Existence and multiplicity of solutions for fractional $p(x)$-Kirchhoff-type problems

1.   Introduction and main results

In ^[1], Kirchhoff studied a stationary version of the equation

where $\rho_0, p_1, h_1, L$ and $E_0$ are constants. Such equation extends the classical D'Alembert wave equation by considering the effects of the changes in the length of the string during the vibrations. It is worthwhile to note that the Eq (1.1) received much attention only after Lions ^[2] put forward an abstract framework to the Eq (1.1). After this work, various equations of Kirchhoff-type have been studied extensively. For instance, many researchers have studied the Kirchhoff-type equations involving the $p$-Laplacian, which can be found in ^[3,4,5,6], $p(x)$-Laplacian (see, for example, ^{[7,8,9,10,11]}) and fractional $p(x)$-Laplacian (see ^{[12,13,14,15]}).

Recently, lots of researchers have been interested in the Kirchhoff-type equations involving the $p$-Laplacian (see ^[16,17]).In ^[5], Liu proved the existence of infinite solutions for the $p$-Kirchhoff-type problems via the fountain theorem. Since the infimum of its principal eigenvalue is zero, the $p$-Laplacian is not homogenous, and generally it does not have the alleged first eigenvalue. Hence, more and more attention has been given to partial differential equations with nonstandard growth conditions. Dai and Hao ^[18] investigated the existence and multiplicity of solutions to Kirchhoff-type problems associated with the $p(x)$-Laplacian via a direct variational approach. The $p(x)$-Laplacian has more complex nonlinear properties than the $p$-Laplacian, and we can refer to ^[11,19] for more details about it.

In the last few years, many researchers have tended to focus on the fractional $p(x)$-Kirchhoff-type problems. Kaufmann, Rossi and Vidal ^[20] introduced the fractional $p(x)$-Laplacian $\vartriangle_{p(x)} v = \text{div}(|\bigtriangledown v|^{p(x)-2})$ (see, for example, ^[21,22]). In ^[23], the authors investigated the fractional $p(x)$-Laplace operator and associated fundamental properties about new fractional Sobolev spaces with variable exponents. In ^[14], by using dint of the variational methods, Azroul et al. investigated the existence of solutions for the Kirchhoff-type problems involving fractional $p(x)$-Laplacian as follows:

where $M\in\mathcal{Q}_1$, i.e., $M$ satisfies the following: there exist $0 < a_1\leq a_2$ and $\beta > 1$ such that

In ^[24], applying the symmetric mountain pass theorem, Azroul, Benkirane and Shimi resolved the existence solutions to the following Kirchhoff-type problems involving fractional $p(x, \cdot)$-Laplacian in $\mathbb{R}^d$:

where $M\in\mathcal{Q}_2$, i.e., the continuous function $M:\mathbb{R}^+_0: = [0, +\infty)\rightarrow \mathbb{R}^+_0$ satisfies the following conditions:

$(M_1)$: Let $\epsilon_0 > 0$ and $\alpha\in(1, (p^*_s)_-/p_+)$. Suppose that

where

and $(p^*_s)_-$, $p_+$ and $p_-$ will be introduced in Section 2.

$(M_2):$ let $\epsilon > 0$. Suppose $l = l(\epsilon) > 0$ such that

By comparison their definitions, we find that the condition of $\mathcal{Q}_2$ is weaker than $\mathcal{Q}_1$. Hence the spontaneous question is to show which results we can obtain if we replace $M\in\mathcal{Q}_1$ by $M\in\mathcal{Q}_2$ in ^[14].

Inspired by the fractional Sobolev spaces with variable exponents (for more details see ^[23]) and the papers referred to above, we aim to deal with the existence and multiplicity solutions to the fractional $p(x)$-Kirchhoff-type problem $(P^s_{M})$ which is introduced in ^[14], where

$\bullet$ $M\in\mathcal{Q}_2$ and $\lambda$ is a positive real constant;

$\bullet$ $Q: = \mathbb{R}^{2d}\backslash(\Omega^c\times\Omega^c)$, where $\Omega\in \mathbb{R}^d$ is a Lipschitz bounded open domain and $\Omega^c = \mathbb{R}^d\setminus\Omega, \, \, d\geq3$;

$\bullet$ the continuous functions $r:\overline{\Omega}\rightarrow (1, +\infty)$ and $p:\overline{Q}\rightarrow (1, +\infty)$ are bounded;

$\bullet$ for $s\in(0, 1)$, the operator $(-\Delta_{p(x)})^s$ is the following fractional $p(x)$-Laplacian

where p.v. stands for Cauchy principle value for brevity.

We introduce our main conclusions and results as follows:

Theorem 1.1. For the continuous function $r:\overline{\Omega}\rightarrow (1, +\infty)$, let

Suppose that $M$ satisfies (1.3) and $r\in C_+(\overline{\Omega})$ such that

then problem $(P^s_M)$ has a nontrivial weak solution, if there is $\lambda_1 > 0$ such that

Theorem 1.2. Suppose that $p\in C_+(\overline{Q})$ is symmetric with $sp_+ < d$ and $s\in(0, 1)$. Let $M\in\mathcal{Q}_2$, $r\in C_+(\overline{Q})$ with

Then problem $(P^s_M)$ has a sequence $\{u_n\}_n$ of nontrivial solutions, if there is a constant $c_1 > 0$ such that

Remark 1.3. We discuss the $p(x)$-Kirchhoff-type problem $(P^s_M)$ in two situations: if $r$ satisfies (1.5), we apply the direct variational methods; we utilize the symmetric mountain pass theorem if $r$ satisfies (1.6).

The paper is organized as follows: we introduce the fractional Sobolev spaces with variable exponents and some necessary properties of variable Lebesgue spaces in Section 2. In the end, Section 3 gives the proofs of Theorems 1.1 and 1.2.

2.   Preliminaries

In this section, we present some useful properties of the fractional Sobolev spaces with variable exponents and the generalized Lebesgue spaces. We can refer to ^{[17,18,20,25,26]} and the references therein for more details.

We always suppose that $\Omega$ is a Lipschitz bounded open domain in $\mathbb{R}^d$ and $\overline{\Omega}$ is a closure of $\Omega$. Consider a continuous function $r:\overline{\Omega}\rightarrow (1, +\infty)$ and set

For a measurable function $v$ and any $r\in C_+(\overline{\Omega})$, the modular functional $\rho_{r(x)}(v)$ is defined by

The variable Lebesgue space is defined as

equipped with the norm

Suppose $r\in C_+(\overline{\Omega})$ such that $\frac{1}{r(x)}+\frac{1}{r'(x)} = 1$, where $r'(x)$ is the conjugate exponent of $r(x)$. Then the Hölder inequality is as follows:

Lemma 2.1 (^[20]). Let $v\in L^{r(x)}$ and $u\in L^{r'(x)}$. There exists a positive constant $c$ such that

Proposition 2.2 (^[27]). Let $v\in L^{r(x)}$. The following properties hold:

(i) $\| v\|_{L^{r(x)}} = 1\ (resp. > 1, < 1)\Leftrightarrow \rho_{r(x)}(v) = 1\ (resp. > 1, < 1);$

(ii) $\| v\|_{L^{r(x)}} < 1\Rightarrow \| v\|^{r_+}_{L^{r(x)}}\leq \rho_{r(x)}(v)\leq \| v\|^{r_-}_{L^{r(x)}};$

(iii) $\|v\|_{L^{r(x)}} > 1\Rightarrow \| v\|^{r_-}_{L^{r(x)}}\leq \rho_{r(x)}(v)\leq \| v\|^{r_+}_{L^{r(x)}};$

(iv)${\lim\limits_{k \to +\infty}}\|v_k-v\|_{L^{r(x)}} = 0\Leftrightarrow {\lim\limits_{k \to +\infty}}\rho_{r(x)}(v_k-v) = 0$.

We show the following proposition, which is from Theorems 1.6 and 1.10 in ^[26].

Proposition 2.3. Suppose $1 < r_-\leq r(x)\leq r_+ < \infty$; then, $\big(L^{r(x)}, \|\cdot\|_{L^{r(x)}}\big)$ is a reflexive uniformly convex and separable Banach space.

Let the continuous function $p:\overline{Q}\rightarrow (1, +\infty)$ be bounded. We set

and $p$ is symmetric, if $p(x, y)$ satisfies the following

We assume that

Throughout this paper, $s\in(0, 1)$ and the fractional Sobolev space with variable exponents is defined in ^[14] given by

The norm of $S$ is as follows

where $[v]_S$ is defined by

Also, $(S, \|\cdot\|_S)$ is a separable reflexive Banach space which is introduced in ^[14].

Now, we denote the linear subspace of $S$ given by

the modular norm is as follows

We know that $(S_0, \|\cdot\|_{S_0})$ is a separable, reflexive and uniformly convex Banach space (see Lemma 2.3 in ^[14]).

We denote the modular $\rho_{p(x, y)}:S_0\rightarrow \mathbb{R}$ by

where

Similarly to Proposition 2.1, $\rho_{p(x, y)}$ has the following property:

Lemma 2.4 (^[13]). Let $p$ satisfy (2.1) and $s\in(0, 1)$. Suppose $v\in S_0$; we can obtain

(i) $\|v\|_{S_0}\leq1\Rightarrow \|v\|^{p_+}_{S_0}\leq \rho_{p(x, y)}(v)\leq \|v\|^{p_-}_{S_0};$

(ii) $\|v\|_{S_0}\geq1\Rightarrow \| v\|^{p_-}_{S_0}\leq \rho_{p(x, y)}(v)\leq \| v\|^{p_+}_{S_0}.$

We will introduce a continuous compact embedding theorem as follows.

Theorem 2.5 (^[13]). Let $s\in(0, 1)$ and $p$ satisfy $(2.1)$ and $(2.2)$ with $sp_+ < d$. If $r$ satisfies $(1.4)$, i.e.,

Then we have

where $c$ is a positive constant depending on $p$, $s$, $r$, $d$ and $\Omega$. In other words, the embedding $S\hookrightarrow L^{r(x)}$ is continuous and this embedding is compact.

Remark 2.6. (i) Theorem $2.5$ still holds if we replace $S$ by $S_0$.

(ii) Since $1 < r_-\leq r(x) < p^\ast_s(x),$ then according to Theorem $2.5$, we can get that $\|\cdot\|_{S_0}$ and $\|\cdot\|_{S}$ are equivalent on $S_0$.

We need to introduce the functional $\mathcal{L}:S_0\rightarrow S_0^*$ defined by

for all $\varphi\in S_0$, where $S_0^*$ is the dual space of $S_0$.

Lemma 2.7 (^[23]). Suppose that $p$ satisfies $(2.1)$, $(2.2)$ and $s\in(0, 1)$. Then the following results hold:

(i) $\mathcal{L}$ is a bounded and strictly monotone operator;

(ii) $\mathcal{L}$ is a homeomorphism;

(iii) $\mathcal{L}$ is a mapping of type $(S_+)$, i.e., $v_n\rightarrow v$ in $S_0$, if $v_n\rightharpoonup v$ in $S_0$ and $\mathcal{L}$

3.   Proof of the main results

Definition 3.1. We say that $v\in S_0$ is a weak solution of problem $(P^s_M)$, if

for all $\varphi\in S_0$, where

For the purpose of formulating the variational method of problem $(P_M^s)$, we present the functional $I_\lambda:S_0\rightarrow \mathbb{R}$ given by

It is not tough to demonstrate that $I_\lambda\in C^1(S_0, \mathbb{R})$ and $I_\lambda$ is well defined. Moreover, for all $v, \varphi\in S_0$, the Gateaux derivative of $I_\lambda$ is introduced by

Thus, the weak solutions of $(P^s_M)$ correspond to the critical points of $I_\lambda$.

To prove Theorem 1.1, we need to introduce the next result:

Lemma 3.2. For $\lambda\in\mathbb{R}$, the functional $I_\lambda$ is coercive on $S_0$.

Proof. We assume $\|v\|_{S_0} > 1$. By $(M_2)$ we obtain that

According to Remark 2.6 (i), we have

where $c$ is a positive constant depending on $d, \, s, \, p, \, r$ and $\Omega$. It follows from (1.5) that

Next, we prove Theorem 1.1.

Proof of Theorem 1.1: According to Lemma 3.2, we know that $I_\lambda$ is coercive on $S_0$. Moreover, $I_\lambda$ is weakly lower semi-continuous on $S_0$. Applying Theorem 1.2 in ^[28], we find that there exists $\bar{v}_1\in S_0$ which is a global minimizer of $I_\lambda$; thus, the problem $(P_M^s)$ has a weak solution.

Now, we claim that the weak solution $\bar{v}_1$ is nontrivial for all $\lambda$ large enough. Indeed, let $\delta_0 > 1$ and $|\Omega_1| > 0$, where $\Omega_1$ is an open subset of $\Omega$. Suppose $\eta_0\in C^\infty_0(\overline{\Omega})$ such that $\eta_0(x)$ satisfies

Then we have that

where $c_3$ is a constant. Therefore, we get

if the nonnegative $\lambda_1$ is large enough. The proof is now complete.

We say that $I_\lambda$ satisfies $(Ce)_c$-condition for any $c\in\mathbb{R}$ if every sequence $\{v_n\}$ such that

has a strongly convergent subsequence in $S_0$. In order to prove Theorem 1.2, we need the symmetric mountain pass theorem as follows.

Theorem 3.3 (^[29,30]). For the infinite dimensional Banach space $S$, we define

where $S_2$ is finite dimensional. Suppose $I\in C^1(S, \mathbb{R})$, if $I$ satisfies the following

(1) $I(0) = 0, \, I(-v) = I(v)$ for all $v\in S$;

(2) for all $c > 0$, $I$ satisfies $(Ce)_c$-condition;

(3) suppose constants $\rho, a$ are positive, and we have $I|_{\partial B_\rho\cap Z}\geq a$;

(4) for each finite dimensional subspace $\tilde{S}\subset S$, we obtain $I(v)\leq0$ on $\tilde{S}\backslash B_r$, if $r = r(\tilde{S})$ is a positive constant.

Then $I$ possesses an unbounded sequence of critical values.

The following result shows that the functional $I_\lambda$ satisfies the geometrical condition of the mountain pass.

Lemma 3.4. Let $c_1 > 0$ and $v\in S_0$ with $\|v\|_{S_0} = \rho > 0$. Then for each $\lambda\in (0, c_1)$, we can choose $a > 0$ such that $I_\lambda|_{\partial B_\rho\cap Z}\geq a$.

Proof. Suppose $v\in S_0$ and $\rho\in(0, 1)$ such that $\|u\|_{S_0} = \rho$. It follows from Theorem 2.5, $(M_1)$, (1.6) and (1.7) that

Thus, choosing $\rho$ even smaller, we have

since $\alpha p_+ < r_- < r_+$.

Lemma 3.5. For each finite dimensional subspace $\widetilde{S}\subset S_0$, $v\in \widetilde{S}$ and $\lambda\in\mathbb{R}$, there exists $r = r(\widetilde{S}) > 0$ such that

where $\|v\|_{S_0}\geq r$.

Proof. Suppose $\phi\in C_0^\infty(\Omega)$ with $\phi > 0$. According to ($M_1$), one must have

Therefore, by (3.1), we have

It follows from (1.6) that

Thus, there exists a large $t > 1$ such that

The statement holds.

Lemma 3.6. The functional $I_\lambda$ satisfies condition $(Ce)_c$ in $S_0$.

Proof. For all $c\in \mathbb{R}$, suppose a sequence $\{v_n\}\subset S_0$ such that

First, we claim that $\{v_n\}\subset S_0$ is bounded. Arguing by contrary, passing eventually to a subsequence, still denote by ${v_n}$, we assume that $\lim\limits_{n\rightarrow +\infty}\|v_n\|_{S_0} = +\infty$. Hence, for all $n$, we can consider that $\|v_n\|_{S_0} > 1$. According to (3.2), Lemma 2.4, $(M_1)$ and $(M_2)$, we get that

when $n$ is large enough. Dividing the above inequality by $\|v_n\|_{S_0}$. According to $\alpha p_+ < r_- < r_+$, we have $-(\frac{\lambda}{r_-}-\frac{\lambda}{\alpha p_+}) > 0$. By passing to the limit as $n\rightarrow +\infty$, we can get a contradiction.

Since $S_0$ is reflexive, then we can assume that $v_n\rightharpoonup\bar{v}$ in $S_0$. It follows from (3.2) that

that is

Moreover, due to $r(x) < p_s^*(x)$ for all $x\in\overline{\Omega}$ and Remark 2.6 (i), we can conclude that $v_n\rightarrow \bar{v}$ in $L^{r(x)}$. Hence according to Lemma 2.1 and the proof of Theorem 3.1 in ^[14], we have that

Since $\{v_n\}$ is bounded in $S_0$, if necessary, we can suppose that

If $t_2 = 0$, then $\{v_n\}\rightarrow \bar{v} = 0$ in $S_0$ and the proof is complete. If $t_2 > 0$, according to the continuous function $M$, we have

Thus, for $n$ large enough and $M\in\mathcal{Q}_2$, we get that

Combining (3.3)–(3.5), we deduce that

Using (3.6), Lemma 2.7 (iii) and $v_n\rightharpoonup\bar{v}$ in $S_0$, we obtain that

Moreover, according to (3.2), we have

This completes the proof of Lemma 3.6.

Proof of Theorem 1.2: Clearly, $I_\lambda(0) = 0$ and $I_\lambda(-v) = I_\lambda(v)$. According to Theorem 3.3 and Lemmas 3.4–3.6, we deduce that Theorem 1.2 holds.

Acknowledgments

This project was supported by the Hunan Provincial Natural Science Foundation (Nos. 2022JJ40145 and 2022JJ40146).

Conflict of interest

The authors declare that there is no conflict of interest.