Infinitely many solutions for a class of fractional Robin problems with variable exponents

1.   Introduction

Fractional Sobolev spaces have major applications to various nonlinear problems, including phase transitions, thin obstacle problem, anomalous diffusion, crystal dislocation, semipermeable membranes and flame propagation, ultra-relativistic limits of quantum mechanics, minimal surfaces, water waves, etc. For more details, we refer the readers to Di Nezza, Palatucci and Valdinoci ^[21]. More recently, the works of Caffarelli et al. ^[9,10,11], led to a large amount of papers involving the fractional diffusion operator $(-\Delta)^{s}$ ($0 < s < 1$). The cited results turn out to be very fruitful in order to recover an elliptic PDE approach in a nonlocal framework, and they have recently been used very often, see ^{[1,6,7,16,18,24,26,27]}. We mention that there are also a great number of results which do not survive in the fractional framework, such as the ones mentioned in ^[13,14].

On the other hand, the study of PDE's involving variable exponents has become very attractive in recent decades, see ^{[15,17,19,23,25,30]} and the references therein.

It is therefore a natural question to see which results "survive" when the $p(x)-$Laplacian is replaced by the fractional $p(x)-$Laplacian.

As far as we know, the first result about the fractional Sobolev spaces with variable exponent of the form $W^ {s, q(\cdot), p(\cdot, \cdot)}(\Omega)$ and the fractional $p(x)-$Laplacian is obtained by Kaufmann-Rossi-Vidal in ^[22]. In particular it is shown that theses spaces are compactly embedded into variable exponent Lebesgue spaces. They also study the existence existence of solution for nonlocal problems involving the fractional $p(\cdot, \cdot)-$Laplacian. Bahrouni-Radulescu ^[2] obtained some further qualitative properties of the fractional Sobolev spaces and the fractional $p(\cdot, \cdot)-$Laplacian. Further developments have been done by Bahrouni, Ho, Biswas, Chung, Zhang, see ^{[3,4,5,8,12,20,29]}.

The main goal of this paper is to study the existence of infinitely many solutions for fractional $p(\cdot, \cdot)-$Laplacian equation with nonlocal Robin boundary condition. Precisely, we deal with the following problem

where $\Omega\subset \mathbb{R}^{N}$, $N > 1$, is a bounded domain with Lipschitz boundary, $a, q\in L^{\infty}(\Omega)$, $s\in (0, 1)$, $p\colon \mathbb{R}^{2N} \to (1, +\infty)$ is a symmetric, continuous function bounded away from $1$, $\overline{p}(\cdot) = p(\cdot, \cdot)$, $\beta \in L^{\infty}(\mathbb{R}^{N}\setminus \Omega)$ with $\beta \geq 0$ in $\mathbb{R}^{N}\setminus \Omega$ and $\left(-\Delta \right)^s_{p(\cdot, \cdot)}$ stands for the fractional $p(\cdot, \cdot)-$Laplacian which is given by

Furthermore, $\mathcal{N}_{s, p(\cdot, \cdot)}$ is defined by

and denotes the nonlocal normal $p(\cdot, \cdot)$-derivative (or $p(\cdot, \cdot)-$Neumann boundary condition) and describes the natural Neumann boundary condition in presence of the fractional $p(\cdot, \cdot)-$Laplacian. We would like to mention that the nonlocal normal derivative was introduced for the first time by A. Bahrouni, V. Radulescu and P. Winkert in ^[5]. This paper can be considered as a continuation of this study. Precisely, using variational methods, we will prove the existence of infinitely many solutions of Eq (1.1).

Now, we recall some results obtained by U. Kaufmann et al. ^[22]. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{N}$, $s\in (0, 1)$, $q\in C(\overline{\Omega}, \mathbb{R})$, and $p\in C(\overline{\Omega}\times \overline{\Omega}, \mathbb{R})$. Throughout this paper, we assume that

and

We define the fractional Sobolev space with variable exponents $W^{s, q(\cdot), p(\cdot, \cdot)}(\Omega)$ as

Let

be the corresponding variable exponent Gagliardo seminorm. For brevity, we denote $W^{s, q(\cdot), p(\cdot, \cdot)}(\Omega)$ by $E$ for a general $q\in C(\overline{\Omega}, \mathbb{R})$ satisfying $\text { (Q') }$ and by $W^{s, p(\cdot, \cdot)}(\Omega)$ when $q(x) = p(x, x)$ on $\overline{\Omega}$. We equip $E$ with the norm

Then, $E$ becomes a reflexive and separable Banach space.

Now, we are ready to recall a crucial theorem which prove some embedding results was obtained in ^[22] for the case $q(x) > p(x, x)$ on $\overline{\Omega}$ and then was refined in ^[20,29].

Theorem 1.1. Let $\Omega \subset \mathbb{R}^{N}$ be a bounded Lipschitz domain and let $s\in (0, 1)$. Let $p\in C(\overline{\Omega}\times \overline{\Omega}, \mathbb{R})$ and $q\in C(\overline{\Omega}, \mathbb{R})$ satisfy $\text { (P') }$ and $\text { (Q') }$ with $q(x)\geq p(x, x)$ for all $x\in \overline{\Omega}$. Let $r\in C(\overline{\Omega}, \mathbb{R})$ satisfy

Then, there exists a constant $C = C(N, s, p, q, r, \Omega)$ such that

Thus, $E$ is continuously embedded in $L^{r(\cdot)}(\Omega)$. Moreover, this embedding is compact.

From Theorem 1.1 and using assumptions $\text { (P') }$ and $\text { (Q') }$ with $q(x)\geq p(x, x)$ for all $x\in \overline{\Omega}$, we can deduce that spaces $E$ and $W^{s, p(\cdot, \cdot)}(\Omega)$ actually coincide. Evidently, $E$ is not suitable for studying the fractional $p(., .)-$Laplacian problem with Robin boundary condition and hence, we need to introduce another space as our solution space.

We suppose the following assumptions:

(A) $a\in L^{\infty}(\Omega)$ and $a > 0$ in $\Omega$.

(S) $s\in \mathbb{R}$ with $s \in (0, 1)$;

(P) $p\colon \mathbb{R}^{2N} \to (1, +\infty)$ is a symmetric, continuous function bounded away from $1$, that is,

with

and $sp^+ < N$;

($\beta$) $\beta \in L^{\infty}(\mathbb{R}^{N}\setminus \Omega)$ and $\beta \geq 0$ in $\mathbb{R}^{N}\setminus \Omega$;

Let $u \colon \mathbb{R}^N\to \mathbb{R}$ be a measurable function and let $\overline{p}(x) = p(x, x)$ for all $x \in \mathbb{R}^{2N}$. We set

where $\mathcal{C}\Omega = \mathbb{R}^N\setminus\Omega$ and

$(X, \|\cdot\|_X)$ is a reflexive and separable Banach space, see ^[5]. Let us recall the compact embedding result introduced in ^[22].

Proposition 1.2. Assume that (S), (P) and ($\beta$) hold. Then, for any $r\in C(\overline{\Omega})$ with $1 < r(x) < p_{s}^{*}(x)$ for all $x\in \overline{\Omega}$, there exists a constant $\alpha > 0$ such that

Moreover, this embedding is compact.

Now we give our main result.

Theorem 1.3. Assume that $q(x)\in (1, p^{-})$, for all $x\in \Omega$ and conditions $(A)$, $(S)$, $(P)$ and $(\beta)$ are fulfilled. Then problem $(1.1)$ has infinitely many solutions.

This paper is organized as follows. In Section $2$ we recall some definitions and fundamental properties of the spaces $L^{p(\cdot)}(\Omega)$ and $W^{ 1, p(\cdot)}(\Omega)$. In Section $3$ we give the proof of Theorem 1.3.

2.   Variable exponent spaces and preliminary results

In this section, we recall some definition and basic properties concerning the basic function spaces with variable exponent. We refer to ^{[5,15,17,23,25,30]} and the references therein.

We start by giving a bounded Lipschitz domain $\Omega\subset \mathbb{R}^{N}$. Next, we consider the following set

For any $p\in C_+(\overline\Omega)$, denote

and recall the variable exponent Lebesgue space $L^{p(\cdot)}(\Omega)$ as

which is endowed with the following Luxemburg norm

It is well known that $(L^{p(\cdot)}(\Omega), \|\cdot\|_{L^{p(\cdot)}(\Omega)})$ is a separable reflexive Banach space.

The variable exponent Sobolev space $W^{1, {p(\cdot)}}(\Omega)$ is defined by

with the norm

Let $L^{q(\cdot)}(\Omega)$ be the conjugate space of $L^{p(\cdot)}(\Omega)$, that is, $1/p(x)+1/q(x) = 1$ for all $x \in \overline{\Omega}$. If $u\in L^{p(\cdot)}(\Omega)$ and $v\in L^{q(\cdot)}(\Omega)$, then the Hölder-type inequality

is satisfied.

Defining the modular function $\rho\colon L^{{p(\cdot)}}(\Omega) \to \mathbb{R}$ by

Then, we have the following crucial result which will be useful in the sequel.

Proposition 2.1. Assume that $u \in L^{p(\cdot)}(\Omega)$. Then:

(i) $\|u\|_{p(\cdot)} < 1( = 1, > 1) \Longleftrightarrow \rho(u) < 1( = 1, 1)$;

(ii) $\|u\|_{p(\cdot)} > 1\ \Rightarrow \ \|u\|_{p(\cdot)}^{p^{-}}\leq \rho(u) \leq \|u\|_{p(\cdot)}^{p^{+}}$;

(iii) $\|u\|_{p(\cdot)} < 1 \ \Rightarrow \ \|u\|_{p(\cdot)}^{p^{+}}\leq \rho(u) \leq \|u\|_{p(\cdot)}^{p^{-}}$\.

Proposition 2.2. Assume that $u, u_{n}\in L^{p(\cdot)}(\Omega)$ with $n\in \mathbb{N}$. Then the following statements are equivalent:

(i) $\lim_{n\to +\infty} \|u_{n}-u\|_{p(\cdot)} = 0$;

(ii) $\lim_{n\to +\infty} \rho(u_{n}-u) = 0$;

(iii) $u_{n}(x)\to u(x)$ a. e. in $\Omega$ and $\lim_{n\to +\infty}\rho(u_{n}) = \rho(u).$

Now, we introduce the variational setting for problem (1.1). We define the functional $I\colon X\to \mathbb{R}$ by

which is well defined and of class $C^{1}$ on $X$. Clearly, the weak solutions of our main problem (1.1) are exactly the critical points of the Euler-Lagrange functional $I$.

3.   Proof of main result

In this section, we investigate the existence of infinitely many solutions for problem (1.1). It is known that, by ^[19], there exist $\left(e_{n}\right)\subset X$ and $e_{n}^{\ast}\subset X^{\ast}$ such that

It follows that

For any integer $k\geq1$, denote

Consider now the functional

where

and

An important ingredient in the proof of Theorem 1.3 is the following version of the fountain theorem, see Zou ^[31].

Theorem 3.1. Suppose that the functional $I_{\lambda}$ defined above satisfies the following conditions:

$\left(T_{1}\right) \ \ I_{\lambda}$ maps bounded sets to bounded sets uniformly for $\lambda \in \left[1, 2\right]$. Furthermore, $I_{\lambda}(-u) = I_{\lambda}(u)$ for all $\left(\lambda, u\right)\in \left[1, 2\right]\times X;$

$\left(T_{2}\right) \ \ K(u)\geq 0, \ \ K(u)\rightarrow \infty$ as $\left\|u\right\|\rightarrow \infty$ on any finite dimensional subspace of $X$;

$\left(T_{3}\right)$ there exist $\rho_{k} > r_{k} > 0$ such that

Then there exist a sequence of real numbers $(\lambda_{n})$ converging to 1 and $u(\lambda_{n})\in Y_{n}$ such that $I_{\lambda_{n}}'|_{Y_{n}}\left(u_{\lambda_{n}}\right) = 0$ and $\left(I_{\lambda_{n}}\right)(u\left(\lambda_{n}\right))\rightarrow c_{k}\in \left[d_{k}(2), b_{k}(1)\right]$ as $n \rightarrow \infty$. In particular, fixed $k\in \mathbb{N}$, if $\left(u(\lambda_{n})\right)$ has a convergent subsequence to $u_{k}$, then $I_{1}$ has infinitely many nontrivial critical points $\left(u_{k}\right)\subset{X}\backslash \left\{0\right\}$ satisfying $I_{1}\left(u_{k}\right)\rightarrow 0^{-}$ as $k\rightarrow \infty$.

We start with the following auxiliary property.

Lemma 3.2. Suppose that condition $(A)$ is satisfied. Then

Proof. It is easy to see that $0 < \beta_{k+1}\leq\beta_{k}$, so that $\beta_{k}\rightarrow \beta\geq 0$ as $k\rightarrow +\infty$. For every $k\geq 0$, by definition of $\beta_{k}$, there exists $u_{k}\in Z_{k}$ such that $\left\|u_{k}\right\| = 1$ and $\int_{\Omega}a(x) \frac{\left|u_{k}\right|^{q(x)}}{q(x)}dx > \frac{\beta_{k}}{2}$. Since $u_{k}\in Z_{k}$, it follows that $u_{k}\rightharpoonup 0$ in $X$. From Proposition 1.2, we deduce that $\int_{\Omega}a(x) \frac{\left|u_{k}\right|^{q(x)}}{q(x)}dx \rightarrow 0$ as $k\rightarrow +\infty$. Thus, $\beta = 0$ and the proof is complete.

Next, we prove the coercivity of $K$ on finite dimensional subspaces of $X$.

Lemma 3.3. Suppose that conditions of Theorem 1.3 are fulfilled. Then $K(u)\rightarrow +\infty$ as $\|u\|\rightarrow +\infty$ on any finite dimensional subspace of $X.$

Proof. Let $F$ be a finite dimensional subspace of $X$. Put

First we show that there exists $\epsilon_{1} > 0$ such that

Arguing by contradiction, for any positive integer $n$, there exists $u_{n}\in{ F\backslash \left\{0\right\}}$ such that

Set $v_{n}(x) = \frac{u_{n}(x)}{\left\|u_{n}\right\|}\in F\backslash \left\{0\right\}$. Then $\left\|v_{n}\right\| = 1$ for all $n\in\mathbb{N}$ and

We may assume, up to a subsequence, that $v_{n}\rightarrow v_{0}$ in $X$ for some $v_{0}\in F$. Then $\left\|v_{0}\right\| = 1$ and, by Proposition 1.2,

Claim: There exists $\gamma_{0} > 0$ such that

Otherwise, we have

It follows that

Hence $v_{0} = 0$, which contradicts $\left\|v_{0}\right\| = 1.$

Set

and

By (3.2) and (3.4), we obtain

for large enough $n$. Consequently, for all large $n$, we have

which is a contradiction to (3.3). Therefore (3.1) holds. For the $\epsilon_{1}$ given in (3.1), let

Then

Using $\left(B\right)$ and (3.5), for any $u\in F\backslash \left\{0\right\}$ with $\|u\|\geq 1$, we infer that

This shows that $K(u)\rightarrow \infty$ as $\left\|u\right\|\rightarrow \infty$ on any finite dimensional subspace of $X$ and this gives the proof of our desired result.

Lemma 3.4. Suppose that the conditions of Theorem 1.3 are satisfied. Then there exists a sequence $\rho_{k}\rightarrow 0^{+}$ as $k\rightarrow +\infty$ such that

and

Proof. By Propositions 1.2 and 2.1, we deduce that for any $u\in Z_{k}$ with $\|u\| < 1$, we have

We denote $\rho_{k} = (\frac{3^{p^{+}-1}(4p^{+})\beta_{k}}{q^{-}})^{\frac{1}{p^{+}-q^{+}}}$. By Lemma 3.2 we deduce that $\rho_{k}\rightarrow 0$ as $k\rightarrow +\infty$. Then there exists $k_{1}\in \mathbb{N}$ such that $\rho_{k}\leq \frac{1}{3^{p^{+}-1}p^{+}}$ for all $k\geq k_{1}$. Relation (3.6) implies that

Furthermore, by (3.6), we have

Since $\beta_{k}\rightarrow 0$ as $k\rightarrow +\infty$, we deduce that

This completes the proof.

Lemma 3.5. Assume that hypotheses of Theorem 1.3 are fulfilled. Then, for the sequence obtained in Lemma 3.4, there exists $0 < r_{k} < \rho_{k}$ for all $k\in \mathbb{N}$ such that

Proof. Let $u\in Y_{k}$ with $\|u\| < 1$ and $\lambda \in [1, 2]$. By $(A)$, $(P)$ and (3.1), there exists $\epsilon_{k} > 0$ such that

Since $0 < q^{-} < q^{+} < p^{-} < p^{+}$, we deduce that for small $\|u\| = r_{k}$ we have

This completes the proof of our lemma.

Proof of Theorem 1.3 completed. It is cleat that condition $\left(T_{1}\right)$ in Theorem 3.1 holds. Combining Lemmas 3.3, 3.4 and 3.5, we concludethat conditions $\left(T_{2}\right)$ and $\left(T_{3}\right)$ in Theorem 3.1 are satisfied. Then, by Theorem 3.1 there exist $\lambda_{n}\rightarrow 1$ and $u(\lambda_{n})\in Y_{n}$ such that

as $n\rightarrow +\infty.$

For the sake of notational simplicity, we always set in what follows $u_{n} = u\left(\lambda_{n}\right)$ for all $n\in \mathbb{N}.$

Claim: the sequence $(u_{n})$ is bounded in $X.$

Otherwise, we can assume that $(u_{n})$ is unbounded in $X$. Without loss of generality, we can assume that $\|u_{n}\| > 1$ for all $n\geq1$.

First, we can observe that there exists $c > 0$ such that for large enough $n$,

Using relation (3.7), we have

From Proposition 2.1, relation (3.8) and since $q^{+} < p^{-}$, we get that $(u_{n})$ is bounded in $X$. This proves that our claim is true. So, by Proposition 1.2 and up to a subsequence, we suppose that

and

In what follows we show that

Recalling that $(u_{n})$ is a bounded sequence, we get

Hence, (3.9) and Proposition 1.2 give as $n\rightarrow +\infty$

where $A(x, y) = (u_n(x)-u(x)-u_n(y)+u(y))$.

We have for all $n\in \mathbb{N}$

and

Therefore

and

Let us now recall the Simon inequalities [28 formula 2.2]

for all $x, y\in \mathbb{R}^{N}$, where $c_{p}$ and $C_{p}$ are positive constants depending only on $p$. Combining (3.10), (3.11), (3.12) and (3.13), we conclude that

Now, by Theorem 3.1, we conclude the proof of Theorem 1.3.

Conflict of interest

The author declares no conflict of interest.