Existence and multiplicity of solutions for a Schrödinger type equations involving the fractional $p(x)$-Laplacian

1.   Introduction

In this paper, we are concerned with the existence and multiplicity of nontrivial solutions for the following nonlinear Schrödinger type equation involving fractional $p(x)$-Laplacian:

where $(-\Delta_{p(x)})^{s}$ is the fractional $p(x)$-Laplacian operator, $s\in (0, 1)$ and potential function $V(x)$ satisfies the following conditions:

$(V_{1})$ $V(x)\in C(\mathbb{R}^N, \mathbb{R})$, $\inf_{x\in \mathbb{R}^N}V(x)\geq V_0 > 0$;

$(V_{2})$ For every constant $M > 0$, the Lebesgue measure of the set $\{x\in \mathbb{R}^N: V(x)\leq M\}$ is finite.

The nonlocal operator $(-\Delta_{p(x)})^{s}$ is defined as

where $\text{P.V.}$ denotes the Cauchy principle value and for brevity. Notice that the operator $(-\Delta_{p(x)})^{s}$ is the fractional version of the well known $p(x)$-Laplacian operator $\Delta_{p(x)}u = \text{div}(|\nabla u|^{p(x)-2}u)$, which was first introduced by Kaufmann, Rossi and Vidal in ^[24]. Some properties of fractional Sobolev space with variable exponent and the existence and multiple results of elliptic equations with fractional $p(x)$-Laplace operator are studied in ^{[1,3,5,8,14,24,38]}.

In recent years, problems involving nonlocal operators have gained a lot of attention due to their occurrence in real world applications, such as the thin obstacle problem, optimization, finance, phase transitions and also in pure mathematical research, such as minimal surfaces, conservation laws etc. The celebrated work of Nezza et al. ^[33] provides the necessary functional set-up to study these nonlocal operator problems using variational methods. We refer ^[30,34] and references therein for more details on problems involving fractional Laplace operator. In (1.1), when $p(\cdot) = p$ (constant), $(-\Delta_{p(x)})^{s}$ reduce to the usual fractional $p$-Laplace operator. In ^{[9,19,28,31,32]}, the authors studied various aspects, viz., existence, multiplicity and regularity of the solutions of the nonlinear elliptic type problems involving fractional $p$-Laplace operator.

When $s\equiv1$, problem (1.1) becomes the following $p(x)$-Laplacian equation:

These equations involving the $p(x)$-Laplacian arise in the modeling of electrorheological fluids and image restorations among other problems in physics and engineering, see ^{[12,13,25,29,36]}. Different from the Laplacian $\Delta$ and the $p$-Laplacian $\Delta_p$, the $p(x)$-Laplacian is nonlinear and nonhomogeneous. It is worth pointing out that Eq (1.2) received much attention after Kovacik and Rakosnik ^[25] set up the variable exponent Soboev space. For example, in ^[16], Fan considered a constrained minimization problem involving $p(x)$-Laplacian in $\mathbb{R}^N$, and in ^[17] considered $p(x)$-Laplacian equations in $\mathbb{R}^N$ with periodic data and non-periodic perturbations. Moreover, some other nonlinear problems with variable exponent can be found in ^{[2,4,10,15,18,20,21,22,23,26,27,36]} and references therein.

A natural question is what results can be recovered when the $p(x)$-Laplacian operator is replaced by the fractional $p(x)$-Laplacian of the form $(-\Delta_{p(x)})^{s}$. To our best knowledge, Kaufmann et al. ^[24] and Del Pezzo et al. ^[14] first introduced some results on fractional Sobolev spaces with variable exponent $W^{s, q(x), p(x, y)}(\Omega)$ and the fractional $p(x)$-Laplacian. Then, the authors established compact embedding theorems of these spaces into variable exponent Lebesgue spaces. As an application, they also prove an existence result for nonlocal problems involving the fractional $p(x)$-Laplacian. In ^[8], Bahrouni et al. obtained some further qualitative properties of the variable exponent fractional Sobolev space $W^{s, q(x), p(x, y)}(\Omega)$ and the fractional $p(x)$-Laplacian operator $(-\Delta_{p(x)})^{s}$. After that, some studies on such problems are performed by using different approaches, see ^[1,3,5,11] and references therein.

Motivated by the results on the $p(x)$-Laplacian problem and some results on the theory of fractional Sobolev spaces with variable exponent in ^{[1,7,8,14,24]}, we study the existence and multiplicity of weak solutions for the problem (1.1) via variational techniques and fountain theorem. Moreover, we show that the equation has a sequence of solutions with high energies. To the best of the author's knowledge, the present paper seems to be the first to study the infinitely solutions to the Schrödinger type problem with fractional $p(x)$-Laplacian operator. In order to state the main results, we introduce some basic definitions of fractional Sobolev space with variable exponent.

Throughout this paper, we assume that the continuous function $p:\mathbb{R}^{N}\times\mathbb{R}^{N}\to (1, \infty)$ satisfies

where $p^*_{s}(x)$ is the so-called critical exponent in fractional Sobolev space with variable exponent. Moreover, we make the following assumptions:

$(P_{1})$ $1 < p^{-}: = \inf\limits_{\mathbb{R}^{N}\times \mathbb{R}^{N}}p(x, y)\leq p(x, y)\leq p^{+}: = \sup\limits_{\mathbb{R}^{N}\times \mathbb{R}^{N}}p(x, y) < +\infty$;

$(P_{2})$ $p$ is symmetric, i.e., $p(x, y) = p(y, x)$ for all $(x, y)\in \mathbb{R}^{N}\times\mathbb{R}^{N}$.

Let $F(x, t) = \int_0^t f(x, \tau)d\tau$. Assume that

$(f_1)$ $f\in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R})$ satisfies $f(x, t)t \geq 0$ for all $(x, t)\in \mathbb{R}^N\times(0, +\infty)$ and

with $p(x) \leq q(x) \ll p^*(x)$ for all $x\in \mathbb{R}^N$.

$(f_2)$There exist $C_{0} > 0$ and $\mu > p^+$ such that

$(f_3)$ $\limsup\limits_{|t| \to 0} \frac{f(x, t)t}{|t|^{p^+}} = 0$ uniformly for $x\in \mathbb{R}^N$.

$(f_4)$ There exist two constants $C_{1}$, $C_{2} > 0$ such that

where $G(x, t) : = tf(x, t) - p^-F(x, t)$ and $H(x, t) : = tf(x, t) - p^+F(x, t)$.

$(f_5)$ $f(x, -t) = -f(x, t)$ for all $(x, t)\in \mathbb{R}^N\times\mathbb{R}$.

The main result of this paper is as follows.

Theorem 1.1. Assume that $(V_{1})$, $(V_2)$, $(P_1)$, $(P_2)$ and $(f_1)$–$(f_5)$ hold. Then the problem (1.1) has infinitely many solution $\{u_k\}$ satisfying

as $n\to\infty$, where $\Phi:W^{s, p(x, y)}(\mathbb{R}^{N})\to \mathbb{R}$ is the energy functional corresponding to problem (1.1).

Remark 1.2. Let us consider

where $q(x)\in C_+(\mathbb{R}^N)$ satisfying $q(x)\ll p^*_{s}(x)$ and $p^{+} < q^{-}$. It is easy to check that $(f_1)$–$(f_3)$ and $(f_5)$ hold. For $(f_4)$, since $F(x, t) = \frac{|t|^{q(x)}}{q(x)}$, $f(x, t)t = |t|^{q(x)}$ and $G(x, t) = (1- \frac{ p^-}{q(x)}) |t|^{q(x)}$, $H(x, t) = (1-\frac{p^+}{q(x)})|t|^{q(x)}$, we get that $G(x, t)$ is nondecreasing in $t\geq 0$. Moreover, in view of $G$, $H\geq 0$, we know that

Choosing $C_{1} = \frac{ q^+ - p^-}{ q^{-} - p^{+} }$, we obtain $G(x, t)\leq C_{1}H(x, t)$, that is, $(f_4)$ holds.

Remark 1.3. Condition $(f_1)$ means that $f(x, t)$ is subcritical in the variable sense. Different from things in constant case (i.e. $p^+ = p^-$), we need $q(x)\ll p^*(x)$. Condition $(f_5)$ assures the energy functional $\Phi$ is an even functional. So this condition is necessary for us to take advantage of the fountain geometry. Furthermore, It's known that $(f_4)$ is much weaker than the Ambrosetti-Rabinowitz type condition in the constant exponent case ($p^+ = p^-$).

This paper is organized as follows. In Section 2, certain basic results on fractional Sobolev spaces with variable exponent are stated, and abstract critical point theory is presented based on fountain theorem. Moreover, under condition $(V_{1})$ and $(V_2)$, we could get some compact embedding theorems. In Section 3, under various conditions on the nonlinear growth term $f$, the compactness condition for the energy functional $\Phi$ is obtained. The existence and multiplicity of nontrivial solutions for the problem (1.1) are established by the fountain theorem without the $(AR)$-condition.

Notation. For two functions $a(x)$, $b(x)\in C(\mathbb{R}^{N})$, $a(x)\ll b(x)$ means that $\inf_{x\in\mathbb{R}^{N}}(b(x)-a(x)) > 0$; "$\rightharpoonup$", "$\rightarrow$" denoted the weak convergence and strong convergence in a Banach space respectively; "$\hookrightarrow$", "$\hookrightarrow\hookrightarrow$" will be used to denote continuous embedding and compact embedding between spaces respectively. Moreover, we use $C$, $C_{i}$ ($i = 1, 2, \cdots$) to denote some generic positive constants.

2.   Fractional Sobolev spaces with variable exponent

In this section, we recall some definitions and basic properties of the variable exponent Lebesgue space $L^{p(\cdot)}(\mathbb{R}^{N})$ and the fractional Sobolev space with variable exponent $W^{s, q(\cdot), p(\cdot, \cdot)}(\mathbb{R}^{N})$, which will be treated in the next section.

Set $C_{+}(\mathbb{R}^{N}): = \{q(x)\in C(\mathbb{R}^{N}): \inf_{x\in \mathbb{R}^{N}}q(x) > 1\}$. For any $q\in C_{+}(\mathbb{R}^{N})$, we define

For any $q(x)\in C_{+}(\mathbb{R}^{N})$, we introduce the variable exponent Lebesgue space

endowed with the Luxemburg norm

Lemma 2.1 (^[20,21]). The space $L^{q(\cdot)}(\mathbb{R}^{N})$ is a separable, reflexive and uniformly convex Banach space, and its conjugate space is $L^{\widehat{q}(\cdot)}(\mathbb{R}^{N})$, where $\frac{1}{q(x)}+\frac{1}{\widehat{q}(x)} = 1$. For any $u\in L^{q(\cdot)}(\mathbb{R}^{N})$, $v\in L^{\widehat{q}(\cdot)}(\mathbb{R}^{N})$, we have the following Hölder type inequality

Lemma 2.2. If $\frac{1}{q(x)}+\frac{1}{r(x)}+\frac{1}{t(x)} = 1$, then for any $u\in L^{q(\cdot)}(\mathbb{R}^{N})$, $v\in L^{r(\cdot)}(\mathbb{R}^{N})$ and $w\in L^{t(\cdot)}(\mathbb{R}^{N})$, we have

The modular of the space $L^{q(\cdot)}(\mathbb{R}^{N})$, which is the mapping $\rho_{q(\cdot)}:\, L^{q(\cdot)}(\mathbb{R}^{N})\to \mathbb{R}$ defined by

Then, we have the following well-known results.

Lemma 2.3 (^[20,21]). If $u$, $u_{n}\in L^{q(\cdot)}(\mathbb{R}^{N})$, then

(1) $\|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})} > 1 ( = 1, < 1)$ if and only if $\rho_{q(\cdot)}(u) > 1 ( = 1, < 1\, \, resp.);$

(2) if $\|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})} < 1 \Longrightarrow \|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})}^{q^+} \leq \rho_{q(\cdot)}(u)\leq \|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})}^{q^-};$

(3) if $\|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})} > 1\Longrightarrow \|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})}^{q^-}\leq \rho_{q(\cdot)}(u)\leq \|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})}^{q^+};$

(4) for any $u_n\in L^{q(\cdot)}(\mathbb{R}^{N})$, $\rho_{q(\cdot)}(u_n)\to 0 \Longleftrightarrow \|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})}\to 0$ as $n\to \infty;$

(5) for any $u_n\in L^{q(\cdot)}(\mathbb{R}^{N})$, $\rho_{q(\cdot)}(u_n)\to \infty \Longleftrightarrow \|u\|_{L^{q(\cdot)}(\mathbb{R}^{N})} \to\infty$ as $n\to \infty.$

Remark 2.4. As a consequence of (2) and (3), for all $u\in L^{q(\cdot)}(\mathbb{R}^{N})$, we have

Let $0 < s < 1$ and assume that $p\in C(\mathbb{R}^{N}\times\mathbb{R}^{N}, (1, +\infty))$ satisfy $(P_1)$ and $(P_2)$. For $q\in C_+(\mathbb{R}^{N})$, the fractional Sobolev space with variable exponent $X: = W^{s, q(\cdot), p(\cdot, \cdot)}(\mathbb{R}^{N})$ is defined as follows

Let

be the variable exponent Gagliardo seminorm and define

On $X$, we shall sometimes work with the norm

where

It is not difficult to see that $\|\cdot\|_{\rho, X}$ is an equivalent norm of $\|\cdot\|_{X}$ with the relation

The following relations between the norm $\|\cdot\|_{\rho, X}$ and the modular $\rho(\cdot)$ can be easily obtained from their definitions.

Proposition 2.5. On $X$ it holds that

$({\rm{i}})$ for $u\in X\backslash\{0\}$, $\lambda = \|u\|_{\rho, X}$ if and only if $\rho(\frac{u}{\lambda}) = 1$;

$({\rm{ii}})$ $\rho(u) > 1$ $( = 1; < 1)$ if and only if $\|u\|_{\rho, X} > 1$ $( = 1; < 1)$, respectively;

${({\rm{iii}})}$ if $\|u\|_{\rho, X}\geq 1$, then $\|u\|_{\rho, X}^{p^-}\leq \rho(u)\leq\|u\|_{\rho, X}^{p^+}$;

$({\rm{iv}})$ if $\|u\|_{\rho, X} < 1$, then $\|u\|_{\rho, X}^{p^+}\leq \rho(u)\leq \|u\|_{\rho, X}^{p^-}$.

Proof. The proof is similar to [21, Theorem 3.1] and the details are omitted.

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For the bounded domain $\Omega\subset \mathbb{R}^{N}$, the following main embedding result was obtained in [24, Theorem 1.1].

Theorem 2.6. Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ and assume that $p$, $q$, $s$ be as above such that $q(x) > p(x, x)$ for all $x\in \overline{\Omega}$. Then, it holds that

for any $\beta\in C_+(\overline{\Omega})$ with $\beta(x) < p^*_s(x)$ for all $x\in \overline\Omega$.

Remark 2.7. (i) It is worth pointing out that in existing articles ^[5,8,24] working on $W^{s, q(\cdot), p(\cdot, \cdot)}(\Omega)$, the function $q$ is actually assumed that $q(x) > p(x, x)$ for all $x\in \overline\Omega$ due to some technical reason. Such spaces are actually not a generalization of the fractional Sobolev space $W^{s, p}(\Omega)$.

(ii) We could like to mention that the Theorem 2.6 is holds if $\Omega$ is bounded and $q(x)\geq p(x, x)$ for all $x\in\overline{\Omega}$, and $\beta\in C_+(\overline{\Omega})$ with $\beta(x) < p^*_s(x)$ for all $x\in \overline\Omega$. Some detail see ^[38].

In what follows, for brevity, in some places we write $p(x)$ instead of $p(x, x)$ and in this sense, $p\in C_+(\mathbb{R}^{N})$. If $q(x) = p(x) = p(x, x)$, we denote $W^{s, q(\cdot), p(\cdot, \cdot)}(\mathbb{R}^{N})$ by $W^{s, p(\cdot, \cdot)}(\mathbb{R}^{N})$. Moreover, we have the following embeddings.

Theorem 2.8. Let $s\in (0, 1)$. Assume that $p\in C_{+}(\mathbb{R}^{N}\times \mathbb{R}^{N})$ be a uniformly continuous and satisfying the conditions $(P_1)$ and $(P_2)$. Then

(i) $W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)\hookrightarrow L^{r(\cdot)}(\mathbb{R}^N)$ for any uniform continuous function $r\in C_+(\mathbb{R}^N)$ satisfying $p(x)\leq r(x)\ll p_{s}^{*}(x)$ for all $x\in\mathbb{R}^N$;

(ii) $W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)\hookrightarrow\hookrightarrow L_{Loc}^{r(\cdot)}(\mathbb{R}^N)$ for any uniform continuous function $r\in C_+(\mathbb{R}^N)$ with $r(x) < p_s^{*}(x)$ for all $x\in\mathbb{R}^N.$

Proof. (ⅰ) It suffices to prove for the case $p(x)\ll r(x)$. Decompose $\mathbb{R}^N$ by cubes $B_i$, ($i = 0, \, 1, \, 2, \, \cdots)$ with sides of length $\varepsilon > 0$ and parallel to the coordinate axes, where $B_{0}$ is the cube centered at the origin.

By the uniform continuity of $p$ and $r$, we can choose $\varepsilon$ sufficiently small and $t\in (0, s)$ such that

where $p_{i}^{-}: = \inf\limits_{(x, y)\in B_{i}\times B_{i}}p(x, y)$, $r_i^-: = \inf\limits_{x\in B_i}r(x)$, $r_i^+: = \sup\limits_{x\in B_i}r(x)$ and $(p_{i}^-)_{t}^{*} = \frac{N p_{i}^{-}}{N-t p_{i}^{-}}$.

Let $u\in W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)\setminus \{0\}$. Set $v: = \frac{u}{\|u\|_{\rho, X}}$. Then, by Proposition 2.5, we get

So, for each $i\in\mathbb{N}$, we have that

Now, we claim that there exists a constant $C = C(p^+, p^-, s, t, \varepsilon, B_{0}) > 0$ such that

where

with

Indeed, for any $r\in C_{+}(B_{i})$, from Corollary 3.3.4 of ^[12], for each $i\in \mathbb{N}$, we get

and

Let $\lambda > 0$ be such that

Then, taking $K = \sup\limits_{B_{i}\times B_{i}}\{1, |x-y|^{s-t}\}\in [1, +\infty)$, for $\widehat{\lambda} = K\lambda$, we have

which implies that

Taking infimum over all $\lambda$, (2.7) and (2.8) imply that

Together (2.6) with (2.9), there exists a constant $C > 0$ such that

Arguing as in that obtained (2.10), we get

where

Using the same methods of Theorems 5.4 and 6.5 in ^[33], there exists an extension function $\widetilde{v}\in W^{t, p_i^-}(\mathbb{R}^N)$ of $v$ such that $\widetilde{v} = v$ on $B_i$,

and

where $\omega_{N}$ is the volume of the unit ball in $\mathbb{R}^{N}$. So, by (2.13), the space $W^{t, p_{i}^{-}}(\mathbb{R}^N)$ is continuously embedded in $L^{q}(\mathbb{R}^{N})$ for any $q\in [\, p^{-}_{i}\, , \, (p^{-}_{i})^{*}_{t}]$, i.e., there exists a constant $C = C(N, p^{-}_{i}, t) > 0$ such that

Thus, combining (2.14) with (2.12), (2.11) and (2.5) that there exists $C > 0$ such that (2.4) holds.

If $\|v\|_{L^{r(\cdot)}(B_i)}\geq 1$. Invoking Lemma 2.3 and Proposition 2.5 with taking (2.4) into account, we have

If $\|v\|_{L^{r(\cdot)}(Q_i)} < 1$. Invoking Lemma 2.3 and Proposition 2.5 with taking (2.4) into account again, we have

Then, taking (2.15) with (2.16), for any $i\in\mathbb{N}$, we have

Summing up the last inequality over all $i\in\mathbb{N}$, combining with (2.2), there exists a constant $C > 0$ such that

Thus, $W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)\subset L^{r(\cdot)}(\mathbb{R}^N)$ and hence, $W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)\hookrightarrow L^{r(\cdot)}(\mathbb{R}^N)$ due to the closed graph theorem. The proof of assertion (ⅰ) is complete.

(ⅱ) Let $B$ be any ball in $\mathbb{R}^N$. Let $u_n\rightharpoonup 0$ in $W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)$ and thus $u_n\rightharpoonup 0$ in $W^{s, p(\cdot, \cdot)}(B)$. Invoking Theorem 2.6, we have $u_n\to 0$ in $L^{r(\cdot)}(B).$ This completed the proof of Theorem 2.8.

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Now, define the following linear subspace

with the norm

Under the conditions $(V_1)$ and $(V_2)$, $E$ is continuously embedded in $W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)$ as a closed subspace, and $E\hookrightarrow L^{p(\cdot)} (\mathbb{R}^N)$, $E\hookrightarrow L^{q(x)}(\mathbb{R}^N)$ if $q(x)\in C_+(\mathbb{R}^N)$ satisfies $p(x) \leq q(x) \ll p^*_{s}(x)$ for all $x\in \mathbb{R}^{N}$. Therefore, $E$ is also a separable reflexive Banach space. It is easy to see that with the norm $\|\cdot\|_{E}$, the following proposition remains valid.

Lemma 2.9. The functional $\psi: E\to \mathbb{R}$ defined by

has the following properties:

$({\rm{i}})$ If $\|u\|_{E}\geq 1$, then $\|u\|_{E}^{p^-}\leq \psi(u)\leq\|u\|_{E}^{p^+}$.

$({\rm{ii}})$ If $\|u\|_{E}\leq 1$, then $\|u\|_{E}^{p^+}\leq \psi(u)\leq \|u\|_{E}^{p^-}$.

Proof. We first prove the pair of inequalities. Indeed, for any $\lambda\in (0, 1)$ it is easy to see that

Now, if $\| u\|_{E} > 1$, we have $0 < \frac{1}{\| u\|_{E}} < 1$ and $\psi(\frac{1}{\| u\|_{E}}u) = 1$. Taking $\lambda = \frac{1}{\| u\|_{E}}$ in (2.17), we get

This completes the proof of Lemma 2.9 (ⅰ). The proof of the second is essentially the same.

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Lemma 2.10. Assume that $V$ satisfies $(V_{1})$ and $(V_2)$. Then

${\rm(i)}$ $E\hookrightarrow\hookrightarrow L^{p(x)}(\mathbb{R}^N)$;

${\rm(ii)}$ $E\hookrightarrow\hookrightarrow L^{\beta(x)}(\mathbb{R}^N)$ for any $\beta(x)\in C_+(\mathbb{R}^N)$ with $p(x) < \beta(x) \ll p_{s}^*(x)$.

Proof. (ⅰ) Assume $u_n\rightharpoonup 0$ in $E$. We will show that $u_n \to 0$ in $L^{p(x)}(\mathbb{R}^N)$, that is, $\int_{\mathbb{R}^N} |u_n(x)|^{p(x)}dx \to 0$ as $n\to \infty$. For any given $R\in (0, +\infty)$, we write

Since $E\hookrightarrow W^{s, p(\cdot, \cdot)}(\mathbb{R}^N)$ and $W^{s, p(\cdot, \cdot)}(B_{R}(0))\hookrightarrow\hookrightarrow L^{p(x)}(B_{R}(0))$, we have $E\hookrightarrow\hookrightarrow L^{p(x)}(B_{R}(0))$, which implies $I_{1}(u_{n})\to 0$ as $n\to \infty$.

Next, we need to show that for any $\varepsilon > 0$, there exists $R > 0$ such that

Note that $\|u_{n}\|_{E} < +\infty$. Given $\varepsilon > 0$, set $M = \frac{2}{\varepsilon}\sup\limits_{n}\left(\|u_{n}\|_{E}^{p^{+}} +\|u_{n }\|_{E}^{p^{-}}\right).$ Denote

and

Then, using Lemma 2.9, we have

On the other hand, by Hölder inequality and Theorem 2.6, we get

where the number $\alpha\in (1, +\infty)$ such that $p(x) < \alpha p(x) < p^*_{s}(x)$ for all $x\in \mathbb{R}^{N}$.

From $(V_2)$, we can choose $R$ large enough such that

Then, (2.20) and (2.21) imply that

It follows from (2.19) and (2.22) that (2.18) holds and completes the proof of (ⅰ).

(ⅱ) Let $u_n\rightharpoonup 0$ in $E$. We need to show $u_n\to 0$ in $L^{\beta(x)}(\mathbb{R}^N)$. That is

Since $E\hookrightarrow L^{p^{*}_{s}(x)}(\mathbb{R}^N)$ is continuous and $\{u_{n}\}$ is bounded in $E$, we have

Since $q(x) < \beta(x)\ll p^*_{s}(x)$, there exists a function $\gamma\in C(\mathbb{R}^{N}, (0, 1))$ such that

Then, by Lemma 2.1 and Remark 2.4, we get

Therefore, from (2.24), (2.25) and (ⅰ), we get that $u_n\to 0$ in $L^{\beta(x)}(\mathbb{R}^N)$, and the proof of (ⅱ) is completed.

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Remark 2.11. From Lemma 2.10, we know that the conditions $(V_{1})$ and $(V_{2})$ play an important role in enables $E$ to be compactly embedded into $L^{\beta(x)}(\mathbb{R}^N)$ type spaces.

3.   Existence of weak solutions

In this section, the proof of the existence and multiplicity of nontrivial solutions for (1.1) by applying the fountain theorem under some assumptions on $f$.

Equation (1.1) has a variational structure and its associated energy functional $\Phi: E \to R$ is defined by

Under the assumptions $(f_1)$–$(f_3)$, $\Phi$ is of class $C^{1}(E, \mathbb{R})$. We say that $u\in E$ is a weak solution of (1.1), if

for all $v\in E$, where

Clearly, the critical points of $\Phi$ are exactly the weak solutions of problem (1.1).

Define the functional $\Psi:E\to \mathbb{R}$ by

Then, $\Psi\in C^{1}(E, \mathbb{R})$ and its Fréchet derivative is

According to the analogous arguments in [8, Lemma 4.2], the following lemma is easily checked, so we omit the proof.

Lemma 3.1. Assume that $(V_1)$ and $(P_1)$, $(P_2)$ hold. Then, the functional $\Psi:E\to \mathbb{R}$ is convex and weakly lower semicontinuous on $E$. Moreover, the operator $\Psi'$ is a mapping of $(S_+)$-type, that is, $u_n \rightharpoonup u$ and $\limsup_{n\to\infty}\langle \Psi'(u_n)-\Psi'(u), u_n-u\rangle\leq 0$ implies $u_n\to u$ strongly in $E$ as $n\to \infty$.

Definition 3.2. For $c\in \mathbb{R}$, we say that $\Phi$ satisfies the $(C)_{c}$-condition if for any sequence $\{u_{n}\}\subset E$ with

there is a subsequence $\{u_{n}\}$ such that $\{u_{n}\}$ converges strongly in $E$.

Let $W$ be a reflexive and separable Banach space. It is well-known that there exist $\{e_i\}_{i = 1}^\infty \subset W$ and $\{f_i^{*}\}_{i = 1}^\infty \subset W^*$ such that

and

Let $W_i = \text{span} \{e_i\}$, then $W = \overline { \bigoplus_{i = 1}^{\infty} W_i}$. Now we define

Then, we will use the following fountain theorem ^[6] (see also ^[35]) to prove our result.

Theorem 3.3 (Fountain theorem). Let $W$ be a real reflexive Banach space, $\mathcal{I}\in C^{1}(W, \mathbb{R})$ satisfies the $(C)$-condition, $\mathcal{I}(-u) = \mathcal{I}(u)$. If for each sufficiently large $k\in \mathbb{N}$, there exist $\rho_{k} > \delta_{k} > 0$ such that the following conditions hold:

${\rm(i)}$ $a_k : = \inf\{ \mathcal{I}(u): u\in Z_k, \|u\|_{W} = \delta_{k}\} \to \infty\ \mathit{\text{as}}\ k\to \infty$;

${\rm(ii)}$ $b_k : = \max\{ \mathcal{I}(u) : u\in Y_k, \|u\|_{W} = \rho_{k}\} \leq 0$.

Then $\mathcal{I}$ has a sequence of critical points $\{u_{k}\}\subset W$ such that $\mathcal{I}(u_{k})\to +\infty$ as $k\to+\infty$.

In the following, for the reflexive and separable Banach space $E$, define $Y_{k}$ and $Z_{k}$ as in (3.1), we will show that the energy functional $\Phi$ satisfies the geometric structure. We now give a useful lemma.

Lemma 3.4. Let $q(x)\in C_+(\mathbb{R}^N)$ with $p(x) \leq q(x) \ll p^*(x)$ and denote

Then $\alpha_k\to 0$ as $k\to \infty$.

Proof. Suppose to the contrary that there exist $\varepsilon_{0}$, $k_{0} > 0$, and the sequence $\{u_{k}\}\subset Z_{k}$ such that

for all $k\geq k_{0}$. Since $\{u_{k}\}$ is bounded in $E$, there exists $u\in E$ such that $u_{k}\rightharpoonup u$ in $E$ as $k\to \infty$ and

Hence, we get $u = 0$. However, we obtain that

which provides a contradiction. Thus, we have proved that $\alpha_{k}\to 0$ as $k\to \infty$.

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Lemma 3.5. Under the assumptions of Theorem 1.1, the geometry conditions of the fountain theorem hold, that is, (ⅰ) and (ⅱ) of Theorem 3.3 hold.

Proof. (ⅰ) By $(f_1)$ and $(f_3)$, for any $\varepsilon > 0$, there exists a $C(\varepsilon) > 0$ such that

Let

Then, by Lemma 3.4, we obtain $\theta_{k}\to 0^{+}$ and $\eta_{k}\to 0^{+}$ as $k\to \infty$. So, for any $u\in Z_k$ with $\|u\|_{E} = \delta_k > 1$, from (3.2), (3.3), Remark 2.4 and Lemma 2.9, we get

Let $\varepsilon > 0$ small enough such that $\varepsilon (\theta_{k}\|u\|_{E})^{p^+} \leq \frac{1}{2p^+}\|u\|_{E}^{p^-}$. If $\|u\|_{E}^{q^-}\geq \|u\|_{E}^{q^+}$, let $\delta_k = (\frac{1}{4p^{+}C\eta_k^{q^-}}) ^ {\frac{1}{q^{-}-p^{-}}}$, for sufficiently large $k$,

Now $\eta_k \to 0$ and $q^- > p^+$ implies that

If $\|u\|_{E}^{q^-} < \|u\|_{E}^{q^+}$, we can similarly derive that $\inf_{u\in Z_{k}, \|u\|_{E} = \delta_{k}}\Phi(u)\to +\infty$ as $k\to \infty$, hence (ⅰ) is satisfied.

(ⅱ) By $(f_2)$ and $(f_3)$, for any $\varepsilon > 0$, there exists a $C(\varepsilon) > 0$ such that

From (3.4) and Lemma 2.9, for some $v\in Y_{k}$ with $\|v\|_{E} = 1$ and $t > 1$, we have

due to $\mu > p^+$ and all norms on $Y_{k}$ are equivalent. So there exists $\rho_{k} > \delta_{k}$ such that $t = \rho_{k}$ concludes $\Phi(tv)\leq 0$, and then

Hence (ⅱ) is satisfied.

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Lemma 3.6. Assume that $(f_1)$, $(f_2)$ and $(f_4)$ hold. Then $(C)_{c}$-sequence of $\Phi$ is bounded.

Proof. Suppose that $\{u_{n}\}\subset E$ is a $(C)_{c}$-sequence for $\Phi$, that is,

which show that

where $o_{n}(1)\to 0$ as $n\to \infty$. We now prove that $\{u_{n}\}$ is bounded in $E$. We argue by contradiction. Suppose that the sequence $\{u_{n}\}$ is unbounded in $E$, let $w_{n} = \frac{u_n}{ \|u_n\|_{E} }$, then $w_{n}\in E$ with $\|w_{n}\|_{E} = 1$. Hence, up to a subsequence, still denoted by itself, there exists a function $w\in E$ such that

If $w = 0$, we can define a sequence $\{t_{n}\}\subset\mathbb{R}$, as argued in ^[37], such that

Then, for any $L > 1$, and $n$ large enough, we have

Moreover, from $(f_1)$ and (3.6), we get $\int_{\mathbb{R}^N} F(x, Lw_n)dx \to 0$. Hence, this and (3.7) imply that $\Phi(t_nu_{n}) \to\infty$ as $n\to \infty$ by the fact $L$ can be large arbitrarily.

Noting that $\Phi(0) = 0$ and $\Phi(u_n)\to c$, then $t_n\in (0, 1)$ when $n$ is large enough. Hence, $\langle\Phi'(t_nu_{n}), t_nu_{n}\rangle \to 0$ and

Moreover,

This and (3.8) deduce that

On the other hand, in view of $(f_4)$, there exist two constants $C_1$, $C_2 > 0$ such that

So, (3.9) and (3.10) imply that

which is contradictory.

If $w\neq 0$. Assume $\|u_{n}\|_{E} > 1$, by $\langle\Phi'(u_{n}), u_{n}\rangle = o_{n}(1)$ and Lemma 2.9, we have

Denote $\Omega_{0} = \{x\in \mathbb{R}^{N}:w(x) = 0\}$. Then, for $x\in \mathbb{R}^{N}\backslash\Omega_{0}$, we have $|u_n| = |w_{n}|\|u_{n}\|_{E}\to +\infty$ as $n\to \infty$. Hence, by $(f_2)$, we have

Combining (3.11) and (3.12), we obtain a contradiction. Therefore, $\{u_{n}\}$ is bounded in $E$ and the proof is complete.

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Lemma 3.7. Assume that conditions $(f_1)$–$(f_4)$ hold. Then $\Phi$ satisfies $(C)_{c}$-condition, that is, for all $c\in \mathbb{R}$, any $(C)_{c}$-sequence of $\Phi$ has a convergent subsequence.

Proof. Let $\{u_{n}\}\subset E$ be a $(C)_{c}$-sequence of $\Phi$. According to Lemma 3.6, we deduce that $\{u_{n}\}$ is bounded in $E$. Up to a subsequence, we may assume that $u_n\rightharpoonup u$ weakly in $E$, $u_n\to u$ strongly in $L^{\beta(x)}(\mathbb{R}^N)$ for $p(x) \leq \beta(x) \ll p^*_{s}(x)$, and $u_n(x)\to u(x)$ a.e. in $\mathbb{R}^N$.

Using (2.1) and Lemma 2.1, we have

Hence, $(f_1)$ and (3.13) give that

By the boundedness of $\{u_{n}\}$ in $E$, we have that

and

Therefore, we can deduce from (3.14)–(3.16) and $u_n\to u$ in $L^{q(x)}(\mathbb{R}^N)$ that

Since $\Phi'(u_{n})\to 0$ in $E^{*}$ and $u_n\rightharpoonup u$ in $E$, we have

That is,

Then, (3.17), (3.18) and $\Psi'$ is a $(S_+)$-type operator imply that $u_{n}\to u$ in $E$. This completes the proof.

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Proof of Theorem 1.1. According to Lemma 3.7 and $(f_5)$, $\Phi$ is an even functional and satisfies $(C)_{c}$-condition for all $c\in \mathbb{R}$. Lemma 3.5 implies that the functional $\Phi$ has the fountain theorem geometry conditions. So, from Theorem 3.3 we deduce that $\Phi$ has a sequence of critical points $\{u_{k}\}$ with $\Phi(u_{k})\to +\infty$ and Theorem 1.1 follows.

4.   Conclusions

This paper considers a class of Schrödinger equations involving the fractional $p(x)$-Laplacian in the whole space $\mathbb{R}^{N}$. We use the variational method and the fountain theorem to prove that this nonlocal problem has infinitely many high-energy solutions.

Conflict of interest

The author declares there is no conflict of interest.