Existence results of nontrivial solutions for a new $p(x)$-biharmonic problem with weight function

1.   Introduction

In this paper, we consider the following $p(x)$-biharmonic problem with weight function

where $a\geq b > 0$ are constants, $\Omega\subset\mathbb{R}^N(N\geq3)$ is a bounded domain with smooth boundary $\partial\Omega$, $p$, $q$$\in C(\overline{\Omega})$ with $1 < p^-: = \inf\limits_{x\in\overline{\Omega}}p(x)\leq p^+: = \sup\limits_{x\in\overline{\Omega}}p(x) < N$, $\lambda > 0$ is a real parameter, and $\Delta^2_{p(x)}u = \Delta(|\Delta u|^{p(x)-2}\Delta u)$ is called the $p(x)$-biharmonic operator of fourth order.

Recently, the problems of variable exponent growth have been extensively studied, we can refer to ^{[1,5,7,10,11,14,16,25]}. Literature ^[41] and ^[36] are the two closest in time, in literature ^[41], the authors studied the regularity for minimizers for functionals of double phase with variable exponents, the main purpose of this paper is to provide a regularity theorem for minimizers of a class of integral functionals of the calculus of variations called of double phase type with variable exponents, and authors also proved the existence of a local minimum value of the problem. It is worth mentioning that the existence and multiplicity results of solutions for many problems with variable exponents can be found in ^[36]. In particular, variable exponent space plays an important role in the studies of electrorheological fluids ^[42,43], thermotropic fluids ^[8], and image processing ^[2,12]. The major difference with the constant exponent is that the $p(x)$-biharmonic operator has inhomogeneity, it brings a lot of difficulties, such as many classical theorems, like the Lagrange multiplier theorem, can't be used. The study of variable exponent problems can be traced back to the article published by Orlic in 1931 ^[40]. However, after this article, the authors did not go further. During this period, many authors carried out a series of research work ^[38,39], such as module space ^[37], etc., and obtained many important properties. Until the 1990s, a lot of work on variable exponent space was carried out on the basis of an article published by two authors, O. Kovacik and J. Rakosnik in 1991, in which the basic theory of variable exponent Lebesgue space and Sobolev space ^[27] is established. Later, authors such as Fan and Zhao ^[21] further promoted it. Up to now, more and more people have studied it. In addition, Kirchhoff differential equation, as a typical partial differential equation, has been extensively studied, and its source can be traced back to 1883 ^[30]. Kirchhoff type differential equation, a special partial differential equation, is a mathematical model proposed by Kirchhoff in 1883 when he studied the free vibration of elastic string. Such model was used to study the variation rule of transverse vibration of telescopic steel wire rope. It is well known that Kirchhoff have created the following model and generalized the d 'Alembert wave equation

where $\rho, \rho_0, h, E, L$ are constants, the problem (1.1) is related to the stationary problem of Kirchhoff's model. Its theoretical and practical applications are very broad, such as Non-Newtonian mechanics, medicine, economics, ecology and other fields are involved. The original Kirchhoff's model was utilized to study the problems in one-dimensional case. In 1955, Berger ^[9] studied the following von Karman plate equation in two-dimensional case

where $\Delta^2$ is called the biharmonic operator. It also plays an important role in engineering, physics and material mechanics. Therefore, in recent decades, researchers have also done a lot of researches on the existence and properties of solutions of biharmonic equations, for example, in ^[4], the authors used Nehari manifold method and fiber mapping to obtain the existence of two solutions. Furthermore, many authors have extended the $p$-biharmonic problems to the $p(x)$-biharmonic problems, we refer to ^{[3,6,17,24,28,29,31,35]}.

In ^[17], the author considered the following problem

where $\Omega\in\mathbb{R}(N\geq2)$ is a bounded domain with a smooth boundary $\partial\Omega$, $p$ is a continuous function on $\overline{\Omega}$ such that $1 < p^-: = \inf\limits_{x\in\overline{\Omega}}p(x)\leq p^+: = \sup\limits_{x\in\overline{\Omega}}p(x) < +\infty$, the continuous function $M:\mathbb{R}^+\rightarrow \mathbb{R}^+$ and the Carath$\acute{e}$dory function $f:\Omega\times\mathbb{R}$ satisfy some conditions. They established the existence and multiplicity of solutions for problem (1.3) by using variational method and the theory of the variable exponent Sobolev spaces.

Replacing $\left(a-b\int_\Omega\frac{1}{p(x)}|\Delta u|^{p(x)}dx\right)\Delta^2_{p(x)}u$ with $(\Delta(|x|^{p(x)}|\Delta u|^{p(x)-2}\Delta u)$ on the left-hand side of problem (1.1), in ^[35], existence results for the following perturbed weighted $p(x)$-biharmonic problem with Navier boundary conditions were considered. The author proved the problem has an eigenvalue when there exists $\lambda^* > 0$ such that $\lambda\in(0, \lambda^*)$ and some other hypotheses are satisfied. The other main result of this problem is at least two non-trivial non-negative weak solutions were obtained provided $|V|_{L^\infty(\Omega)} < M$.

In ^[29], the author considered a class of $p(x)$-biharmonic operators with weights

where $\lambda, \mu$ are positive real numbers, $p_1, p_2, q$ and $\alpha$ are continuous functions on $\overline{\Omega}$, $V_1$ and $V_2$ are weight functions in generalized Lebesgue spaces $L^{s_1(x)}(\Omega)$ and $L^{s_2(x)}(\Omega)$ such that $V_1$ may change sign in $\Omega$ and $V_2 > 0$ on $\Omega$, respectively. The author established the existence results by using variational approaches and Ekeland's variational principle.

In 2020, a class of new problems with nonlocal term were considered in ^[26]

where the nonlocal term is $a-b\int_\Omega\frac{1}{p(x)}|\nabla u|^2dx$. The key point in the main results of this literature is to show that the energy functional $J$ has a Mountain Pass energy $c$. For this type of problems, they have been extensively studied as constant exponent case when $p(x) = 2$, we can refer to ^[13,32,33]. It is worth mentioning that concerning the Kirchhoff type problem, the authors studied the $a+b$ type problem in literature ^[44,45]. Precisely, Tang and Cheng in ^[44] proposed a new approach to recover the compactness for the Palais-Smale sequences; Moreover, Tang and Chen in ^[45] proposed a new approach to recover the compactness for the minimizing sequences.

Inspired by the above literature ^{[17,26,29,35]}, in this paper, we study the existence and multiplicity of solutions for a class of $p(x)$-biharmonic problems with weight function and negative nonlocal term.

To prove our first main result, as shown below, we assume the following assumptions:

($M_0$) There exists a function $s\in C(\overline{\Omega})$ for all $x\in{\overline{\Omega}}$, then $1 < q(x) < \alpha(x) < p(x)\leq \frac{N}{2} < s(x)$ hold.

($M$) $V\in L^{s(x)}(\Omega)$, and $V > 0$ in $\Omega_0\subset\subset\Omega$, with $|\Omega_0| > 0.$

($M_1$) Assume that $1 < \alpha < \min\{\frac{n}{p^+}, \frac{np^-}{p^+(n-p^-)}\}$ and $V:\Omega\rightarrow [0, \infty)$ belongs to $L^\infty(\Omega)$, denote

($M_2$) For some $x_0\in\Omega$ and $0 < r < R < \infty$ with $\overline{B_R(x_0)}\subset\Omega$ we have

($M_3$) $q\in C_+(\overline{\Omega})$ is such that $1 < q(x) < p^*(x) \ \ \forall x\in\overline{\Omega}$ and

The theorem to be proved is as follows:

Theorem 1.1. Assume that $a\geq b > 0$, $p$, $q$$\in C(\overline{\Omega})$, ($M$) and ($M_1$)–($M_3$) are satisfied and

Then there exist constants $\lambda_\ast, \ \zeta > 0$, such that for any $\lambda\in(0, \lambda_\ast)$, problem (1.1) has at least three non-trivial solutions provided $|V|_{L^\infty(\Omega)} < \zeta$.

Remark 1.2. Compared with the most of existing work involving $p(x)$-biharmonic operaters, the main difference between them and our problem is that the equation in the present paper contains negative nonlocal term and weight function. In addition, different from other papers ^{[3,24,28,31,35]} dealing with $p(x)$-biharmonic operators, the present paper gives the existence of three nontrivial solutions.

Theorem 1.1 cannot be completed without the proof of the $(PS)_c$ condition, and since problem (1.1) contains a negative nonlocal term, the general method cannot be adopted. Therefore, to complete the proof and solve this difficulty, we use the method in ^[35]. In addition, a class of problems (4.1), with nonlinear terms and negative nonlocal terms are considered, which is a supplement to problem (1.1).

In Section 2, we introduce the preliminary knowledge that will be involved later, and prove the $(PS)_c$ condition. In Section 3, Theorem 1.1 is proved by proving some lemmas. In Section 4, we supplement the problem (1.1), and an infinite number of solutions are obtained by using the symmetric mountain pass theorem.

2.   Preliminaries and functional framework

In order to investigate Eq (1.1) we first review some conclusions about the generalized Lebesgue-Sobolev spaces $W^{1, p(x)}(\Omega)$ and $L^{1, p(x)}(\Omega)$. Let $\Omega$ be a bounded domain of $\mathbb{R}^N$, denote $C_+(\overline{\Omega}) = \{p(x):p(x)\in C(\overline{\Omega}), p(x) > 1, \ \mathrm{for \ all} \ x\in {\overline{\Omega}}\}$ and $p^- = \inf_\Omega p(x)\leq p(x)\leq p^+ = \sup_\Omega p(x) < N$.

For any $p\in C_+(\overline{\Omega})$, we introduce the Lebesgue space of variable exponent endowed with the Luxembourg norm

The Luxembourg norm is as follows

which is a separable and reflexive Banach space. We can refer to ^[21] for more relevant knowledge.

No doubt, when $p(x)\equiv p$, the space $L^{p(x)}(\Omega)$ is reduced to the classical Lebesgue space $L^p(\Omega)$ and the norm $|u|_{p(x)}$ reduces to the standard norm in $L^p(\Omega)$

For any positive integer $k$ and a given multi-index $\alpha = (\alpha_1, \alpha_1, ..., \alpha_n)$, we define the generalized Sobolev space as

where $|\alpha| = \Sigma^n_{i = 1}\alpha_i$ and $D^\alpha u = \partial^{|\alpha|}u/\partial^{\alpha_1}x_1\cdot\cdot\cdot\partial^{\alpha_N}x_N$.

Endowed with the following norm

We define the space $W^{k, p(x)}_0(\Omega)$ as the closure of $C^\infty_0(\Omega)$ in $W^{k, p(x)}(\Omega)$, with respect to the norm $||\cdot||_{k, p(x)}$. They are separable and reflexive Banach spaces, in addition, we also have similar properties, and the following are some propositions concerning these properties, we refer the reader to ^[22,27,40].

Proposition 2.1. (See ^[21], Pages 430–431) The space $(L^{p(x)}(\Omega), |.|_{p(x)})$ is separable, uniformly convex, reflexive, and its conjugate space is $(L^{p'(x)}(\Omega), |.|_{p'(x)})$, where $p'(x)$ is the conjugate function of $p(x)$, i.e,

For all $u\in L^{p(x)}(\Omega), v\in L^{p'(x)}(\Omega)$, the Hölder type inequality

holds.

Proposition 2.2. (See ^[27], Theorem 2.1) Let $p, q, s:\Omega\rightarrow [1, \infty]$ be measurable functions such that

Let $f\in L^{p(x)}(\Omega), g\in L^{q(x)}(\Omega)$. Then, $fg\in L^{s(x)}(\Omega) \ with$

Proposition 2.3.(See ^[20], Proposition 2.5) Let $h_1, h_2$ and $h_3:\Omega\rightarrow (1, \infty)$ be Lipschitz continuous functions such that $1/h_1(x)+1/h_2(x)+1/h_3(x) = 1$, then for any $u\in L^{h_1(x)} (\Omega), v\in L^{h_2(x)}(\Omega)$ and $w\in L^{h_3(x)}(\Omega)$ the following inequality holds

Proposition 2.4.(See ^[21], Theorem 1.11) Let $r, s:\Omega\rightarrow [1, \infty)$ be measurable functions such that $r(x)\leq s(x)$, then the embedding $L^{s(x)}(\Omega)\hookrightarrow L^{r(x)}(\Omega)$is continuous.

Proposition 2.5. (See ^[21], Theorem 2.3) For $p, r\in C_+(\overline{\Omega})$ such that $r(x)\leq p^*_k(x)$ for all $x\in {\overline{\Omega}}$, there is a continuous embedding

If we replace $\leq$ with $<$, the embedding is compact.

Proposition 2.6. (See ^[18], Lemma 2.1) Let $p$ and $q$ be measuable functions such that $p\in L^\infty(\Omega)$, and $1\leq p(x)q(x)\leq\infty$, for a.e. $x\in\Omega$. Let $u\in L^{q(x)}(\Omega), u\not\neq0$. Then

Definition 2.1. (See ^[28], Definition 2.3) Assume that spaces $E$, $F$ are Banach spaces, we define the norm on the space $X: = E\bigcap F$ as $||u||_X = ||u||_E+||u||_F$, $X^*$ its dual space and $< \cdot, \cdot >$ denote the duality product.

We also need some properties about the space $X: = W^{1, p(x)}_0(\Omega)\bigcap W^{2, p(x)}(\Omega).$ Denote by $W^{k, p(x)}_0(\Omega)$ the closure of $C^\infty_0(\Omega)$ in $W^{k, p(x)}(\Omega)$. Note that the weak solutions of problem (1.1) are considered in the generalized Sobolev space $X$ endowed with the norm

According to ^[23], the norm $||u||_{2, p(x)}$ is equivalent to the norm $|\Delta u|_{p(x)}$ in the space $X$. Consequently, the norm $||u||_{2, p(x)}, |\Delta u|_{p(x)}$ and $||u||$ are equivalent.

We define the relevant modular on the space $L^{p(x)}(\Omega)$ and give the basic properties needed in this paper as following

Proposition 2.7.(See ^[6], Proposition 3.2) Suppose that $u_n, u\in X$ and $p_+ < \infty$. Then the following properties hold:

(1) $||u|| < 1( = 1, > 1)\Leftrightarrow \rho(u) < 1( = 1. > 1)$;

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

(3) $||u|| < 1\Rightarrow ||u||^{p^+}\leq\rho(u)\leq||u||^{p^-}$;

(4) $||u||\rightarrow 0(\rightarrow \infty)\Leftrightarrow\rho(u)\rightarrow 0(\rightarrow \infty)$.

Moreover, we define

Let for any $u\in X, L(u) = \int_\Omega\frac{1}{p(x)}|\Delta u|^{p(x)}dx$, and $A: = L':X\rightarrow X^*$, then

Lemma 2.1. (See ^[6], Theorem 3.4)

(1) $L':X\rightarrow X^*$ is a bounded homeomorphism and strictly monotone operator.

(2) $L'$ is a mapping of type $S_+$, namely, if $u_n\rightharpoonup u$ and $\limsup\langle L(u_n)-L(u), u_n-u\rangle\leq0$, imply $u_n\rightarrow u$ (strongly) in $X$.

Remark 2.1. (See ^[28], Remark 3.2) We denote by $s'(x)$ the conjugate of the functions $s(x)$, $r(x): = \frac{s(x)q(x)}{s(x)-q(x)}$. Then the following embedding properties hold.

Under assumption ($M_0$) and ($M$), we have $\max(r(x), s'(x)q(x)) < p^*(x)$, for all $x\in\overline{\Omega}$. It follows that the embeddings $X\hookrightarrow L^{s'(x)q(x)}(\Omega)$ and $X\hookrightarrow L^{r(x)}(\Omega)$ are compact and continuous.

Under Remark 2.1, we have for all $u\in X$

and

A function $u\in X$ is a weak solution of (1.1), if

where $\varphi\in X$.

Next, we will prove Theorem 1.1 using the variational method. Define the energy functional $J:X\rightarrow \mathbb{R}$ associated with problem (1.1) by

for all $u\in X$ is well defined and of $C^1$ class on $X$. Also, we have

for all $u, v\in X$. Therefore, we can find the weak solutions of problem (1.1) as the critical points of functional $J$. To simplify the notation, we are going to represent the norm $X$ by $||.||$ instead of $||.||_X$.

Related definitions of $(PS)_c$ have been given in Definition 3.1 of literature ^[26], where if any sequence $u_n\in X$ satisfying:

has a convergent subsequence, it is said to satisfy the $(PS)_c$ condition.

Next, we prove that the functional $J$ satisfies the compactness condition.

Lemma 2.2. Assume that $a\geq b > 0, p, q\in C(\overline{\Omega})$, then the functional $J(u)$ satisfies the $(PS)_c$ condition at the level $c < \frac{a^2}{2b}$.

Proof. The proof we're going to do is divided into two steps.

Step 1. We prove that $\{u_n\}$ is bounded in $X$. Let $\{u_n\}\subset X$ be a $(PS)_c$ sequence for $J$ such that $c < \frac{a^2}{2b}$.

Our main tool in this step is based on proving by contradiction, we assume that, passing eventually to a subsequence, still denote by $\{u_n\}$, we have $||u_n||\rightarrow \infty$ as $n\rightarrow \infty$. By (2.4), for $n$ large enough, we can get

By the above last inequality, we have

Dividing the above inequality by $||u_n||^{2^{p^-}}$, and since (1.4) holds, passing to the limit as $n\rightarrow \infty$, we get a contradiction. Thus, $\{u_n\}$ is bounded in $X$.

Step 2. In the second part, we will show that $\{u_n\}$ has a convergent subsequence in $X$. According to Proposition 2.5, the embedding

is compact, where $1\leq h(x) < p^*(x)$. Passing, if necessary, to a subsequence, there exists $u\in X$ such that

From Hölder's inequality and (2.7), we can inferred that

Thus,

Due to (2.6), we have

Therefore

Hence, we can infer from (2.8) that

Since $\{u_n\}$ is bounded in $X$, passing to a subsequence, if necessary, we may assume that

If $t_0 = 0$, then $\{u_n\}$ strongly converges to $u = 0$ in $X$ and the proof is finished. If $t_0 > 0$, we will discuss the following two cases respectively:

Case 1. If $t_0\neq \frac{a}{b}$ then $a-b\int_\Omega\frac{1}{p(x)}|\Delta u_n|^{p(x)}dx\rightarrow 0$ is not true and no subsequence of $\{a-b\int_\Omega\frac{1}{p(x)}|\Delta u_n|^{p(x)}dx\}$ converges to zero. Therefore, there exists $\delta > 0$ such that $\left|a-b\int_\Omega\frac{1}{p(x)}|\Delta u_n|^{p(x)}dx\right| > \delta > 0$ when $n$ is large enough. Obviously it is concluded that

Case 2. If $t_0 = \frac{a}{b}$ then $a-b\int_\Omega\frac{1}{p(x)}|\Delta u_n|^{p(x)}dx\rightarrow 0.$

Set

Then

From the above equation, we have

We complete the proof with such a lemma:

Lemma 2.3. Let $u_n, u\in X$ such that (2.7) holds. Then, passing to a subsequence, if necessary, the following properties hold:

$(i) \int_\Omega V(x)(|u_n|^{q(x)-2}u_n-|u|^{q(x)-2}u)vdx\rightarrow 0$ ($n\rightarrow \infty$);

$(ii) \langle\varphi'(u_n)-\varphi'(u), v\rangle\rightarrow 0$, ($n\rightarrow \infty$) $v\in X$.

Proof. Due to (2.7), we have $u_n\rightarrow u$ in $L^{q(x)}(\Omega)$. Then we get

So from the Hölder inequality, we have

The proof of (ii) can also be obtained by slightly modifying the proof above. So we will not prove it in detail here. So we end up with both $||\varphi'(u_n)-\varphi'(u)||_{X^*}\rightarrow 0$ and $\varphi'(u_n)\rightarrow \varphi'(u)$.

Next, we will make the proof for Case 2:

According to Lemma 2.3 and since

and

so there are $\varphi'(u_n)\rightarrow 0 \ (n\rightarrow \infty)$, i.e.,

and so

by the fundamental lemma of the variational method (see^[47]). It follows that $u = 0$. So

So when $t_0 = \frac{a}{b}$, we have

This is a contradiction since $J(u_n)\rightarrow c < \frac{a^2}{2b}$, then $a-b\int_\Omega\frac{1}{p(x)}|\Delta u_n|^{p(x)}dx\rightarrow 0$ is not true. And it's similar to Case 1, we have that

From what has been discussed above, it can be inferred that

Thus, from Lemma 2.1, we can deduce that $u_n\rightarrow u$ strongly in $X$ as $n\rightarrow \infty$, which implied that $J$ satisfies the $(PS)_c$ condition.

3.   Proof of Theorem 1.1

We prove the conditions for satisfying the Mountain Pass theorem (see ^[47]) by proving the following lemma.

Lemma 3.1. The following assertions hold:

(i) There exists $\zeta > 0$ such that for any $|V|_{L^\infty(\Omega)} < \zeta$, there exist $\rho, \gamma > 0$ such that $J(u)\geq\gamma, \forall u\in X$ with $||u|| = \rho$;

(ii) There exists $\psi\in X, \psi > 0$ such that $\lim\limits_{t\rightarrow \infty}J(t\psi) = -\infty$;

(iii) There exists $\omega\in X, \omega > 0$ such that $J(t\omega) < 0$ for all $t > 0$ small enough.

Proof. ($i$) Define the function $q_1:\overline{B_r(x_0)}\rightarrow (1, +\infty)$ by $q_1(x) = q(x)$ $\forall x\in\overline{B_r(x_0)}$ and the function $q_2:\overline{\Omega\setminus B_R(x_0)}\rightarrow (1, \infty)$ by $q_2(x) = q(x)$ $\forall x\in\overline{\Omega\setminus B_R(x_0)}$.

Set $q^-_1 = \min\limits_{x\in\overline{B_r(x_0)}}q_1(x), \ q^+_1 = \max\limits_{x\in\overline{B_r(x_0)}}q_1(x), \ q^-_2 = \min\limits_{x\in\overline{\Omega\setminus B_R(x_0)}}q_2(x)$ and $q^+_2 = \min\limits_{x\in\overline{\Omega\setminus B_R(x_0)}}q_2(x)$. Due to the condition ($M$), we have

Thus, $X$ is continuously embedded in $L^{q_i^\pm}$ ($i = 1, 2$).

Then, there exist two positive constants $C_1, C_2$ such that

and

The inequalities (3.1) and (3.2) have been proved in ^[15].

Applying the inequalities and Propositions described above, we obtain

for all $u\in X$ with $||u|| < 1$. Therefore, we infer that any $\zeta$ satisfies

($ii$) Let $\psi\in C^\infty_0(\Omega), \psi\geq0$ and let $x_1\in\Omega\setminus B_R(x_0)$ and $\varepsilon > 0$ such that $\psi(x) > 0$ for any $x\in B_\varepsilon(x_1)\subset \Omega\setminus B_R(x_0)$. For any $t > 1$, we have

Since $\alpha p^+ < q^-_2$ we infer that $\lim\limits_{t\rightarrow \infty}J(t\psi) = -\infty$.

($iii$) Let $\omega\in C^\infty_0(\Omega), \omega\geq0$ and there exist $x_2\in B_r(x_0)$ and $\varepsilon > 0$ such that for any $x\in B_\varepsilon(x_2)\subset B_r(x_0)$ we have $\omega(x) > 0$. Let $0 < t < 1$, we obtain

Thus, we have $J(t\omega) < 0$ for any $0 < t < \delta^{\frac{1}{\alpha p^–q^+_1}}$; where

By using Ekeland variational principle in reference ^[19], we can similarly reach the following conclusions:

There exists a sequence $\{u_n\}\subset B_\rho(0)$ such that

It has been proved by Lemma 2.2, the sequence $\{u_n\}$ converges strongly to some $u_2$ as $n\rightarrow \infty$. Moreover, since $J\in C^1(X, \mathbb{R})$, by (3.3) it follows that $J'(u_2) = 0$. Thus, $u_2$ is a non-trivial weak solution of problem (1.1) with negative energy $J(u_2) = \underline{c} < 0$.

Finally, since $J(u_1) = \overline{c} > 0 > \underline{c} = J(u_2)$, we can point out the fact that $u_1\neq u_2$.

Let

then $M < +\infty$, hence

Applying Ekland's variational principle on space $X$ for $-J(u)$, there exists the $(PS)_{-M}$ sequence of $-J(u)$, so there is the $(PS)_M$ sequence of $J(u)$.

Since there is a global minimum, we can define a minimization sequence and prove its convergence. And then we get our third solution.

4.   Final comments

When $\lambda V(x)|u|^{q(x)-2}u$ is replaced by $\lambda f(x, u)$, the problem (1.1) becomes

where $f$ satisfies the following properties:

($f_1$) $f:\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ satisfies the Carath$\acute{e}$dory condition and there exists a constant $C_1\geq0$ such that

for all $(x, t)\in\Omega\times\mathbb{R}$ with $q(x)\in C_+(\overline{\Omega})$ and $q(x) < p^*_2(x)$ for all $x\in\overline{\Omega}$ where $p^*_2(x): = \frac{Np(x)}{N-2p(x)}$ if $p(x) < \frac{N}{2}$, $p^*_2(x) = \infty$ if $p(x)\geq\frac{N}{2}$.

($f_2$) There exists $k > 0, \theta > \frac{2(p^+)^2}{p^-}$ such that for all $x\in\Omega$ and all $t\in\mathbb{R}$ with $|t|\geq k$,

where $F(x, t) = \int^t_0f(x, s)ds.$

($f_3$) $f(x, t) = o(|t|^{2p^+-1})$ as $t\rightarrow 0$ uniformly with respect to $x\in\Omega$, with $q^- > 2(p^+)^2$.

($f_4$) $f(x, -t) = -f(x, t)$ for all $x\in\Omega$ and $t\in\mathbb{R}$.

Theorem 4.1. When $\lambda V(x)|u|^{q(x)-2}u$ is replaced by $\lambda f(x, u)$, if ($f_1$), ($f_2$), ($f_4$) hold, and $q^- > 2(p^+)^2 > 2p^+$, then problem (4.1) has a sequence of weak solutions $\{\pm u_k\}$ such that $I(\pm u_k)\rightarrow -\infty$ as $k\rightarrow +\infty$.

By referring to the methods and conditions in ^[17], we can obtain the existence of two related theorems for this problem, the main conclusion is that by using the symmetric mountain pass theorem, we can conclude that there are a series of weak solutions for the problem (4.1). The method of proving $(PS)_c$ condition is similar to Lemma 2.2.

Definition 4.1. A function $u\in X$ is a weak solution of (4.1), if

where $\varphi\in X$.

Define the energy functional $I:X\rightarrow \mathbb{R}$ associated with problem (4.1) by

for all $u\in X$, $I(u)$ is well defined and of $C^1$ class on $X$. Moreover, we have

for all $u, v\in X$.

Lemma A. (Symmetric mountain pass theorem, See ^[47]) Let $X = Y\bigoplus Z$ be an infinite dimensional Banach space, where $Y$ is finite dimensional, and $I\in C^1(X, \mathbb{R})$ satisfies the $(PS)_c$ condition as well as the following properties:

($i$) $I(u) = 0$ and there exist two constants $r, \alpha > 0$ such that $I|_{\partial B_r}\geq \alpha$.

($ii$) $I$ is even.

($iii$) for all finite dimensional subspace $\widetilde{X}\subset X$ there exists $R = R(\widetilde{X}) > 0$ such that $I(u)\leq 0$ for all $u\in X\setminus B_R(\widetilde{X})$, where $B_R(\widetilde{X}) = \{u\in\widetilde{X}:||u|| < R\}$.

Then $I$ has an unbounded sequence of critical points.

Proof of Theorem 4.1. The method and steps of proving that problem (4.1) satisfies the condition of $(PS)_c$ are consistent with Lemma 2.2, from ($f_1$), Propositions 2.1 and 2.5, we deduce easily that

Hence, we can deduce that $I$ satisfies the $(PS)_c$ condition.

Due to ($f_4$), and $I$ is an even functional and satisfies the $(PS)_c$ condition. We will show that $I$ satisfies the conditions of Lemma A.

($i$) Obviously, $I(0) = 0$. Since $p^+ < (p^+)^2 < q^- < q(x) < p^*_2(x), \ X\hookrightarrow L^{2p^+}(\Omega), \ X\hookrightarrow L^{q(x)}(\Omega)$ and then there exist $C_3, C_4 > 0$ such that

By ($f_1$) and ($f_3$), we have

Let $r\in(0, 1)$ and $u\in X$ be such that $||u|| = r$. Thus, by considering (4.4), Propositions 2.5 and 2.7, hypothesis that $1 < p^- < p(x) < p^+ < 2p^- < q^- < q(x) < p^*(x)$, we have

Set

so for any $\lambda\in(0, \lambda^*)$, there exists $\alpha > 0$ such that $u\in X$ with $||u|| = r$, we have $I(u)\geq\alpha > 0$.

($ii$) It is clear that $I$ is even.

($iii$) By($f_2$), we have that

Let $R = R(\widetilde{X}) > 1$, for all $u\in\widetilde{X}$ with $||u|| > R$. By (4.5), we have

Thus, all norms on the finite dimensional space $\widetilde{X}$ are equivalent, so there exists $C_W > 0$ such that

Therefore, we obtain

Due to $\theta > p^+$, it follows that for some $||u|| > R$ large enough, we can infer that $I(u)\leq0$. Therefore, the conclusion of Theorem 4.1 can be obtained by using the symmetric mountain pass theorem.

This paper can be regarded as an extension of literature ^[26]. In addition, for this kind of negative nonlocal term problem, in recent literature ^[46], the author studied a class of variable exponent $p(x)$-Kirchhoff type problem with convection. In this paper, Galerkin method, pseudomonotone operators and a fixed-point argument were used to study the existence of the solution to this problem. In ^[34], the author studied the multiplicity results of a class of $p(x)$-Choquard equations with nonlocal and nondegenerate Kirchhoff terms by using truncation argument and Krasnoselskiis genus method.

5.   Conclusions

In this paper, we study a class of $p(x)$-biharmonic problems with negative nonlocal terms and weight function. Using the mountain pass theorem and the Ekeland's variational principle, we proved the existence of at least three solutions. We also proved the existence of infinite solutions when the nonlinear term is a general function.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 11661021, 11861021); Science and Technology Foundation of Guizhou Province (No. KY[2017]1084) and Natural Science Research Project of Guizhou Minzu University (No. GZMUZK[2021]YB20). Authors are grateful to the referees for their very constructive comments and valuable suggestions.

Conflict of interest

All authors declare no conflicts of interest in this paper.