xi ,\eta )|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta +{\int }_{\Lambda }{\displaystyle \frac{t^{r\left(\xi \right)}{\left|\varphi \left(\xi \right)\right|}^{r\left(\xi \right)}}{r\left(\xi \right)}}d\xi \hfill \\ & \le & t^{min(p^{-},r^{-})}\left({\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{+}}+{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\xi \right)}d\xi \right).\hfill \end{array}$$

(8)

Since $min(p^{-},r^{-})>1-m^{-}$, and $\lambda \ge 0$, then, from hypothesis $\left(H_3\right)$, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(t\varphi \right)& \le & t^{min(p^{-},r^{-})}\left({\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{+}}+{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\xi \right)}d\xi \right)-t^{1-m^{-}}{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)\varphi {\left(\xi \right)}^{1-m\left(\xi \right)}}{1-m\left(\xi \right)}}d\xi ,\hfill \\ & \le & t^{1-m^{-}}\left(t^{min(p^{-},r^{-})-1+m^{-}}{A}_1-B_1\right),\hfill \end{array}$$

where

$$A_1={\displaystyle \frac{a_2{C}_A}{p^{-}}}{\left|\right|\varphi \left|\right|}^{p^{+}}+{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\left|\right|\varphi \left|\right|}^{p^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\xi \right)}d\xi ,\mathrm{and}{B}_1={\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)\varphi {\left(\xi \right)}^{1-m\left(\xi \right)}}{1-m\left(\xi \right)}}d\xi .$$

Then,

$${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0,\mathrm{f}\mathrm{o}\mathrm{r}0<t<min\left(1,{\left({\displaystyle \frac{B_1}{A_1}}\right)}^{{\displaystyle \frac{1}{min(p^{-},r^{-})-1+m^{-}}}}\right).$$

□

We note that, from Lemma 3, the infimum of ${\mathcal{I}}_{\lambda }$ in E exists. So, let

$${\displaystyle l=\underset{\varphi \in E}{inf}{\mathcal{I}}_{\lambda }\left(\varphi \right).}$$

Lemma 4. 

Assume that hypotheses $\left(A_1\right)-\left(A_3\right)$ and $\left(H_1\right)-\left(H_3\right)$ hold. Then, for all $\lambda \ge 0$, there exists ${\psi }_{*}\in E$ such that ${\mathcal{I}}_{\lambda }\left({\psi }_{*}\right)=l<0$.

Proof. 

Let ${\left\{{\psi }_n\right\}}_{n\in \mathbb{N}}$ be a minimizing sequence for ${\mathcal{I}}_{\lambda }$; this means that ${\mathcal{I}}_{\lambda }\left({\psi }_n\right)\to l$ as $n\to 0$. Since ${\mathcal{I}}_{\lambda }$ is coercive, then the sequence $\left({\psi }_n\right)$ is bounded in $W_0^{s,r\left(\xi \right),p(\xi ,\eta )}(\Lambda )$. Since the space E is reflexive, there exists a sub-sequence also denoted by ${\psi }_n$ and ${\psi }_{*}\in E$ such that

$$\left\{\begin{array}{c}{\psi }_n\rightharpoonup {\psi }_{*}weaklyinE;\hfill \\ {\psi }_n\to {\psi }_{*}stronglyin{L}^{\alpha \left(\xi \right)}(\Lambda ),1\le \alpha \left(\xi \right)<p^{*}\left(\xi \right);\hfill \\ {\psi }_n\to {\psi }_{*}a.ein\Lambda .\hfill \end{array}\right.$$

From Equation (8), we know that

$$\begin{array}{ccc}\hfill \mathcal{J}\left(\varphi \right)& \le & a_2{C}_A{\int }_{\Lambda \times \Lambda }{\displaystyle \frac{{|\varphi \left(\xi \right)-\varphi \left(\eta \right)|}^{p(\xi ,\eta )}}{{p(\xi ,\eta )|\xi -\eta |}^{N+sp(\xi ,\eta )}}}d\xi d\eta +{\int }_{\Lambda }{\displaystyle \frac{{\left|\varphi \left(\xi \right)\right|}^{r\left(\xi \right)}}{r\left(\xi \right)}}d\xi .\hfill \end{array}$$

Thus, using Fatou’s Lemma, and the fact ${\psi }_n\to {\psi }_{*}$ a.e in $\Lambda$, we obtain

$$\mathcal{J}\left({\psi }_{*}\right)\le \underset{n\to \infty }{lim}inf\mathcal{J}\left({\psi }_n\right).$$

(9)

Next, we will prove that

$$\underset{n\to +\infty }{lim}{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}|{\psi }_n{|}^{1-m\left(\xi \right)}d\xi ={\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}{\left|{\psi }_{*}\right|}^{1-m\left(\xi \right)}d\xi .$$

(10)

From Theorem 1 and the fact that $\left({\psi }_n\right)$ is bounded in E, we conclude that $\left\{{\psi }_n\right\}$ is also bounded in $L^{t\left(\xi \right)}(\Lambda )$. Thus, using Vitali’s theorem (see [22], p. 113), it suffices to prove that the set

$$\left\{{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}{\left|{\psi }_n\right|}^{1-m\left(\xi \right)}d\xi ,n\in \mathbb{N}\right\},$$

is equi-absolutely continuous.

Let $ϵ>0$; then, from Proposition 2 and using the absolute continuity of ${\int }_{\Lambda }{\left|\mu \left(\xi \right)\right|}^{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}d\xi$, there exist $\alpha ,\tau >0$, such that, for every ${\Lambda }_2\subset \Lambda$ with $|{\Lambda }_2|<\tau$, we have

$${\left|\mu \right|}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}^{\alpha }\le {\int }_{{\Lambda }_2}{\left|\mu \left(\xi \right)\right|}^{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}d\xi \le {ϵ}^{\alpha }.$$

So, from Proposition 3, we obtain

$${\int }_{{\Lambda }_2}{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}|{\psi }_n{|}^{1-m\left(\xi \right)}d\xi \le {\displaystyle \frac{1}{1-m^{+}}}{|\mu |}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}} \Vert {\psi }_n{|}^{1-m\left(\xi \right)}{|}_{{\displaystyle \frac{t\left(\xi \right)}{1-m\left(\xi \right)}}}\le {\displaystyle \frac{1}{1-m^{+}}}{|\mu |}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}\left(|{\psi }_n{|}_{t\left(\xi \right)}^{1-m^{-}},{|{\psi }_n|}_{t\left(\xi \right)}^{1-m^{+}}\right).$$

Thus,

$${\int }_{{\Lambda }_2}{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}{|{\psi }_n|}^{1-m\left(\xi \right)}d\xi <{\displaystyle \frac{ϵ}{1-m^{+}}}\left(|{\psi }_n{|}_{t\left(\xi \right)}^{1-m^{-}},{|{\psi }_n|}_{t\left(\xi \right)}^{1-m^{+}}\right).$$

Since $t\left(\xi \right)<p^{*}\left(\xi \right)$, then $(|{\psi }_n{|}_{t\left(\xi \right)})$ is bounded, and this facts implies that (10) is valid.

Finally, let us prove that

$$\underset{n\to \infty }{lim}{\int }_{\Lambda }F(\xi ,{\psi }_n\left(\xi \right))d\xi ={\int }_{\Lambda }F(\xi ,{\psi }_{*}\left(\xi \right))d\xi .$$

(11)

Let $ϵ>0$; then, from $\left(H_3\right)$, there exists $c_{ϵ}>0$ such that

$$|F(\xi ,{\psi }_n\left(\xi \right))|\le {\displaystyle \frac{c_{ϵ}}{v^{-}}}|\sigma \left(\xi \right)\Vert {\psi }_n{|}^{v\left(\xi \right)}.$$

(12)

Since $S\prime \left(\xi \right)v\left(\xi \right)<p^{*}\left(\xi \right)$, then, from the Sobolev embedding, we deduce that ${\psi }_n\to {\psi }_{*}$ strongly in $L^{S\prime \left(\xi \right)v\left(\xi \right)}(\Lambda )$. Moreover, up to a sub-sequence, we have ${\psi }_n\to {\psi }_{*}$ a.e in $\Lambda$, and there exists $\mu \in L^{S\prime \left(\xi \right)v\left(\xi \right)}(\Lambda )$ such that $|{\psi }_n\left(\xi \right)|\le \mu \left(\xi \right)$. Thus, from Equation (12) and Proposition 1, we obtain

$$|F(\xi ,{\psi }_n\left(\xi \right))|\le {\displaystyle \frac{c_{ϵ}}{v^{-}}}{\left|\sigma \left(\xi \right)\right|\left|\mu \left(\xi \right)\right|}^{v\left(\xi \right)},$$

and

$${\int }_{\Lambda }|F(\xi ,{\psi }_n\left(\xi \right))|d\xi \le {\displaystyle \frac{c_{ϵ}}{v^{-}}}{\left|\sigma \right|}_{S\left(\xi \right)}{\left|\mu \right|}_{S\prime \left(\xi \right)}^{v\left(\xi \right)}.$$

Hence, Proposition 3 implies that Equation (11) is valid.

By combining Equations (9) and (10) with Equation (11), we obtain the weakly lower semi-continuity of the operator ${\mathcal{I}}_{\lambda }$. So, we obtain

$$l\le {\mathcal{I}}_{\lambda }\left({\psi }_{*}\right)\le \underset{n\to \infty }{lim}inf{\mathcal{I}}_{\lambda }\left({\psi }_n\right)=l$$

Thus, ${\mathcal{I}}_{\lambda }\left({\psi }_{*}\right)=l$. □

Proof of Theorem 2. 

From Lemma 4, we deduce that ${\psi }_{*}$ is a global minimizer for ${\mathcal{I}}_{\lambda }$; this means that, for all $t\in \mathbb{R}$, and for all $\varphi \in E$, we have

$$0\le {\mathcal{I}}_{\lambda }({\psi }_{*}+t\varphi )-{\mathcal{I}}_{\lambda }\left({\psi }_{*}\right).$$

(13)

Moreover, ${\psi }_{*}$ satisfies ${\mathcal{I}}_{\lambda }\left({\psi }_{*}\right)<0$, which implies that ${\psi }_{*}$ is non-trivial.

From Equation (13), for any $t>0$ and any arbitrary $\varphi \in E$, we obtain

$${\displaystyle 0\le \underset{t\to 0}{lim}\frac{{\mathcal{I}}_{\lambda }({\psi }_{*}+t\varphi )-{\mathcal{I}}_{\lambda }\left({\psi }_{*}\right)}{t}.}$$

This implies that

$$\begin{array}{ccc}& & {\int }_{\Lambda \times \Lambda }{A}_{(\xi ,\eta )}({\psi }_{*}\left(\xi \right)-{\psi }_{*}\left(\eta \right))(\varphi \left(\xi \right)-\varphi \left(\eta \right))\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\left|{\psi }_{*}\left(\xi \right)\right|}^{r\left(\xi \right)-2}{\psi }_{*}\left(\xi \right)\varphi \left(\xi \right)d\xi \hfill \\ & -& {\int }_{\Lambda }\mu \left(\xi \right){\left|{\psi }_{*}\right|}^{-m\left(\xi \right)}\varphi \left(\xi \right)d\xi -\lambda {\int }_{\Lambda }f(\xi ,{\psi }_{*}\left(\xi \right))\varphi \left(\xi \right)d\xi \ge 0.\hfill \end{array}$$

Since $\varphi$ is arbitrary, we can replace $\varphi$ with $-\varphi$ in the last inequality, so we have

$$\begin{array}{ccc}& & {\int }_{\Lambda \times \Lambda }{A}_{(\xi ,\eta )}({\psi }_{*}\left(\xi \right)-{\psi }_{*}\left(\eta \right))(\varphi \left(\xi \right)-\varphi \left(\eta \right))\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\left|{\psi }_{*}\left(\xi \right)\right|}^{r\left(\xi \right)-2}{\psi }_{*}\left(\xi \right)\varphi \left(\xi \right)d\xi \hfill \\ & -& {\int }_{\Lambda }\mu \left(\xi \right){\left|{\psi }_{*}\right|}^{-m\left(\xi \right)}\varphi \left(\xi \right)d\xi -\lambda {\int }_{\Lambda }f(\xi ,{\psi }_{*}\left(\xi \right))\varphi \left(\xi \right)d\xi \le 0.\hfill \end{array}$$

Hence, we deduce that

$$\begin{array}{ccc}& & {\int }_{\Lambda \times \Lambda }{A}_{(\xi ,\eta )}({\psi }_{*}\left(\xi \right)-{\psi }_{*}\left(\eta \right))(\varphi \left(\xi \right)-\varphi \left(\eta \right))\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\left|{\psi }_{*}\left(\xi \right)\right|}^{r\left(\xi \right)-2}{\psi }_{*}\left(\xi \right)\varphi \left(\xi \right)d\xi \hfill \\ & -& {\int }_{\Lambda }\mu \left(\xi \right){\left|{\psi }_{*}\right|}^{-m\left(\xi \right)}\varphi \left(\xi \right)d\xi -\lambda {\int }_{\Lambda }f(\xi ,{\psi }_{*}\left(\xi \right))\varphi \left(\xi \right)d\xi =0.\hfill \end{array}$$

Finally, we conclude that ${\psi }_{*}$ is a non-trivial solution of problem (1). □

4. Existence Result for $\mathit{\lambda }<\mathbf{0}$

In this section, we present and prove the second main result of this work concerning the existence of a weak solution for problem (1). To this aim, we assume the following supplementary hypothesis.

(H₄) 

There exists $A>0$ such that

$${\int }_{\Lambda }F(x,t)dx\ge 0,\forall t>A.$$

The second main result of this paper is the following theorem.

Theorem 3. 

Assume that hypotheses $\left(A_1\right)$–$\left(A_3\right)$ and $\left(H_1\right)$–$\left(H_4\right)$ hold. Then, for any $\lambda <0,$ the problem (1) admits a non-trivial weak solution.

As in Section 3, to prove Theorem 3, we need to prove three lemmas. In the first lemma, we show the coercivity of the energy functional ${\mathcal{I}}_{\lambda }$.

Lemma 5. 

Under the same hypotheses as Theorem 3, if $\lambda <0$, the functional ${\mathcal{I}}_{\lambda }$ is coercive in E.

Proof. 

Let $\varphi \in E$ with $\left|\right|\varphi \left|\right|>max(1,A)$, where A is defined in hypothesis $\left(H_4\right)$; then, from Equations (5) and (6), we obtain

$$\begin{array}{c}\hfill {\int }_{\Lambda \times \Lambda }{\widehat{A}}_{(\xi ,\eta )}(\varphi \left(\xi \right)-\varphi \left(\eta \right))\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\displaystyle \frac{{\left|\varphi \left(\xi \right)\right|}^{r\left(\xi \right)}}{r\left(\xi \right)}}d\xi \ge min({\displaystyle \frac{a_1{c}_A}{p^{+}}},{\displaystyle \frac{1}{r^{+}}}){\left|\right|\varphi \left|\right|}^{p^{-}},\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}{\varphi }^{1-m\left(\xi \right)}d\xi \le {\displaystyle \frac{1}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}{\left(\right|\varphi |}_{t\left(\xi \right)}^{1-m^{+}}+{\left|\varphi \right|}_{t\left(\xi \right)}^{1-m^{-}}).\end{array}$$

Hence, we deduce that

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(\varphi \right)& \ge & {\int }_{\Lambda \times \Lambda }{\widehat{A}}_{(\xi ,\eta )}(\varphi \left(\xi \right)-\varphi \left(\eta \right))\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\displaystyle \frac{{\left|\varphi \left(\xi \right)\right|}^{r\left(\xi \right)}}{r\left(\xi \right)}}d\xi -{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)}{1-m\left(\xi \right)}}{\varphi }^{1-m\left(\xi \right)}d\xi \hfill \\ & \ge & min({\displaystyle \frac{a_1{c}_A}{p^{+}}},{\displaystyle \frac{1}{r^{+}}}){\left|\right|\varphi \left|\right|}^{p^{-}}-{\displaystyle \frac{c}{1-m^{+}}}{\left|\mu \right|}_{{\displaystyle \frac{t\left(\xi \right)}{t\left(\xi \right)+m\left(\xi \right)-1}}}{\left(\right|\left|\varphi \right||}^{1-m^{+}}+{\left|\right|\varphi \left|\right|}^{1-m^{-}}).\hfill \end{array}$$

Since we have $1-m^{-}<p^{-},$ then ${\mathcal{I}}_{\lambda }\left(\varphi \right)\to \infty$ as $\left|\right|\varphi \left|\right|\to \infty$. That is, ${\mathcal{I}}_{\lambda }$ is coercive on E. □

Lemma 6. 

Under the same hypotheses as Theorem 3, if $\lambda <0$, there exists a non-trivial non-negative function ϕ in E such that ${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0$, provided that $t\in (0,1)$ is small enough.

Proof. 

Let $0<t<1$, $\varphi \in C_0^{\infty }(\Lambda )$ such that $supp\left(\phi \right)\subset {\Lambda }_1\subset \subset \Lambda ,\varphi =1$ in a subset $\Lambda \prime \subset supp\left(\varphi \right)$, $0\le \varphi \le 1$ in $\Lambda$. Then, from Equation (7), we obtain

$${\int }_{\Lambda }F(\xi ,t\varphi \left(\xi \right))d\xi \le c{t}^{v^{-}}{\left|\sigma \right|}_{S\left(\xi \right)}{\left(\right|\left|\varphi \right||}^{v^{+}}+{\left|\right|\varphi \left|\right|}^{v^{-}}).$$

Moreover, from Equation (8), we have

$$\begin{array}{ccc}& & {\int }_{\Lambda \times \Lambda }{\widehat{A}}_{(\xi ,\eta )}\left(t\left(\varphi \right(\xi )-\varphi (\eta \left)\right)\right)\kappa (\xi ,\eta )d\xi d\eta +{\int }_{\Lambda }{\displaystyle \frac{t^{r\left(\xi \right)}{\left|\varphi \left(\xi \right)\right|}^{r\left(\xi \right)}}{r\left(\xi \right)}}d\xi \hfill \\ & \le & t^{min(p^{-},r^{-})}\left({\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{+}}+{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\xi \right)}d\xi \right).\hfill \end{array}$$

So, by combining the above equations, we conclude that

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(t\varphi \right)& \le & t^{min(p^{-},r^{-})}\left({\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{+}}+{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\xi \right)}d\xi \right)+c{t}^{v^{-}}{\left|\sigma \right|}_{S\left(\xi \right)}{\left(\right|\left|\varphi \right||}^{v^{+}}+{\left|\right|\varphi \left|\right|}^{v^{-}})\hfill \\ & -& t^{1-m^{-}}{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)\varphi {\left(\xi \right)}^{1-m\left(\xi \right)}}{1-m\left(\xi \right)}}d\xi .\hfill \end{array}$$

Since $1-m^{-}<v^{-}<p^{-}$, then we obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_{\lambda }\left(t\varphi \right)& \le & t^{min(v^{-},r^{-})}\left[{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{+}}+{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\Vert \varphi \Vert }^{p^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\xi \right)}d\xi +{c\left|\sigma \right|}_{S\left(\xi \right)}{(\Vert \varphi \Vert }^{v^{+}}+{\Vert \varphi \Vert }^{v^{-}})\right]\hfill \\ & -& t^{1-m^{-}}{\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)\varphi {\left(\xi \right)}^{1-m\left(\xi \right)}}{1-m\left(\xi \right)}}d\xi ,\hfill \\ & \le & t^{1-m^{-}}\left(t^{min(v^{-},r^{-})-1+m^{-}}{A}_2-B_2\right),\hfill \end{array}$$

where

$$A_2={\displaystyle \frac{a_2{C}_A}{p^{-}}}{\left|\right|\varphi \left|\right|}^{p^{+}}+{\displaystyle \frac{a_2{C}_A}{p^{-}}}{\left|\right|\varphi \left|\right|}^{p^{-}}+{\displaystyle \frac{1}{r^{-}}}{\int }_{\Lambda }{\left|\varphi \right|}^{r\left(\xi \right)}d\xi +{c\left|\sigma \right|}_{S\left(\xi \right)}{\left(\right|\left|\varphi \right||}^{v^{+}}+{\left|\right|\varphi \left|\right|}^{v^{-}}),$$

and

$$B_2={\int }_{\Lambda }{\displaystyle \frac{\mu \left(\xi \right)\varphi {\left(\xi \right)}^{1-m\left(\xi \right)}}{1-m\left(\xi \right)}}d\xi .$$

Thus, we deduce that

$${\mathcal{I}}_{\lambda }\left(t\varphi \right)<0,for0<t<min\left(1,{\left({\displaystyle \frac{B_2}{A_2}}\right)}^{{\displaystyle \frac{1}{min(v^{-},r^{-})-1+m^{-}}}}\right).$$

□

Put ${\displaystyle l_1=\underset{\varphi \in E}{inf}{\mathcal{I}}_{\lambda }\left(\varphi \right)}$. Then, we have the following result.

Proposition 4. 

Under the same hypotheses as Theorem 3, if $\lambda <0$, then there exists ${\psi }_1\in E$ such that

$${\mathcal{I}}_{\lambda }\left({\psi }_1\right)=l_1<0.$$

Proof. 

The proof of Proposition 4 is very similar to that of Proposition 4, so we omit it here. Moreover, the rest of the proof of Theorem 3 is very similar to that of Theorem 2, so we also omit it here. □

Singular-type problems have attracted considerable interest, particularly in recent years, with increasing attention paid to methods incorporating fractional operators. Based on the results discussed above, several future research areas can be considered, including the following.

• The first possibility is to study a problem with a singular Kirchhoff-type non-linearity with a double phase.
• The second possibility is to extend the problem (1) to a similar problem by changing a Dirichlet boundary condition to a Neumann boundary condition or a Styklov boundary condition.