A variational inequality involving nonlocal elliptic operators

Abstract

In this paper, we study the existence of solutions to a variational inequality involving nonlocal elliptic operators, and the problem studied here is not variational in nature. The proof of the main result is based on Schauder’s fixed point theorem combined with adequate variational arguments and a penalization technique.

1 Introduction

In the last years, great attention has been devoted to the study of fractional and nonlocal problems. However, the interest in nonlocal elliptic problems goes beyond the mathematical curiosity. Indeed, this type of problems arises in a quite natural way in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, as they are the typical outcome of stochastic stabilization of Lévy processes, see [1–4] and the references therein. The literature on nonlocal operators and their applications is very interesting and quite large, we refer the interested readers to [5–13].

In [14] the existence of two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators is obtained by using critical point theory for non-differentiable functionals. In [15] the Lewy-Stampacchia type estimates for variational inequalities driven by nonlocal operators are discussed. In [16] the obstacle problem is considered for a linear elliptic operator perturbed by a nonlinearity having at most a quadratic growth on the gradient, and the existence of weak solutions of the variational inequality is obtained by using a penalization method, Schauder’s fixed point theorem and a priori estimates. In [17], the existence of nontrivial solutions for a semilinear elliptic variational inequalities with gradient-dependent nonlinearity is discussed by using variational methods combined with a penalization technique and an iterative scheme.

Motivated by the above mentioned works, we study a variational inequality involving nonlocal elliptic operators. To introduce our problem precisely, we first give some notation. Let Ω be a bounded domain of $\mathbb{R}^{N}$ with smooth boundary ∂Ω. Denote $\mathcal{Q}=\mathbb{R}^{2N}\backslash O$, where $O=\mathcal {C}(\Omega)\times\mathcal{C}(\Omega)\subset{\mathbb{R}}^{2N}$ and $\mathcal{C}(\Omega)=\mathbb{R}^{N}\backslash\Omega$. W is a linear space of Lebesgue measurable functions from $\mathbb{R}^{N}$ to $\mathbb {R}$ such that the restriction to Ω of any function u in W belongs to $L^{p}(\Omega)$ and $\int_{\mathcal{Q}}|u(x)-u(y)|^{p}|x-y|^{-N-ps}\,dx\,dy<\infty$. The space W is equipped with the norm

$$\begin{aligned} \|u\|_{W}=\|u\|_{L^{p}(\Omega)}+ \biggl(\int_{\mathcal {Q}}\bigl|u(x)-u(y)\bigr|^{p}|x-y|^{-N-ps}\,dx\,dy \biggr)^{1/p}. \end{aligned}$$

Then $(W,\|\cdot\|_{W})$ is a uniformly convex Banach space, see [18]. We shall work on the closed linear subspace

$$\begin{aligned} W_{0}= \bigl\{ u\in W :u(x)=0 \mbox{ a.e. in } \mathbb{R}^{N}\backslash \Omega \bigr\} . \end{aligned}$$

A norm of $W_{0}$ is given by $\|u\|= ( \iint_{\mathcal {Q}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy )^{1/p}$ for all $u\in W_{0}$. Set $L_{0}^{p}(\Omega)=\{u\in L^{p}(\Omega): u=0 \mbox{ a.e. on } \mathbb{R}^{N}\setminus \Omega \}$. Obviously, $(L^{p}_{0}(\Omega),\|\cdot\|_{L^{p}(\Omega)})$ is a Banach space.

In the present paper, we adapt the methods in [16] and [17] in order to prove the existence of solutions of the following variational inequality:

$$\begin{aligned} &u\in W_{0},\quad u\leq\psi \mbox{ a.e. in } \Omega, \\ &\int_{\mathcal{Q}}\frac {a(x,y,u(x),u(y))|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}} \bigl(\varphi(x)-u(x)- \varphi(y)+u(y)\bigr)\,dx\,dy \\ &\quad\geq\int_{\Omega}f(\varphi-u)\,dx, \quad \forall\varphi\in W_{0}, \varphi\leq\psi \mbox{ a.e. in } \Omega, \end{aligned}$$

(1.1)

where $N>sp$, $1< p<\infty$, $\psi:\Omega\rightarrow[0,\infty)$ is the obstacle with $\psi\in L^{p}(\Omega)$, and functions a, f satisfy the following assumptions:

• (a) $a:\mathcal{Q}\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ is a Carathéodory function, i.e., for each $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}$, $a(x,y,t_{1},t_{2})$ is measurable with respect to $(x,y)\in\mathcal{Q}$; for each $(x,y)\in \mathcal{Q}$, $a(x,y,t_{1},t_{2})$ is continuous with respect to $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}$. There exist two constants $a_{0},a_{1}>0$ such that $a_{0}\leq a(x,y,t_{1},t_{2})\leq a_{1}$ for all $(x,y,t_{1},t_{2})\in\mathcal{Q}\times\mathbb{R}\times\mathbb{R}$.

• (f) $f:\Omega\rightarrow\mathbb{R}$ is a measurable function with $f\in L^{p^{\prime}}(\Omega)$ and $p^{\prime}=p/(p-1)$.

Remark 1.1

If $a\equiv1$, the variational inequality (1.1) is governed by the following fractional p-Laplacian problem:

$$\begin{aligned} &(-\Delta)^{s}_{p}u=f \quad\mbox{in } \Omega, \\ &u=0 \quad\mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{aligned}$$

where $N> ps$ with $s\in(0,1)$, and $(-\Delta)^{s}_{p}$ is the fractional p-Laplacian defined for each $x\in\mathbb{R}^{N}$ as

$$\begin{aligned} (-\Delta)_{p}^{s}v(x)=2\lim_{\varepsilon\searrow0}\int _{\mathbb {R}^{N}\setminus B_{\varepsilon}(x)}\frac {|v(x)-v(y)|^{p-2}(v(x)-v(y))}{|x-y|^{N+ps}}\,dx\,dy, \end{aligned}$$

along any $v\in C_{0}^{\infty}(\mathbb{R}^{N})$, where $B_{\varepsilon}(x)$ denotes the ball in $\mathbb{R}^{N}$ with radius $\varepsilon>0$ centered at $x\in\mathbb{R}^{N}$. Actually, we can define the fractional p-Laplacian as follows:

$$\begin{aligned} \bigl\langle (-\Delta)_{p}^{s} u,v\bigr\rangle = \iint_{\mathcal{Q}}\frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}\,dx\,dy \end{aligned}$$

for all $u,v\in W_{0}$. For more details about the fractional p-Laplacian, we refer to [12] and [18]. When $a\not\equiv1$, it is natural to consider the nonlocal operator $\mathcal{L}$ defined as

$$\begin{aligned} \langle\mathcal{L}u,v\rangle=\int_{\mathcal{Q}}\frac {a(x,y,u(x),u(y))|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}\,dx\,dy \end{aligned}$$

for all $u,v\in W_{0}$. Hence, the variational inequality (1.1) can be written as

$$\begin{aligned} \langle\mathcal{L}u,\varphi-u\rangle\geq\int_{\Omega}f( \varphi-u)\,dx. \end{aligned}$$

Note that the function a depends on the state variable u. Generally, problem (1.1) is not variational.

Firstly, we ‘freeze’ the state variable u on the function a, that is, we fix $v\in L_{0}^{p}(\Omega)$. Let $\varepsilon\in(0,1)$ be fixed and consider the following penalization problem:

$$\begin{aligned} &\int_{\mathcal{Q}}\frac {a(x,y,v(x),v(y))|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}} \bigl(\varphi(x)- \varphi(y)\bigr)\,dx\,dy \\ &\quad{}+\frac{1}{\varepsilon}\int_{\Omega}\bigl|(u-\psi )^{+}\bigr|^{p-1}\varphi \,dx =\int_{\Omega}f\varphi \,dx, \quad \forall\varphi\in W_{0}, \end{aligned}$$

(1.2)

where $u^{+}=\max\{u,0\}$. To get the solutions of problem (1.2), we apply variational methods to the energy functional

$$\begin{aligned} I(u)=\frac{1}{p}\iint_{\mathcal{Q}}\frac {a(x,y,v(x),v(y))|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy + \frac{1}{\varepsilon p}\int_{\Omega}\bigl((u-\psi)^{+} \bigr)^{p}\,dx-\int_{\mathbb{R}^{N}}fu\,dx, \end{aligned}$$

associated to (1.2), for all $u\in W_{0}$. Note that the condition $f\in L^{p^{\prime}}(\Omega)$ implies that $I\in C^{1}(W_{0},\mathbb{R})$ and

$$\begin{aligned} \bigl\langle I^{\prime}(u),\varphi\bigr\rangle ={}&\iint_{\mathcal{Q}} \frac {a(x,y,v(x),v(y))|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\bigl(\varphi (x)-\varphi(y)\bigr)\,dx\,dy \\ &{}+\frac{1}{\varepsilon}\int_{\Omega}\bigl((u-\psi)^{+} \bigr)^{p-1}\varphi \,dx-\int_{\Omega}f\varphi \,dx \end{aligned}$$

for all $\varphi\in W_{0}$. Hence a critical point of functional I is a solution of penalization problem (1.2). Note that such penalization technique can be found in [19], and we study the penalized problem (1.2) through variational methods, see [20], Theorem 1.21.

Secondly, for $v\in L_{0}^{p}(\Omega)$ fixed, we deduce from (1.2) that there exists $u_{v}\in W_{0}$ with $u_{v}\leq\psi$ a.e. in Ω such that

$$\begin{aligned} &\int_{\mathcal{Q}}\frac {a(x,y,v(x),v(y))|u_{v}(x)-u_{v}(y)|^{p-2}(u_{v}(x)-u_{v}(y))}{|x-y|^{N+ps}} \bigl( \varphi(x)-u_{v}(x)-\varphi(y)+u_{v}(y)\bigr)\,dx\,dy \\ &\quad\geq\int_{\Omega}f(\varphi-u_{v})\,dx, \quad \forall\varphi\in W_{0}, \varphi \leq\psi. \end{aligned}$$

(1.3)

Finally, starting from (1.3) and using Schauder’s fixed point theorem (see [21], Theorem 3.21), we get our main result.

Theorem 1.1

Let $\psi\in L^{p}(\Omega)$, (a) and (f) hold. Then problem (1.1) has a solution in $W_{0}$.

2 The associated problem (1.3)

We start by proving some auxiliary results which will be useful in establishing the existence result of problem (1.3). In the following, we shortly denote by $\|\cdot\|_{q}$ the norm of $L^{q}(\Omega)$.

Lemma 2.1

For each $v\in L_{0}^{p}(\Omega)$ and $\varepsilon\in(0,1)$, problem (1.2) has a solution in $W_{0}$.

Proof

Fix $v\in L_{0}^{p}(\Omega)$ and $\varepsilon\in(0,1)$. By (a), the Hölder inequality and the continuous embedding from $W_{0}$ to $L^{p}(\Omega)$ (see [18], Lemma 2.3), it follows that for each $u\in W_{0}$ we have

$$\begin{aligned} I(u)&\geq\frac{a_{0}}{p}\iint_{\mathcal{Q}}\frac {|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy-\int _{\Omega}f u\,dx \\ &\geq \frac{a_{0}}{p}\iint_{\mathcal{Q}}\frac {|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy-\|f \|_{p^{\prime}}\|u\|_{p}\geq\frac {a_{0}}{p}\|u\|^{p}-C^{*} \|f\|_{p^{\prime}}\|u\|, \end{aligned}$$

where $C^{*}>0$ is the embedding constant of $W_{0}\hookrightarrow L^{p}(\Omega)$. The above estimate shows that I is coercive on $W_{0}$, being $1< p<\infty$.

On the other hand, it is easy to verify that I is convex and, consequently, weakly lower semi-continuous. Hence, we conclude via the direct method of variational methods (see [20], Theorem 1.2) that there exists a global minimum point of I in $W_{0}$ and, consequently, a solution of problem (1.2). □

Lemma 2.2

Let $v\in L_{0}^{p}(\Omega)$ and let $u_{v}^{\varepsilon}$ be the solution of (1.2) given by Lemma  2.1. If $\psi\in L^{p}(\Omega)$, then the family $\{u_{v}^{\varepsilon}\}_{\varepsilon}$ is bounded in $W_{0}$, i.e., there exists a positive constant C independent of ε and v such that $\|u_{v}^{\varepsilon}\|\leq C$ and $\|(u_{v}^{\varepsilon}-\psi )^{+}\|_{p}\leq C\varepsilon^{1/p}$ for all $\varepsilon>0$.

Proof

Taking $\varphi=u_{v}^{\varepsilon}$ as a test function in (1.2), we have

$$\begin{aligned} &\iint_{\mathcal{Q}}\frac{a(x,y,v(x),v(y))|u_{v}^{\varepsilon}(x)-u_{v}^{\varepsilon}(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy +\frac{1}{\varepsilon}\int _{\Omega}\bigl|\bigl(u _{v}^{\varepsilon}-\psi \bigr)^{+}\bigr|^{p}\,dx =\int_{\Omega}fu_{v}^{\varepsilon}\,dx. \end{aligned}$$

It follows from the Hölder inequality, the embedding $W_{0}\hookrightarrow L^{p}(\Omega)$ and (a) that

$$\begin{aligned} &a_{0}\iint_{\mathcal{Q}}\frac{|u_{v}^{\varepsilon}(x)-u_{v}^{\varepsilon}(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy +\frac{1}{\varepsilon} \int_{\Omega}\bigl|\bigl(u_{v}^{\varepsilon}-\psi \bigr)^{+}\bigr|^{p}\,dx \leq\|f\|_{p^{\prime}}\bigl\| u_{v}^{\varepsilon}\bigr\| _{p}\leq C^{*}\|f\|_{p^{\prime}}\bigl\| u_{v}^{\varepsilon}\bigr\| , \end{aligned}$$

this implies that

$$\begin{aligned} &\iint_{\mathcal{Q}}\frac{|u_{v}^{\varepsilon}(x)-u_{v}^{\varepsilon}(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy +\frac{1}{\varepsilon}\int _{\Omega}\bigl|\bigl(u_{v}^{\varepsilon}-\psi \bigr)^{+}\bigr|^{p}\,dx \leq C, \end{aligned}$$

where $C>0$ denotes various constants independent of ε and v. Hence $\|u_{v}^{\varepsilon}\|\leq C$ and $\|(u_{v}^{\varepsilon}-\psi)^{+}\| _{p}\leq C\varepsilon^{1/p}$. □

Theorem 2.1

Let $v\in L_{0}^{p}(\Omega)$ be fixed and let conditions (a) and (f) hold true. Furthermore, let $\psi\in L^{p}(\Omega)$. Then the variational inequality (1.3) has a unique solution $u_{v}\in W_{0}$.

Proof

By Lemma 2.2, there exist a subsequence of $\{u_{v}^{\varepsilon}\}$ still denoted by $\{u_{v}^{\varepsilon}\}$ and $u_{v}\in W_{0}$ such that $u_{v}^{\varepsilon}\rightharpoonup u_{v}$ in $W_{0}$ as $\varepsilon \rightarrow0$. It follows from $\|(u_{v}^{\varepsilon}-\psi)^{+}\|_{p}\leq C\varepsilon^{1/p}$ that $(u_{v}^{\varepsilon}-\psi)^{+}\rightarrow0$ in $L^{p}(\Omega)$ as $\varepsilon\rightarrow0$. By the compact embedding $W_{0}\hookrightarrow\hookrightarrow L^{p}(\Omega)$ (see [18], Lemma 2.5), we obtain that $u_{v}^{\varepsilon}\rightarrow u_{v}$ strongly in $L^{p}(\Omega)$ as $\varepsilon\rightarrow0$. Furthermore, $(u_{v}^{\varepsilon}-\psi )^{+}\rightarrow(u_{v}-\psi)^{+}$ in $L^{p}(\Omega)$ as $\varepsilon\rightarrow 0$. Hence $(u_{v}-\psi)^{+}=0$ a.e. in Ω, that is, $u_{v}\leq\psi$ a.e. in Ω.

Now we show that $u_{v}^{\varepsilon}\rightarrow u_{v}$ strongly in $W_{0}$ as $\varepsilon\rightarrow0$. For each fixed $\omega\in W_{0}$, we introduce a linear operator $F_{a}(\omega,\cdot):W_{0}\rightarrow\mathbb{R}$ defined by

$$\begin{aligned} F_{a}(\omega,\varphi)=\iint_{\mathcal{Q}}\frac{a(x,y,v(x),v(y))|\omega (x)-\omega(y)|^{p-2}(\omega(x)-\omega(y))}{|x-y|^{N+ps}}\bigl( \varphi (x)-\varphi(y)\bigr)\,dx\,dy \end{aligned}$$

for each $\varphi\in W_{0}$. Obviously, $F_{a}(\omega,\cdot)$ is a continuous linear functional on $W_{0}$ by the following inequality:

$$\begin{aligned} \bigl|F_{a}(\omega,\varphi)\bigr|\leq a_{1}\|\omega\|^{p-1} \|\varphi\|,\quad \forall \varphi\in W_{0}, \end{aligned}$$

thanks to assumption (a) and the Hölder inequality. Hence $u_{v}^{\varepsilon}\rightharpoonup u_{v}$ in $W_{0}$ as $\varepsilon \rightarrow0$ implies that

$$\begin{aligned} \lim_{\varepsilon\rightarrow0}F_{a}\bigl(u_{v},u_{v}^{\varepsilon}-u_{v} \bigr)=0. \end{aligned}$$

(2.1)

Taking $u_{v}^{\varepsilon}-u_{v}$ as a test function in (1.2), we obtain

$$\begin{aligned} F_{a}\bigl(u_{v}^{\varepsilon},u_{v}^{\varepsilon}-u_{v} \bigr)+\frac{1}{\varepsilon}\int_{\Omega}\bigl|\bigl(u_{v}^{\varepsilon}- \psi\bigr)^{+}\bigr|^{p-1}\bigl(u_{v}^{\varepsilon}-u_{v} \bigr)\,dx=\int_{\Omega}f\bigl(u_{v}^{\varepsilon}-u_{v} \bigr)\,dx. \end{aligned}$$

(2.2)

Note that

$$\begin{aligned} &\frac{1}{\varepsilon}\int_{\Omega}\bigl|\bigl(u_{v}^{\varepsilon}- \psi \bigr)^{+}\bigr|^{p-1}\bigl(u_{v}^{\varepsilon}-u_{v} \bigr)\,dx \\ &\quad=\frac{1}{\varepsilon}\int_{\Omega}\bigl|\bigl(u_{v}^{\varepsilon}- \psi\bigr)^{+}\bigr|^{p}\,dx+\frac {1}{\varepsilon}\int_{\Omega}\bigl| \bigl(u_{v}^{\varepsilon}-\psi\bigr)^{+}\bigr|^{p-1} (\psi -u_{v})\,dx\geq0, \end{aligned}$$

by $u_{v}\leq\psi$ a.e. in Ω. Putting this into (2.2), we have

$$\begin{aligned} &F_{a}\bigl(u_{v}^{\varepsilon},u_{v}^{\varepsilon}-u_{v} \bigr)\leq\int_{\Omega}f\bigl(u_{v}^{\varepsilon}-u_{v} \bigr)\,dx. \end{aligned}$$

(2.3)

Using $u_{v}^{\varepsilon}\rightarrow u_{v}$ strongly in $L^{p}(\Omega)$, (2.1) and (2.3), we deduce

$$\begin{aligned} \limsup_{\varepsilon\rightarrow0}\bigl[F_{a} \bigl(u_{v}^{\varepsilon},u_{v}^{\varepsilon}-u_{v}\bigr)-F_{a}\bigl(u_{v},u_{v}^{\varepsilon}-u_{v} \bigr)\bigr]\leq0. \end{aligned}$$

(2.4)

On the other hand, by the following well-known inequalities (see [18]):

$$\begin{aligned} |\xi-\eta|^{p}\leq \left \{ \textstyle\begin{array}{@{}l@{\quad}l} C_{p} (|\xi|^{p-2}\xi-|\eta|^{p-2}\eta )\cdot(\xi -\eta) & \mbox{for }p\geq2,\\ \widetilde{C}_{p} [ (|\xi|^{p-2}\xi-|\eta|^{p-2}\eta )\cdot(\xi-\eta) ]^{p/2}(|\xi|^{p}+|\eta|^{p})^{(2-p)/2} & \mbox{for }1< p< 2 \end{array}\displaystyle \displaystyle \displaystyle \right . \end{aligned}$$

for all $\xi,\eta\in\mathbb{R}^{N}$, where $C_{p}$ and $\widetilde{C}_{p}$ are positive constants depending only on p, we get

$$\begin{aligned} \bigl[F_{a}\bigl(u_{v}^{\varepsilon},u_{v}^{\varepsilon}-u_{v} \bigr)-F_{a}\bigl(u_{v},u_{v}^{\varepsilon}-u_{v}\bigr)\bigr]\geq0. \end{aligned}$$

Hence, we conclude from (2.4) that $\lim_{\varepsilon\rightarrow0}[F_{a}(u_{v}^{\varepsilon},u_{v}^{\varepsilon}-u_{v})-F_{a}(u_{v},u_{v}^{\varepsilon}-u_{v})]= 0$. A similar discussion as in [18] gives that $u_{v}^{\varepsilon}\rightarrow u_{v}$ strongly in $W_{0}$ as $\varepsilon\rightarrow0$.

Now we are ready to show that $u_{v}$ is a solution of (1.3). Taking $\varphi-u_{v}^{\varepsilon}$, with $\varphi\in W_{0}$ and $\varphi\leq\psi$ a.e. in Ω, as a test function in (1.2), we have

$$\begin{aligned} F_{a}\bigl(u_{v}^{\varepsilon}, \varphi-u_{v}^{\varepsilon}\bigr)+\frac{1}{\varepsilon}\int _{\Omega}\bigl|\bigl(u_{v}^{\varepsilon}-\psi \bigr)^{+}\bigr|^{p-1}\bigl(\varphi-u_{v}^{\varepsilon}\bigr)\,dx = \int_{\Omega}f\bigl(\varphi-u_{v}^{\varepsilon}\bigr)\,dx. \end{aligned}$$

(2.5)

Thanks to the choice of φ, one has $\frac{1}{\varepsilon}\int_{\Omega}|(u_{v}^{\varepsilon}-\psi)^{+}|^{p-1}(\varphi -u_{v}^{\varepsilon})\,dx\leq0$, so that we deduce from (2.5) that

$$\begin{aligned} F_{a}\bigl(u_{v}^{\varepsilon}, \varphi-u_{v}^{\varepsilon}\bigr)\geq\int_{\Omega}f \bigl(\varphi -u_{v}^{\varepsilon}\bigr)\,dx, \quad \forall\varphi\in W_{0}, \varphi\leq\psi. \end{aligned}$$

(2.6)

By the boundedness of a and the stronger convergence of $u_{v}^{\varepsilon}$ in $W_{0}$ and $L^{p}(\Omega)$, we conclude from (2.6) that

$$\begin{aligned} F_{a}(u_{v},\varphi-u_{v})\geq\int _{\Omega}f(\varphi-u_{v})\,dx,\quad \forall\varphi \in W_{0}, \varphi\leq\psi. \end{aligned}$$

This means that $u_{v}$ is a solution of (1.3).

It remains to prove the uniqueness of $u_{v}$. Now we suppose that there exist $u_{v}^{1}$ and $u_{v}^{2}$ such that (1.3) holds, that is,

$$\begin{aligned} F_{a}\bigl(u_{v}^{1}, \varphi-u_{v}^{1}\bigr)\geq\int_{\Omega}f \bigl(\varphi-u_{v}^{1}\bigr)\,dx \end{aligned}$$

(2.7)

and

$$\begin{aligned} F_{a}\bigl(u_{v}^{2}, \varphi-u_{v}^{2}\bigr)\geq\int_{\Omega}f \bigl(\varphi-u_{v}^{2}\bigr)\,dx \end{aligned}$$

(2.8)

for all $\varphi\in W_{0}$ with $\varphi\leq\psi$ a.e. in Ω. Taking $\varphi=u_{v}^{2}$ and $\varphi=u_{v}^{1}$ in (2.7) and (2.8), respectively, we have $F_{a}(u_{v}^{1},u_{v}^{2}-u_{v}^{1})\geq\int_{\Omega}f(u_{v}^{2}-u_{v}^{1})\,dx$ and $F_{a}(u_{v}^{2},u_{v}^{1}-u_{v}^{2})\geq\int_{\Omega}f(u_{v}^{1}-u_{v}^{2})\,dx$. From these two inequalities, we get $F_{a}(u_{v}^{1},u_{v}^{1}-u_{v}^{2})-F_{a}(u_{v}^{2},u_{v}^{1}-u_{v}^{2})\leq0$, this implies that $u_{v}^{1}=u_{v}^{2}$ a.e. in Ω. Thus we complete the proof. □

3 The proof of main result

For each $v\in L_{0}^{p}(\Omega)$, let $u=T(v)\in W_{0}$ be the weak solution of problem (1.3) given by Theorem 2.1. Thus, we can actually introduce an application $T:L_{0}^{p}(\Omega)\rightarrow W_{0}$ associating to each $v\in L_{0}^{p}(\Omega)$, the solution of problem (1.3), $T(v)\in W_{0}$.

Lemma 3.1

There exists a constant $C>0$ such that

$$\begin{aligned} \int_{\mathcal{Q}}\frac{|T(v(x))-T(v(y))|^{p}}{|x-y|^{N+ps}}\,dx \,dy\leq C,\quad \forall v\in L_{0}^{p}(\Omega). \end{aligned}$$

Proof

For each $v\in L_{0}^{p}(\Omega)$, $T(v)$ is a solution of problem (1.3). Let $u_{v}^{\varepsilon}$ be the solution of problem (1.2). By Lemma 2.2 and Theorem 2.1, we have $\|u_{v}^{\varepsilon}\| ^{p}\leq C$ and $u_{v}^{\varepsilon}\rightarrow T(v)$ strongly in $W_{0}$ as $\varepsilon\rightarrow0$, where $C>0$ is a constant independent of ε and v. Hence $\|T(v)\|^{p}\leq C$. □

Remark 3.1

By Lemma 3.1 and the fractional Sobolev embedding $W_{0}\hookrightarrow L^{p}(\Omega)$, it clearly follows that there exists $C_{1}>0$ such that $\int_{\Omega}|T(v)|^{p}\,dx\leq C_{1}$, $\forall v\in L_{0}^{p}(\Omega)$.

Lemma 3.2

The map $T:L_{0}^{p}(\Omega)\rightarrow W_{0}$ is continuous.

Proof

Let $\{v_{n}\}_{n}\subset L_{0}^{p}(\Omega)$ and $v\in L_{0}^{p}(\Omega)$ be such that $v_{n}$ converges to v in $L_{0}^{p}(\Omega)$ as $n\rightarrow\infty$. Without loss of generality, we assume that $v_{n}\rightarrow v$ a.e. in Ω. Set $u_{n}:=T(v_{n})$ for all $n\geq1$. By Lemma 3.1, we have

$$\begin{aligned} \iint_{\mathcal{Q}}\frac{|u_{n}(x)-u_{n}(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy=\int_{\mathcal{Q}} \frac{|T(v_{n}(x))-T(v_{n}(y))|^{p}}{|x-y|^{N+ps}}\,dx\,dy\leq C,\quad \forall n\geq1, \end{aligned}$$

i.e., $\{u_{n}\}_{n}$ is bounded in $W_{0}$. It follows that up to a subsequence we can deduce that $u_{n}$ converges weakly to some u in $W_{0}$. By the compact embedding $W_{0}\hookrightarrow L^{p}(\Omega)$, up to a subsequence, we have $u_{n}\rightarrow u$ strongly in $L^{p}(\Omega)$ and a.e. in Ω. Obviously, we also have $u_{n}$ converges to u strongly in $L_{0}^{p}(\Omega)$. Since $u_{n}$ ($n\geq1$) is a solution of problem (1.3), we have $u_{n}\leq\psi$ a.e. in Ω. Hence, $u\leq\psi$ a.e. in Ω.

On the other hand, for each n, we have

$$\begin{aligned} &\iint_{\mathcal{Q}}\frac {a(x,y,v_{n}(x),v_{n}(y))|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))}{|x-y|^{N+ps}} \\ &\qquad{}\times\bigl(\varphi (x)-u_{n}(x)-\varphi(y)+u_{n}(y)\bigr)\,dx\,dy \\ &\quad\geq\int_{\Omega}f(\varphi-u_{n})\,dx \end{aligned}$$

(3.1)

for all $\varphi\in W_{0}$ with $\varphi\leq\psi$. Taking $\varphi=u$ in (3.1), we obtain

$$\begin{aligned} &\iint_{\mathcal{Q}}\frac {a(x,y,v_{n}(x),v_{n}(y))|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))}{|x-y|^{N+ps}} \\ &\qquad{}\times\bigl(u_{n}(x)-u(x)-u_{n}(y)+u(y) \bigr)\,dx\,dy \\ &\quad\leq\int_{\Omega}f(u_{n}-u)\,dx. \end{aligned}$$

(3.2)

It follows from $u_{n}$ converges to u in $L^{p}(\Omega)$ and (3.2) that

$$\begin{aligned} &\limsup_{n\rightarrow\infty}\iint_{\mathcal{Q}} \frac {a(x,y,v_{n}(x),v_{n}(y))|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))}{|x-y|^{N+ps}} \\ &\qquad{}\times\bigl(u_{n}(x)-u(x)-u_{n}(y)+u(y) \bigr)\,dx\,dy \\ &\quad\leq0. \end{aligned}$$

(3.3)

Note that

$$\biggl\vert a\bigl(x,y,v_{n}(x),v_{n}(y)\bigr) \frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{(N+ps)/p^{\prime}}}\biggr\vert ^{p^{\prime}}\leq a_{1}^{p^{\prime}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\in L^{1}(\mathcal {Q},dx\,dy) $$

and

$$\begin{aligned} &a\bigl(x,y,v_{n}(x),v_{n}(y)\bigr)\frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{(N+ps)/p^{\prime}}}\\ &\quad\rightarrow a\bigl(x,y,v(x),v(y)\bigr)\frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{(N+ps)/p^{\prime}}} \end{aligned}$$

a.e. on $\mathcal{Q}$. Hence the Lebesgue dominated convergence theorem implies that

$$\begin{aligned} &a\bigl(x,y,v_{n}(x),v_{n}(y)\bigr)\frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{(N+ps)/p^{\prime}}}\\ &\quad\rightarrow a\bigl(x,y,v(x),v(y)\bigr)\frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{(N+ps)/p^{\prime}}} \end{aligned}$$

strongly in $L^{p^{\prime}}(\mathcal{Q},dx\,dy)$. Thus, we deduce from the Hölder inequality that

$$\begin{aligned} &\lim_{n\rightarrow\infty}\iint_{\mathcal{Q}} \frac {[a(x,y,v_{n}(x),v_{n}(y))-a(x,y,v(x),v(y))] |u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}} \\ &\quad{} \times\bigl(u_{n}(x)-u(x)-u_{n}(y)+u(y) \bigr)\,dx\,dy=0. \end{aligned}$$

(3.4)

Notice that

$$\begin{aligned} &\lim_{n\rightarrow\infty}\iint_{\mathcal{Q}}\frac {a(x,y,v(x),v(y))|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\\ &\quad{}\times\bigl(u_{n}(x)-u(x)-u_{n}(y)+u(y)\bigr)\,dx\,dy=0. \end{aligned}$$

Hence, from (3.4), it yields

$$\begin{aligned} &\lim_{n\rightarrow\infty}\iint_{\mathcal{Q}}\frac {a(x,y,v_{n}(x),v_{n}(y))|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\\ &\quad{}\times\bigl(u_{n}(x)-u(x)-u_{n}(y)+u(y)\bigr)\,dx\,dy=0. \end{aligned}$$

Therefore, by (3.3) and (3.4), we get $\lim_{n\rightarrow\infty}[F_{a}(u_{n},u_{n}-u)-F_{a}(u,u_{n}-u)]=0$. Further, we can conclude that $u_{n}\rightarrow u$ strongly in $W_{0}$. Consequently, $T:L_{0}^{p}(\Omega)\rightarrow W_{0}$ is continuous. □

Now we define an operator $L:L_{0}^{p}(\Omega)\rightarrow L_{0}^{p}(\Omega)$, with $L=i\circ T$, where $i:W_{0}\rightarrow L_{0}^{p}(\Omega)$ is the inclusion operator. Since $W_{0}$ is compactly embedded in $L_{0}^{p}(\Omega )$, the inclusion operator i is compact. It follows by Lemma 3.2 that the operator $L=i\circ T$ is compact.

Proof of Theorem 1.1

Let $C_{1}$ be the constant given in Remark 3.1, i.e., $\int_{\Omega}|L(v)|^{p}\,dx\leq C_{1}$, $\forall v\in L_{0}^{p}(\Omega)$. Consider the ball $B_{C_{1}}(0):=\{v\in L_{0}^{p}(\Omega):\int_{\Omega}|v|^{p}\,dx\leq C_{1}\}$. Clearly, $B_{C_{1}}(0)$ is a convex closed subset of $L_{0}^{p}(\Omega)$ and $L(B_{C_{1}}(0))\subset B_{C_{1}}(0)$. Moreover, $L(B_{C_{1}}(0))$ is relatively compact in $B_{C_{1}}(0)$.

Hence, by Lemma 3.2, we have $L: B_{C_{1}}(0)\rightarrow B_{C_{1}}(0)$ is a continuous map. Therefore, we can apply Schauder’s fixed point theorem (see [21], Theorem 3.21) to obtain L with a fixed point. This means that problem (1.1) has a solution, and thus the proof of Theorem 1.1 is complete. □