Positive Solutions for Perturbed Fractional p-Laplacian Problems

Abstract

In this article, we consider a class of quasilinear elliptic equations involving the fractional p-Laplacian, in which the nonlinear term satisfies subcritical or critical growth. Based on a fixed point result due to Carl and Heikkilä, we can well overcome the lack of compactness which has been a key difficulty for elliptic equations with critical growth. Moreover, we establish the existence and boundedness of the weak solutions for the above equations.

1. Introduction and Main Result

In recent years, fractional and nonlocal operators have been more and more widely used in modern applied science, such as optimization, plasma physics, finance, population dynamics, soft thin films, geophysical fluid dynamics, phases transitions, game theory, water waves, stratified materials, and so on; for more related applications, see [1,2,3] and references therein.

In this article, we concentrate on the qualitative analysis of positive solutions for the following perturbed fractional p-Laplacian problems:

$$\left\{\begin{array}{cc}{(-\mathrm{\Delta })}_p^{s}u=f(x,u)+\lambda g(x)\hfill & \mathrm{in}\mathrm{\Omega }\hfill \\ u=0\hfill & \mathrm{in}{\mathbb{R}}^N\backslash \mathrm{\Omega }\hfill \end{array}\right.$$

(1)

where $s\in (0,1)$, $p\ge 2$, $N>sp$, $\lambda >0$ is a real parameter, the perturbation term $g(x)$ is nonnegative and $g(x)\not\equiv 0$, and $\mathrm{\Omega }$ is a bounded domain of ${\mathbb{R}}^N$ with Lipschitz boundary. Here, ${(-\mathrm{\Delta })}_p^s$ is the fractional p-Laplace operator defined by

$${(-\mathrm{\Delta })}_p^{s}u(x)=2\underset{ϵ\to 0}{lim}{\int }_{{\mathbb{R}}^N\backslash B_{ϵ}(x)}\frac{{|u(x)-u(y)|}^{p-2}(u(x)-u(y))}{{|x-y|}^{N+sp}}dy,x\in {\mathbb{R}}^N,$$

(2)

where $B_{ϵ}(x)$ denotes the ball of ${\mathbb{R}}^N$ centered at $x\in {\mathbb{R}}^N$, with radius $ϵ>0$. We refer the reader to [4,5] and the references therein for further introduction to the fractional Sobolev spaces theory and the study of fractional p-Laplacian problems based on variational methods.

As in the case of p-Laplacian equations, the existence of solutions for fractional p-Laplacian equations has drawn a lot of attention. More precisely, for the following quasi-linear problems

$$\left\{\begin{array}{cc}{(-\mathrm{\Delta })}_p^{s}u=f(x,u)\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \mathrm{in}{\mathbb{R}}^N\backslash \mathrm{\Omega },\hfill \end{array}\right.$$

(3)

Wu et al. in [6] obtained a new regularity result for problem (3) by establishing a modified Marcinkiewicz interpolation result, and the authors also employed the Stampacchia truncation method to obtain a new regularity result on ${\mathbb{R}}^N$ for problem (3). In addition, by using the Leray-Schauder fixed point theorem, the authors in [6] obtained the existence of solutions to a class of fractional Laplacian problems with weak growth conditions. For the regularity results, we can also refer to [7], although the work in [7] mainly focuses on the case where p is large, and the regularity of the solution is directly inherited from the function embedding itself. Bjorland et al. in [8] obtained some higher regularity with s close to $1^{-}$ by showing that the solutions converged to the solutions with the p-Laplace operator $div(|\nabla u{|}^{p-2}\nabla u)$ whenever $s\to 1^{-}$.

As $p=2,s=1$, Ambrosetti and Rabinowitz made new assumptions about nonlinearity for the first time when they studied the elliptic equation $-\mathrm{\Delta }u=f(x,u)\mathrm{in}\mathrm{\Omega }$, $u=0\mathrm{on}\partial \mathrm{\Omega }$ in [9], where f satisfies

$({\mathcal{A}}_1)$

$f(x,0)=0,{lim}_{u\to 0}\frac{f(x,u)}{u}=0$;

$({\mathcal{A}}_2)$

$|f(x,u)|\le {\vartheta }_1+{\vartheta }_2{|u|}^q,1<q<p^{*}-1:=\frac{N+2}{N-2},{\vartheta }_1,{\vartheta }_2>0$;

(AR)

There exists $\rho >0$ and $\xi >2$ such that $0<\xi F(x,u)\le uf(x,u)$, for $|u|>\rho$, where $F(x,u)={\int }_0^{u}f(x,t)dt$.

Here, the last condition is the well-known (AR) condition, which contributes significantly to the compactness condition of the mountain pass theorem. However, more and more researchers have become aware that the (AR) condition has a certain constraint so that it is difficult to include some important nonlinearities. Therefore, this condition has been weakened or even abandoned in many papers. A weaker condition than the (AR) condition was proposed by Miyagaki and Souto in [10], that is, ${lim}_{u\to \infty }\frac{F(x,u)}{u^2}=+\infty$. Furthermore, the authors used the mountain pass theorem to obtain the existence of nontrivial weak solutions for the problem $-\mathrm{\Delta }u=\lambda f(x,u)\mathrm{in}\mathrm{\Omega }$, $u=0\mathrm{on}\partial \mathrm{\Omega }$ for all $\lambda >0$, in which f also satisfies $({\mathcal{A}}_1)$–$({\mathcal{A}}_2)$ and a monotonicity assumption instead of the (AR) condition. It is worth mentioning that the monotonicity assumption in [10] reads as follows:

$({\mathcal{B}}_1)$

There is ${\eta }_1>0$ such that $\frac{f(x,t)}{t}\mathrm{is}\mathrm{increasing}\mathrm{in}t⩾{\eta }_1\mathrm{and}\mathrm{decreasing}\mathrm{in}t⩽-{\eta }_1,\mathrm{for}\mathrm{all}x\in \mathrm{\Omega }.$

In addition, Willem and Zou in [11] proposed a monotonicity assumption to replace the (AR) condition. More specifically, they considered the following condition:

$({\mathcal{B}}_2)$

$G(x,t)\equiv tf(x,t)-2F(x,t)\mathrm{is}\mathrm{increasing}\mathrm{in}t,\forall x\in \mathrm{\Omega },t\in \mathbb{R}.$

Another monotonicity condition as an effective substitute of (AR) was proposed by Jeanjean in [12]

$({\mathcal{B}}_3)$

There is ${\eta }_2\ge 1$ such that $G(x,\tau t)\le {\eta }_{2}G(x,t),\mathrm{for}\mathrm{all}(x,t)\in \mathrm{\Omega }\times \mathbb{R},\tau \in [0,1].$

Indeed, $({\mathcal{B}}_3)$ implies $({\mathcal{B}}_1)$ if ${\eta }_1=0$; we refer to [13] [Proposition 2.3] for a proof.

Below we present some nonlinear terms that satisfy subcritical or critical growth, which are also instructive for this paper. To our knowledge, an assumption imposed on the nonlinearity in most of the papers is $({\mathcal{A}}_2)$. Along this direction, the following growth condition imposed on the nonlinearity in solving p-Laplacian equations was often used in the literature:

$({\mathcal{C}}_1)$

There are $p<\tau <p^{*}$ and ${\eta }_3>0$ such that $|f(x,t)|⩽{\eta }_3{(1+|t|}^{\tau -1})$, for every $(x,t)\in \mathrm{\Omega }\times \mathbb{R}$.

Furthermore, a more general growth condition was proposed to study the existence of solutions in [14]:

$({\mathcal{C}}_2)$

$f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a continuous function such that

$|f(x,\tau )|\le {\eta (x)|\tau |}^{l-1}+\mu (x){|\tau |}^{m-1}$, for all $(x,\tau )\in {\mathbb{R}}^N\times \mathbb{R}$, where $1\le l\le p\le m<p^{*}=Np/(N-p)$.

Motivated by the above assumptions, we impose the following hypotheses on the nonlinearity f:

$({\mathcal{F}}_1)$

$f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function and fulfills

$$|f(x,t)|\le {a(x)|t|}^{\sigma -1}+b(x){|t|}^{\gamma -1},(x,t)\in \mathrm{\Omega }\times \mathbb{R},$$

(4)

where $p<\sigma <\gamma \le p_s^{*}=Np/(N-sp)$, $0\le a\in L^q(\mathrm{\Omega })\cap L^{\infty }(\mathrm{\Omega })$, $0\le b\in L^{\varrho }(\mathrm{\Omega })\cap L^{\infty }(\mathrm{\Omega })$, $q=Np/[Np-\sigma (N-sp)]$, $\varrho =Np/[Np-\gamma (N-sp)]$.

$({\mathcal{F}}_2)$

For each $x\in \mathrm{\Omega }$, $f(x,t)$ is nondecreasing in t, $f(x,t)\ge 0$ as $t\ge 0$, and $f(x,t)=0$ as $t<0$.

Obviously, $({\mathcal{F}}_2)$ is weaker than $({\mathcal{B}}_1)$ or $({\mathcal{B}}_2)$. However, a simple example $f(x,t)=t^{(p-1)/2}ln(1+t)$ for $t\ge 0$; $f(x,t)=0$ for $x<0$ satisfying $({\mathcal{F}}_2)$ demonstrates that it does not satisfy the (AR) condition or other monotonicity assumptions mentioned in this paper.

It is easy to see that there are a lot of functions satisfying our assumption $({\mathcal{F}}_2)$ that do not satisfy the (AR) condition. Moreover, our condition $({\mathcal{F}}_1)$ for subcritical and critical growth is relatively more general than the usual subcritical one $({\mathcal{C}}_1)$ or $({\mathcal{C}}_2)$. Importantly, our monotonicity condition $({\mathcal{F}}_2)$ is weaker than condition $({\mathcal{B}}_1)$ or $({\mathcal{B}}_2)$, which is one of the advantages that the approach in [15] brings us. For our approach in this paper, we point out that the author in [15] used a non-variational fixed point theorem to find the nontrivial solution of a class of nonhomogeneous fractional p-Laplacian equations. As a result, the approach in [15] has aroused our interest in solving some quasi-linear elliptic equations; we refer the interested reader to [16,17] for more details on the study of nonhomogeneous fractional singular p-Laplacian equations and systems involving critical nonlinearities. Here, we would like to point out a key point that owing to the particularity of this approach, we do not need to appeal to the compactness of the Sobolev embedding theorem, so that we can avoid this difficulty when dealing with the critical case. Guided by this idea, we turn our attention to a class of perturbed fractional p-Laplacian equations. As a matter of fact, we can find a nonnegative weak solution of (1) by using the approach above. Due to the existence of the nonhomogeneous term, we can further obtain the positive solution of problem (1). In practice, we could ignore the nonhomogeneous term if the zero solution can be easily ruled out. Finally, inspired by Theorem 3.1 in [18], we can prove that the weak solution to problem $(1)$ found by the fixed point theorem in [15] is $L^{\infty }$-bounded. To the best of our knowledge, subcritical and critical assumptions for related Laplacian problems in the available literature are often treated separately by various variational methods. In this sense, our result is new, even in the Laplacian setting.

Now, let us introduce the space

$$\mathcal{W}=\{u\mid u:{\mathbb{R}}^N\to \mathbb{R}\mathrm{is}\mathrm{measurable},u\in L^p({\mathbb{R}}^N)\mathrm{and}{\left[u\right]}_{s,p}<\infty \},$$

endowed with the norm ${\parallel u\parallel }_{\mathcal{W}}={(\parallel u\parallel }_{L^p({\mathbb{R}}^N)}^p+{\left[u\right]}_{s,p}^p{)}^{\frac{1}{p}}$, where $L^p({\mathbb{R}}^N)$ is represented as the usual Lebesgue function space, endowed with the norm ${\parallel ·\parallel }_{L^p({\mathbb{R}}^N)}$, and the Gagliardo seminorm ${[·]}_{s,p}$ is defined by

$${\left[u\right]}_{s,p}={({\int \int }_{{\mathbb{R}}^{2N}}\frac{{|u(x)-u(y)|}^p}{{|x-y|}^{N+sp}}dxdy)}^{\frac{1}{p}}.$$

We consider the subspace of $\mathcal{W}$: $\mathcal{P}(\mathrm{\Omega })=\{u\in \mathcal{W}:u=0\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^N\backslash \mathrm{\Omega }\}$, which can be equivalently endowed with the norm $|·|={[·]}_{s,p}$ (see [4] [Theorem 7.1]). Therefore, the space $\mathcal{P}(\mathrm{\Omega })$ is a reflexive Banach space (see [19] [Lemma 2.4]). Next, we set

$$S_{*}:=\underset{\begin{array}{c}0\ne u\in P_{+}(\mathrm{\Omega })\end{array}}{inf}\frac{{\parallel u\parallel }^p}{{\parallel u\parallel }_{L^{p_s^{*}}(\mathrm{\Omega })}^p}.$$

Then, from the continuous embedding $\mathcal{P}(\mathrm{\Omega })\hookrightarrow L^{\alpha }(\mathrm{\Omega })$, for all $1\le \alpha \le p_s^{*}$ (see [19] [Lemma 2.3]), it follows that $S_{*}>0$.

Definition 1.

We say that $u\in \mathcal{P}(\mathrm{\Omega })$ is a (weak) solution of problem (1), if

$${\int \int }_{\mathcal{Q}}\frac{{|u(x)-u(y)|}^{p-2}(u(x)-u(y))(\nu (x)-\nu (y))}{{|x-y|}^{N+sp}}dxdy-{\int }_{\mathrm{\Omega }}\left(f(x,u)+\lambda g(x)\right)\nu (x)dx=0,$$

(5)

for all $\nu \in \mathcal{P}(\mathrm{\Omega })$, where $\mathcal{Q}:={\mathbb{R}}^{2N}\backslash (C\mathrm{\Omega }\times C\mathrm{\Omega })$ and $C\mathrm{\Omega }:={\mathbb{R}}^N\backslash \mathrm{\Omega }$.

The main result in our article is stated as follows:

Theorem 1.

Suppose that f satisfies $({\mathcal{F}}_1)$–$({\mathcal{F}}_2)$, $g\in L^{{\gamma }^{\prime }}(\mathrm{\Omega })$(dual space of $L^{\gamma }(\mathrm{\Omega })$), and $g\not\equiv 0$. Then, there exists ${\lambda }_0>0$ such that for all $\lambda \in (0,{\lambda }_0]$, problem (1) possesses a positive weak solution ${\omega }^{*}$. Furthermore, if $1+\sigma /p>\gamma /p+\gamma /p_s^{*}$ and $\gamma <p_s^{*}$, then the obtained solution is ${\omega }^{*}\in L^{\infty }(\mathrm{\Omega })$.

Remark 1.

If we consider the critical case, i.e., $\gamma =p_s^{*}$, we also obtain the existence of at least a positive solution, but we can not guarantee that the positive solution is $L^{\infty }$-bounded.

Remark 2.

We can only guarantee the existence of a weak solution to problem (1) for small perturbation, i.e., $\lambda \in (0,{\lambda }_0]$. The case of high perturbation ($\lambda >{\lambda }_0$) seems completely open.

Our article is organized as follows. In Section 2, we recall some necessary knowledge along with a description of the main method of this paper. Section 3 is devoted to demonstrating the main result of this article.

Unless otherwise specified, we indicate that ${\mathcal{M}}_k$, $\mathcal{M}$, ${\mathcal{M}}_p$, ${\mathcal{M}}_{p1}$, ${\mathcal{M}}_{p2}$... are positive constants.

2. Preliminaries

In this section, we recall some notations and concepts necessary to introduce a crucial fixed point result. Suppose that $\mathcal{S}$ is a real Banach space, and ${\mathcal{S}}_{+}\ne 0$ is a nonempty subset of $\mathcal{S}$. ${\mathcal{S}}_{+}$ is said to be an order cone if the space ${\mathcal{S}}_{+}$ fulfills the following assumptions:

$(i)$

${\mathcal{S}}_{+}$ is convex and closed.

$(ii)$

If $\xi \ge 0$ and $\rho \in {\mathcal{S}}_{+}$, then $\xi \rho \in {\mathcal{S}}_{+}$.

$(iii)$

If $-\rho \in {\mathcal{S}}_{+}$ and $\rho \in {\mathcal{S}}_{+}$, then $\rho =0$.

If ${\mathcal{S}}_{+}$ is an order cone, then

• $(\mathcal{S},⪯)$ is an ordered Banach space if ${\rho }_t⪯{\rho }_s\leftrightarrow {\rho }_s-{\rho }_t\in {\mathcal{S}}_{+}$.
• $(\mathcal{S},\parallel ·\parallel )$ is a lattice if $sup\{{\rho }_s,{\rho }_t\}$ and $inf\{{\rho }_s,{\rho }_t\}$ exist for all ${\rho }_s$, ${\rho }_t\in \mathcal{S}$ according to ⪯.
• $(\mathcal{S},\parallel ·\parallel )$ is a Banach semilattice if $\parallel {\rho }_s^{\pm }\parallel \le \parallel {\rho }_s\parallel \mathrm{for}\mathrm{all}{\rho }_s\in \mathcal{S}$, where ${\rho }_s^{+}:=sup\{0,{\rho }_s\}$ and ${\rho }_p^{-}:=-inf\{0,{\rho }_s\}$.

Lemma 1.

(See [20]) [Corollary 2.2] Let ${\mathcal{P}}_{+}$ be a reflexive Banach semilattice. Then, any closed ball of ${\mathcal{P}}_{+}$ has a fixed point.

Remark 3.

Assume that $(\mathcal{S},⪯)$ and $({\mathcal{S}}^{\prime },◃)$ are ordered Banach spaces. The operator $\mathcal{I}:\mathcal{S}\to {\mathcal{S}}^{\prime }$ is increasing if and only if there is $\mathcal{I}{u}_p◃\mathcal{I}{u}_t$ for all $u_p$, $u_t\in \mathcal{S}$, $u_p⪯u_t$. Suppose that ${\mathcal{S}}_1$ is a subset of $\mathcal{S}$. We say ${\mathcal{S}}_1$ has a fixed point if any increasing operator $\mathcal{I}:{\mathcal{S}}_1\to {\mathcal{S}}_1$ has a fixed point.

3. Proof of Theorems

Next, in order to apply Lemma 1 to prove the existence of the weak solution, we define the following two operators: $\mathcal{J}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ (the dual space of $\mathcal{P}(\mathrm{\Omega })$) given by

$$⟨\mathcal{J}u,\nu ⟩={\int \int }_{\mathcal{Q}}\frac{{|u(x)-u(y)|}^{p-2}(u(x)-u(y))(\nu (x)-\nu (y))}{{|x-y|}^{N+sp}}dxdy,\forall u,\nu \in \mathcal{P}(\mathrm{\Omega }),$$

(6)

and $\mathcal{K}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ given by

$$⟨\mathcal{K}u,\nu ⟩={\int }_{\mathrm{\Omega }}\left(f(x,u)+\lambda g(x)\right)\nu (x)dx,\forall u,\nu \in \mathcal{P}(\mathrm{\Omega }),$$

(7)

Evidently, the operators $\mathcal{J}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ and $\mathcal{K}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ are well defined.

Lemma 2.

The operator $\mathcal{J}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ is continuous and invertible.

Proof. 

Let $u_n\in \mathcal{P}(\mathrm{\Omega })$ such that $u_n\to u$ in $\mathcal{P}(\mathrm{\Omega })$. For $\nu \in \mathcal{P}(\mathrm{\Omega })$ with $\parallel \nu \parallel \le 1$, using Hölder’s inequality, we can deduce

$|⟨\mathcal{J}u-\mathcal{J}{u}_n,\nu ⟩|$

$$\begin{array}{cc}\hfill & \le {({\int \int }_{\mathcal{Q}}\frac{{||u(x)-u(y)|}^{p-2}(u(x)-u(y))-|u_n(x)-u_n(y){{|}^{p-2}(u_n(x)-u_n(y))|}^{\frac{p}{p-1}}}{{|x-y|}^{N+sp}}dxdy)}^{\frac{p-1}{p}}\hfill \\ \hfill & :=\mathcal{T}\hfill \end{array}$$

(8)

Next, we will employ the following basic inequality to estimate $\mathcal{T}$:

$$||t_1{|}^{k-2}{t}_1-|t_2{|}^{k-2}{t}_2|\le {\mathcal{M}}_k|t_1-t_2|(|t_1|+|t_2{|)}^{k-2},\forall k\ge 2,t_1,t_2\in \mathbb{R}.$$

(9)

By invoking the Hölder inequality and (9), there exists ${\mathcal{M}}_p>0$ such that

$$\begin{array}{c}\hfill {\mathcal{T}}^{\frac{p}{p-1}}\le {\mathcal{M}}_p\parallel u-u_n{\parallel }^{\frac{p}{p-1}}(\parallel u\parallel +\parallel u_n{\parallel )}^{\frac{p(p-2)}{p-1}}.\end{array}$$

(10)

Since $u_n\to u$ in $\mathcal{P}(\mathrm{\Omega })$, (10) yields that

$$\begin{array}{c}\hfill \parallel (\mathcal{J}u-\mathcal{J}{u}_n,\nu ){\parallel }_{{\mathcal{P}}_{+}^{\prime }(\mathrm{\Omega })}=\underset{\nu \in {\mathcal{P}}_{+}(\mathrm{\Omega }),\parallel \nu \parallel \le 1}{sup}|⟨\mathcal{J}u-\mathcal{J}{u}_n,\nu ⟩|\le \mathcal{T}\to 0,\end{array}$$

(11)

as $n\to \infty$. Then, the operator $\mathcal{J}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ is continuous. Notice that $⟨\mathcal{J}u,u⟩={\parallel u\parallel }^p$, $\forall u\in {\mathcal{P}}_{+}(\mathrm{\Omega })$. It follows that ${lim}_{\parallel u\parallel \to \infty }\frac{⟨\mathcal{J}u,u⟩}{\parallel u\parallel }=\infty$. Furthermore, we also need the following inequality:

$$(|t_1{|}^{\mu -2}{t}_1-|t_2{|}^{\mu -2}{t}_2|)(t_1-t_2)\ge {\mathcal{M}}_{\mu }{|t_1-t_2|}^{\mu },\forall \mu \ge 2,t_1,t_2\in \mathbb{R},$$

(12)

from which we can obtain

$$⟨\mathcal{J}{u}_n-\mathcal{J}{u}_k,u_n-u_k⟩>0,\forall u_n,u_k\in \mathcal{P}(\mathrm{\Omega }),u_n\ne u_k.$$

(13)

Therefore, by means of the Minty–Browder Theorem (cf. [21] [Theorem 5.16]), we obtain that the operator $\mathcal{J}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ is reversible. Hence, the proof is complete. □

The next step is to explore the monotonicity of operators. For this purpose, we consider the partial order in $\mathcal{P}(\mathrm{\Omega })$ as follows:

$$u_p,u_q\in \mathcal{P}(\mathrm{\Omega }),u_q⪯u_q\leftrightarrow u_p\le u_q\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega }.$$

(14)

In addition, for any $u_p$, $u_q\in \mathcal{P}(\mathrm{\Omega })$, $sup\{u_p,u_q\}$ and $inf\{u_p,u_q\}$ exist for the partial order ⪯. It is easy to see that $|{\rho }^{\pm }(x)|\le |\rho (x)|$ and $|{\rho }^{\pm }(x)-{\rho }^{\pm }(y)|\le |\rho (x)-\rho (y)|$ are almost everywhere in $\mathrm{\Omega }$, so we have $\parallel {\rho }^{\pm }\parallel \le \parallel \rho \parallel$. To summarize, $(\mathcal{P}(\mathrm{\Omega }),⪯)$ is a reflexive Banach semilattice. Meanwhile, the following partial order is embedded in the dual space ${\mathcal{P}}^{\prime }(\mathrm{\Omega })$ of $\mathcal{P}(\mathrm{\Omega })$,

$${\phi }_p,{\phi }_q\in {\mathcal{P}}^{\prime }(\mathrm{\Omega }),{\phi }_p◃{\phi }_q\leftrightarrow ⟨{\phi }_p,\nu ⟩\le ⟨{\phi }_q,\nu ⟩,\forall \nu \in {\mathcal{P}}_{+}(\mathrm{\Omega }).$$

(15)

where ${\mathcal{P}}_{+}(\mathrm{\Omega }):=\{\nu \in \mathcal{P}(\mathrm{\Omega }):\nu \ge 0\mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega }\}$.

Additionally, we consider the operator ${\mathcal{J}}^{-1}:({\mathcal{P}}^{\prime }(\mathrm{\Omega }),◃)\to (\mathcal{P}(\mathrm{\Omega }),⪯)$ to be monotonically increasing. In fact, we choose ${\nu }_n$, ${\nu }_k\in {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ such that ${\nu }_n$◃${\nu }_k$ and set $u_n={\mathcal{J}}^{-1}{\nu }_n$, $u_k={\mathcal{J}}^{-1}{\nu }_k$; then, for $\forall \phi \in \mathcal{P}(\mathrm{\Omega })$, we have

$$\begin{array}{c}\hfill 0\le ⟨{\nu }_k-{\nu }_n,\phi ⟩=⟨\mathcal{J}{u}_k-\mathcal{J}{u}_n,\phi ⟩\end{array}$$

Taking $\phi ={(u_k-u_n)}^{-}\in \mathcal{P}(\mathrm{\Omega })$, we conclude that

$$0\le ⟨{\nu }_k-{\nu }_n,{(u_k-u_n)}^{-}⟩\le -{\mathcal{M}}_p{\parallel {(u_k-u_n)}^{-}\parallel }^p\le 0,$$

(16)

from which it follows that $u_n\le u_k$ almost everywhere in $\mathrm{\Omega }$, then ${\mathcal{J}}^{-1}{\nu }_n⪯{\mathcal{J}}^{-1}{\nu }_k$; hence, the operator ${\mathcal{J}}^{-1}:({\mathcal{P}}^{\prime }(\mathrm{\Omega }),◃)\to (\mathcal{P}(\mathrm{\Omega }),⪯)$ is increasing. For more computational details, we refer to [16] [Lemma 3.2].

Now, we turn our attention to the operator $\mathcal{K}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ and make a necessary estimate for it. According to the definition of $\mathcal{K}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$, we have

$$⟨\mathcal{K}u,\nu ⟩={\int }_{\mathrm{\Omega }}\left(f(x,u)+\lambda g(x)\right)\nu (x)dx,\forall u,\nu \in \mathcal{P}(\mathrm{\Omega }).$$

(17)

By the Hölder inequality, the Sobolev embedding theorem (cf. [22] [Lemma 2.1]) and $({\mathcal{F}}_1)$, there exists ${\mathcal{M}}_{\gamma }$ such that

$$\begin{array}{cc}\hfill |⟨\mathcal{K}u,\nu ⟩|& \le {\int }_{\mathrm{\Omega }}{(a(x)|u|}^{\sigma -1}+{b(x)|u|}^{\gamma -1}+\lambda g(x))\nu (x)dx\hfill \\ \hfill & \le ({\mathcal{S}}_{*}^{-\sigma /p}\parallel a{\parallel }_{L^q(\mathrm{\Omega })}\parallel u{\parallel }^{\sigma -1}+{\mathcal{S}}_{*}^{-\gamma /p}\parallel b{\parallel }_{L^{\varrho }(\mathrm{\Omega })}\parallel u{\parallel }^{\gamma -1}+\lambda {\mathcal{M}}_{\gamma }\parallel g{\parallel }_{L^{{\gamma }^{\prime }}(\mathrm{\Omega })})\parallel \nu \parallel ,\hfill \end{array}$$

(18)

where ${\mathcal{M}}_{\gamma }$ is the best Sobolev constant of the continuous embedding $\mathcal{P}(\mathrm{\Omega })\hookrightarrow L^{\gamma }(\mathrm{\Omega })$.

When $\gamma =p_s^{*}$, a similar estimate can be calculated; hence, we obtain

$$\begin{array}{cc}\hfill & \left|⟨\mathcal{K}u,\nu ⟩\right|\hfill \\ & \le \left({\mathcal{S}}_{*}^{-\sigma /p}{\parallel a\parallel }_{L^q(\mathrm{\Omega })}{\parallel u\parallel }^{\sigma -1}+{\mathcal{S}}_{*}^{-p_s^{*}/p}{\parallel b\parallel }_{L^{\infty }(\mathrm{\Omega })}{\parallel u\parallel }^{p_s^{*}-1}+\lambda {\mathcal{M}}_{\gamma }{\parallel g\parallel }_{L^{{\gamma }^{\prime }}(\mathrm{\Omega })}\right)\parallel \nu \parallel .\hfill \end{array}$$

(19)

We define a new operator $\mathcal{Z}:={\mathcal{J}}^{-1}\circ \mathcal{K}$. Combining (18) and the definitions of $\mathcal{J}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$ and $\mathcal{K}:\mathcal{P}(\mathrm{\Omega })\to {\mathcal{P}}^{\prime }(\mathrm{\Omega })$, we give the following key result.

Lemma 3.

Under the assumptions of Theorem 1, there exists $\mathcal{M}$ such that for all $0<\lambda \le {\lambda }_0$,

$$\mathcal{Z}(\mathcal{Q}[0,\mathcal{M}])\subset \mathcal{Q}[0,\mathcal{M}],$$

(20)

where $\mathcal{Q}[0,\mathcal{M}]=\{u\in \mathcal{P}(\mathrm{\Omega }):\parallel u\parallel \le \mathcal{M}\}$, and the norm $\parallel ·\parallel$ in $\mathcal{Q}[0,\mathcal{M}]$ is consistent with that in $\mathcal{P}(\mathrm{\Omega })$.

Proof. 

Let u, $\nu \in \mathcal{P}(\mathrm{\Omega })$ with $\mathcal{Z}u=\nu$. Since $\mathcal{Z}:={\mathcal{J}}^{-1}\circ \mathcal{K}$ and $⟨\mathcal{J}\nu ,\nu ⟩={\parallel \nu \parallel }^p$, we have

$${\parallel \mathcal{Z}u\parallel }^p=⟨\mathcal{J}(\mathcal{Z}u),\mathcal{Z}u⟩=⟨\mathcal{K}u,\mathcal{Z}u⟩\le {\parallel \mathcal{K}u\parallel }_{{\mathcal{P}}^{\prime }(\mathrm{\Omega })}\parallel \mathcal{Z}u\parallel .$$

(21)

Combining (18) and (21), we deduce

$$\begin{array}{cc}\hfill {\parallel \mathcal{Z}u\parallel }^{p-1}& \le {\parallel \mathcal{K}u\parallel }_{{\mathcal{P}}^{\prime }(\mathrm{\Omega })}\hfill \\ \hfill & \le {\mathcal{S}}_{*}^{-\sigma /p}{\parallel a\parallel }_{L^q(\mathrm{\Omega })}{\parallel u\parallel }^{\sigma -1}+{\mathcal{S}}_{*}^{-\gamma /p}{\parallel b\parallel }_{L^{\varrho }(\mathrm{\Omega })}{\parallel u\parallel }^{\gamma -1}+\lambda {\mathcal{M}}_{\gamma }{\parallel g\parallel }_{L^{{\gamma }^{\prime }}(\mathrm{\Omega })}.\hfill \end{array}$$

(22)

We consider that $\parallel u\parallel \le \mathcal{M}$, which implies

$$\frac{{\parallel \mathcal{Z}u\parallel }^{p-1}}{{\mathcal{M}}^{p-1}}\le {\mathcal{S}}_{*}^{-\sigma /p}{\parallel a\parallel }_{L^q(\mathrm{\Omega })}{\mathcal{M}}^{\sigma -p}+{\mathcal{S}}_{*}^{-\gamma /p}{\parallel b\parallel }_{L^{\varrho }(\mathrm{\Omega })}{\mathcal{M}}^{\gamma -p}+\frac{\lambda {\mathcal{M}}_{\gamma }{\parallel g\parallel }_{L^{{\gamma }^{\prime }}(\mathrm{\Omega })}}{{\mathcal{M}}^{p-1}}.$$

(23)

Let $\mathcal{M}$ be sufficiently small such that

$${\mathcal{S}}_{*}^{-\sigma /p}{\parallel a\parallel }_{L^q(\mathrm{\Omega })}{\mathcal{M}}^{\sigma -p}+{\mathcal{S}}_{*}^{-\gamma /p}{\parallel b\parallel }_{L^{\varrho }(\mathrm{\Omega })}{\mathcal{M}}^{\gamma -p}\le \frac{1}{2}.$$

(24)

Letting

$${\lambda }_0:=\frac{{\mathcal{M}}^{p-1}}{2{\mathcal{M}}_{\gamma }{\parallel g\parallel }_{L^{{\gamma }^{\prime }}(\mathrm{\Omega })}},$$

(25)

then for all $0<\lambda \le {\lambda }_0$, it is easy to derive that

$$\frac{{\parallel \mathcal{Z}u\parallel }^{p-1}}{{\mathcal{M}}^{p-1}}\le 1.$$

(26)

Note that the result of the Lemma 3 also holds in the case of $\gamma =p_s^{*}$. Hence, the proof is complete. □

Finally, we deal with the monotonicity of operator $\mathcal{Z}:(\mathcal{P}(\mathrm{\Omega }),⪯)\to (\mathcal{P}(\mathrm{\Omega }),⪯)$ and the existence of solutions to problem (1). We say that operator $\mathcal{Z}:(\mathcal{P}(\mathrm{\Omega }),⪯)\to (\mathcal{P}(\mathrm{\Omega }),⪯)$ is increasing if both operators ${\mathcal{J}}^{-1}:({\mathcal{P}}^{\prime }(\mathrm{\Omega }),◃)\to (\mathcal{P}(\mathrm{\Omega }),⪯)$ and $\mathcal{K}:(\mathcal{P}(\mathrm{\Omega }),⪯)\to ({\mathcal{P}}^{\prime }(\mathrm{\Omega }),◃)$ are increasing. Whereas we have proved in Section 3 that operator ${\mathcal{J}}^{-1}:({\mathcal{P}}^{\prime }(\mathrm{\Omega }),◃)\to (\mathcal{P}(\mathrm{\Omega }),⪯)$ is increasing, we now focus on the monotonicity of operator $\mathcal{K}:(\mathcal{P}(\mathrm{\Omega }),⪯)\to ({\mathcal{P}}^{\prime }(\mathrm{\Omega }),◃)$. By condition $({\mathcal{F}}_2)$ and the definition of operator $\mathcal{K}$, we can see that $\mathcal{K}:(\mathcal{P}(\mathrm{\Omega }),⪯)\to ({\mathcal{P}}^{\prime }(\mathrm{\Omega }),◃)$ is increasing. Now, by Lemma 1, $\mathcal{Q}[0,\mathcal{M}]$ has the fixed point property, which means that there exists ${\omega }^{*}\in \mathcal{Q}[0,\mathcal{M}]$ such that $\mathcal{Z}{\omega }^{*}={\omega }^{*}$. Then, we have

$$<\mathcal{J}{\omega }^{*},\nu >=<\mathcal{K}{\omega }^{*},\nu >,\forall \nu \in \mathcal{P}(\mathrm{\Omega }).$$

(27)

Hence, ${\omega }^{*}$ is considered to be a weak solution of problem (1). Furthermore, $g\not\equiv 0$ yields that ${\omega }^{*}$ is a nontrivial weak solution of problem (1). Since the perturbation term $g(x)$ and the nonlinearity $f(x,u)$ are both nonnegative under our assumptions, the weak solution ${\omega }^{*}$ is nonnegative by the comparison principle (cf. [23] [Proposition 2.10]), and then the strong maximum principle (cf. [24] [Lemma 2.3]) implies that ${\omega }^{*}$ is positive. Eventually, similar to the proof of Theorem 3.1 in [18], we are able to obtain that the positive solution ${\omega }^{*}$ is $L^{\infty }$-bounded in the case of $\gamma <p_s^{*}$.

4. Conclusions

In this paper, we solved a class of perturbed fractional p-Laplacian equations satisfying subcritical or critical growth by means of a non-variational fixed point theorem. This approach can well overcome the lack of compactness in the critical case. As a result, we obtained a positive weak solution to problem $(1)$, which can also be applied to solve a much larger class of elliptic problems, which was based on a comparison with the classical elliptic problems explored by variational methods. However, this fixed-point theorem also has its limitations, such as requiring the nonlinearity to be nondecreasing so that it can not effectively exclude the zero solution, so we need to add a perturbation term to exclude this situation.