1 Introduction

In this paper, we consider the following biharmonic system

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^{2}_{p_i(x)}( u_i)+b_i(x)\dfrac{|u_i|^{s_i-2}u_i}{|x|^{2s_i}}=\lambda F_{u_{i}}(x, u_1, \ldots , u_n) &{}\quad in\; \Omega , \\ u_i=\Delta u_i=0 &{}\quad on \;\partial \Omega \end{array} \right. \end{aligned}$$
(1.1)

for \(i=1,\ldots ,n\), where \(\Omega\) is a bounded domain in \({{\mathbb {R}}^{N}}(N\ge 2)\) with smooth boundary and \(\Delta ^2_{p_i(x)}u:=\Delta (\vert \Delta u\vert ^{p_i(x)-2}\Delta u)\) for each \(p_i\in C({\overline{\Omega }})\), \(i=1,\ldots ,n\). For \(i=1,\ldots ,n\), we assume that \(1<s_i<\frac{N}{2}\) and nonnegative function \(b_i\) belongs to \(L^\infty (\Omega )\). \(\lambda\) is a positive parameter and the function

$$\begin{aligned} F:\Omega \times {\mathbb {R}}^n\rightarrow {\mathbb {R}} \end{aligned}$$

is a measurable function with respect to \(x\in \Omega\) for each \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) and is \(C^1\) with respect to \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) for a.e. \(x\in \Omega\); \(F_{u_i}\) denotes the partial derivative of F with respect to \(u_i\).

Problems involving biharmonic operator arise in the study of traveling waves in suspension bridge and the study of static deflection of plate.

Singular boundary value problems arise in the context of chemical heterogeneous catalysts and chemical catalyst kinetics, in the theory of heat conduction in electrical conducting materials, as well as in the study of non-Newtonian fluids and boundary layer phenomena for viscous fluids.

In the last years, study of the biharmonic problems in various spaces is one of the interesting objects; for instance, authors of [10] studied the following fourth-order elliptic problem with Navier boundary conditions

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^{2}_{p}u +\frac{u|u|^{p-2}}{|x{|^{2p}}}= \lambda f(x,u)&{}\quad {\text{in}}\; \Omega , \\ u =\Delta u= 0&{}\quad {\text{on}}\; \partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Delta ^{2}_{p}u= \Delta |\Delta u|^{p-2} \Delta u\) denotes the p-biharmonic operator, \(\Omega\) is a bounded domain in \({\mathbb {R}}^N\) \((N\ge 5)\) containing the origin and with smooth boundary, \(1<p<\frac{N}{2}\), \(\lambda >0\) is a parameter and \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function such that

$$\begin{aligned} |f(x,t)|\leqslant a_1+a_2|t|^{q-1} \qquad \forall (x,t) \in \Omega \times {\mathbb {R}} \end{aligned}$$

for some non-negative constants \(a_1\), \(a_2\) and \(q \in ]p,p^*[\), where

$$\begin{aligned} p^*=\frac{pN}{N-2p}. \end{aligned}$$

The existence of the solutions to the following weighted (p(x), q(x))-Laplacian problem consisting of a singular term

$$\begin{aligned} \left\{ \begin{array}{ll} -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+\frac{u|u|^{s-2}}{|x{|^s}}= \lambda f(x,u)&{}\quad {\text{in}}\; \Omega , \\ u = 0&{}\quad {\text{on}}\; \partial \Omega \end{array} \right. \end{aligned}$$

have been proved [11], where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain with smooth boundary, \(a,b\in L^\infty (\Omega )\) are positive functions with \(a(x)\ge 1\) a.e. on \(\Omega\); \(\lambda >0\) is a real parameter, \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function satisfying the following growth condition

$$\begin{aligned} |f(x,t)|\le \alpha +\beta |t|^{h(x)-1} \end{aligned}$$

for all \((x,t)\in \Omega \times {\mathbb {R}}\) (see also [1, 17] and the references therein). Recently the existence of at least one positive radial solution of the weighted p-biharmonic problem

$$\begin{aligned}&\Delta _{{\mathbb {H}}^n}\big (w(\xi )|\Delta _{{\mathbb {H}}^n}u|^{p-2}\Delta _{{\mathbb {H}}^n}u\big )+ R (\xi )w(\xi ) |u|^{p-2} u\\&\quad =\sum _{i=1}^ma_i (|\xi |_{{\mathbb {H}}^n}) |u|^{q_i-2} u -\sum _{j=1}^k b_j (|\xi |_{{\mathbb {H}}^n}) |u|^{r_j-2}u, \end{aligned}$$

with Navier boundary conditions on a Korányi ball has been proved [19] via a variational principle, where \(w\in A_s\) is a Muckenhoupt weight function and \(\Delta ^2_{{\mathbb {H}}^n, p}\) is the Heisenberg p-biharmonic operator.

Motivated by their works, we study the existence of multiplicity of weak solutions for the problem (1.1), which is consisting of biharmonic type operator and singular terms. Our main tool in this study is the following theorem [2].

Theorem 1.1

Let X be a reflexive real Banach space, \(\Phi :X \rightarrow {\mathbb {R}}\) be a coercive, continuously G\({\hat{a}}\)teaux differentiable and sequentially weakly lower semi-continuous functional whose G\({\hat{a}}\)teaux derivative admits a continuous inverse on \(X^{*}\). Let \(\Psi :X\rightarrow {\mathbb {R}}\) be a continuously G\({\hat{a}}\)teaux differentiable functional whose G\({\hat{a}}\)teaux derivative is compact such that

$$\begin{aligned} \inf _{x\in X} \Phi =\Phi (0)=\Psi (0)=0. \end{aligned}$$

Assume that there exist \(r>0\) and \({\bar{x}}\in X\), with \(r<\Phi ({\bar{x}})\), such that

  1. (i)

    \(\dfrac{sup_{\Phi (x)<r}\Psi (x)}{r}<\dfrac{\Psi ({\bar{x}})}{\Phi ({\bar{x}})}\);

  2. (ii)

    For each

    $$\begin{aligned} \lambda \in \Lambda _{r}:=\left]\dfrac{\Phi ({\bar{x}})}{\Psi ({\bar{x}})}, \dfrac{r}{sup_{\Phi (x)<r}\Psi (x)} \right[, \end{aligned}$$

    the functional \(I_{\lambda }=\Phi -\lambda \Psi\) is coercive.

Then, for each \(\lambda \in \Lambda _{r}\), the functional \(I_\lambda =\Phi -\lambda \Psi\) has at least three distinct critical points in X.

The structure of this paper is the following: In Sect. 2, we present preliminaries and some basic facts. We also introduce a suitable function space for the solution and we prove some remarks which we need for the last section. In Sect. 3, the existence of multiple weak solutions for Problem (1.1) is prove by variational methods and three critical points result mentioned above.

2 Preliminaries

During the note, \(\Omega\) is a bounded domain in \({{\mathbb {R}}^{N}}(N\ge 2)\) with smooth boundary. We suppose that \(1<s_i<\frac{N}{2}\) and \(p_i \in C({\overline{\Omega }})\) for \(i=1,\ldots ,n\), satisfy the following inequalities:

$$\begin{aligned} \max \left\{ 2,\frac{N}{2}\right\}<p_i^{-}:=\inf _{x \in \Omega } p_i(x)\leqslant p_i(x) \leqslant p_i^{+}:= \sup _{x \in \Omega }p_i(x)<+\infty . \end{aligned}$$
(2.1)

Define the variable exponent Lebesgue space \(L^{p_i(x)}(\Omega )\), \(i=1,\ldots ,n\), by

$$\begin{aligned} L^{p_i(x)}(\Omega ):=\left\{ u:\Omega \longrightarrow {\mathbb {R}}: \;\text{u is measurable and}\; \int _{\Omega } \vert u(x) \vert ^{p_i(x)} dx< \infty \right\} . \end{aligned}$$

We difine a norm, the so-called Luxemburg norm, on this space by the formula

$$\begin{aligned} \vert u\vert _{p_i(x)}:=\inf \left\{ \lambda >0:\int _{\Omega } \left| \frac{u(x)}{\lambda }\right| ^{p_i(x)}dx\leqslant 1\right\} . \end{aligned}$$

Let us point out that if \(q(\cdot )\equiv q, \; q\in \{s_i:i=1,\ldots ,n\}\cup \{1\}\), this norm is equal to the standard norm on \(L^q(\Omega )\) that we denote it by \(|\cdot |_q\); that is

$$\begin{aligned} |u|_q=\left( \int _\Omega |u|^qdx\right) ^\frac{1}{q}. \end{aligned}$$

For any \(u\in L^{p_i(x)}(\Omega )\) and \(v \in L^{p_i' (x)}(\Omega )\), where \(L^{p_i' (x)}(\Omega )\) is the conjugate space of \(L^{p_i (x)}(\Omega )\), the Hölder type inequality

$$\begin{aligned} \left| \int _\Omega uv dx\right| \le \left( \frac{1}{p_i^-}+\frac{1}{p_i'^-}\right) |u|_{p_i(x)} |v|_{p_i'(x)} \end{aligned}$$

holds true. The following theorem is in [9, Theorem 2.8].

Theorem 2.1

Assume that \(\Omega\) is a bounded and smooth in \({\mathbb {R}}^N\) and \(p, q \in C_+({\bar{\Omega }})=\{g\in C({\overline{\Omega }}): g^->1\}\). Then

$$\begin{aligned} L^{p(x)}(\Omega ) \hookrightarrow L^{q(x)}(\Omega ) \end{aligned}$$

if and only if \(q(x) \le p(x)\) a.e. \(x \in \Omega\); moreover, there exists constant \(M_q\) such that

$$\begin{aligned} |u|_{q(x)}\le M_q |u|_{p(x)}. \end{aligned}$$
(2.2)

Following the authors of [18], for any \(\kappa >0\), we put

$$\begin{aligned} \kappa ^{\check{r}}:= \left\{ \begin{array}{ll} \kappa ^{r^+} &{}\quad \kappa <1, \\ \kappa ^{r^-} &{}\quad \kappa \ge 1 \end{array}\right. \end{aligned}$$
(2.3)

and

$$\begin{aligned} \kappa ^{{\hat{r}}}:= \left\{ \begin{array}{ll} \kappa ^{r^-}&{}\quad \kappa <1, \\ \kappa ^{r^+} &{}\quad \kappa \ge 1 \end{array}\right. \end{aligned}$$
(2.4)

for \(r\in \{p_i: i=1,\ldots ,n\}\). Then the well-known proposition 2.7 of [7] will be rewritten as follows.

Proposition 2.1

For each \(u\in L^{p(x)}(\Omega )\), we have

$$\begin{aligned} |u|_{p(x)}^{\check{p}}\le \int _\Omega |u(x)|^{p(x)}dx \le |u|_{p(x)}^{{\hat{p}}}. \end{aligned}$$

We denote the first order variable exponent Sobolev space by

$$\begin{aligned} W^{1,p(x)}(\Omega ):=\{u \in L^{p(x)}(\Omega ): |\nabla u| \in L^{p(x)}(\Omega )\}, \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert u\Vert _{1,p(x)}:=|\nabla u| _{p(x)}, \end{aligned}$$

where

$$\begin{aligned} \nabla u=\left( \frac{\partial u}{\partial x_1}(x), \ldots , \frac{\partial u}{\partial x_N}(x)\right) \end{aligned}$$

is the gradient of u at \(x=(x_1, \ldots ,x_n)\) and as usual \(|\nabla u|= \big ( \sum _{i=1}^N |\frac{\partial u}{\partial x_i}|^2 \big )^\frac{1}{2}\). And, the second order variable exponent Sobolev space is defined as follows

$$\begin{aligned} W^{2,p(x)}(\Omega ):=\{u \in L^{p(x)}(\Omega ): |\nabla u|, |\Delta u| \in L^{p(x)}(\Omega )\}, \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u\Vert _{2,p(x)}:=| u| _{p(x)}+|\nabla u| _{p(x)}+|\Delta u| _{p(x)}, \end{aligned}$$

where \(\Delta u=\sum _{i=1}^N \frac{\partial ^2 u}{\partial x_i^2}\) is Laplace operator. It is well known that the spaces \(L^{p(x)}(\Omega )\) and \(W^{m,p(x)}(\Omega ), m=1,2,\) are separable, reflexive and uniform convex Banach spaces [4]. Let \(W_0^{1,p(x)}(\Omega )\) be the closure of \(C^{\infty }_{0}(\Omega )\) in \(W^{1,p(x)}(\Omega )\). We set

$$\begin{aligned} Z:=W^{2,p(x)}(\Omega )\cap W_{0}^{1,p(x)}(\Omega ), \end{aligned}$$

for \(p\in \{p_i: i=1,\ldots ,n\}\), which is a reflexive Banach space with respect to the norm

$$\begin{aligned} \begin{aligned} \Vert u\Vert _Z&:=\Vert u\Vert _{W^{2, p(x)}(\Omega )}+\Vert u\Vert _{W_{0}^{1, p(x)}(\Omega )}\\&=|u|_{p(x)}+| \nabla u|_{p(x)}+|\Delta u|_{p(x)}. \end{aligned} \end{aligned}$$

By using the Poincaré inequality and [21], the norms \(\Vert \cdot \Vert _{Z}\) and

$$\begin{aligned} |\Delta u|_{p(x)}:=\inf \left\{ \mu >0:\int _{\Omega }\left| \dfrac{\Delta u}{\mu }\right| ^{p(x)}dx\le 1\right\} \end{aligned}$$

are equivalent on Z.

Remark 2.1

As a consequence of Theorem 2.1, we have

$$\begin{aligned} W^{m,q(x)}(\Omega )\hookrightarrow W^{m,p(x)}(\Omega ), \end{aligned}$$
(2.5)

if \(p(x) \le q(x)\) a.e. \(x \in \Omega\). In a special case, for \(p_i\), \(i=1,\ldots ,n\), with the condition (2.1),

$$\begin{aligned} W^{2,p_i(x)}(\Omega )\cap W_{0}^{1,p_i(x)}(\Omega )\hookrightarrow W^{2, p_i^{-}}(\Omega )\cap W_{0}^{1, p_i^{-}}(\Omega ) \end{aligned}$$

is embedded continuously and since \(p_i^->\frac{N}{2}\), one has the following compact embedding

$$\begin{aligned} W^{2, p_i^{-}}(\Omega )\cap W_{0}^{1, p_i^{-}}(\Omega ) \hookrightarrow \hookrightarrow C^{0}({\overline{\Omega }}). \end{aligned}$$

Then

$$\begin{aligned} W^{2,p_i(x)}(\Omega )\cap W_{0}^{1,p_i(x)}(\Omega )\hookrightarrow \hookrightarrow C^{0}({\overline{\Omega }}). \end{aligned}$$

So, in particular, there exist positive constants \(k_i>0, \; i=1,\ldots ,n\), such that

$$\begin{aligned} |u|_{\infty }\leqslant k_i |\Delta u|_{p_i(x)}, \end{aligned}$$
(2.6)

for each \(u \in W^{2,p_i(x)}(\Omega )\cap W_{0}^{1,p_i(x)}(\Omega )\) where \(| u |_{\infty }:=\sup _{x \in \Omega }|u(x)|\).

Remark 2.2

From Proposition 2.1, for each \(u\in Z\), we have

$$\begin{aligned} |\Delta u|_{p(x)}^{\check{p}}\le \rho (u):=\int _\Omega |\Delta u(x)|^{p(x)}dx \le |\Delta u|_{p(x)}^{{\hat{p}}}. \end{aligned}$$

Here, we recall the classical Hardy–Rellich inequality mentioned in [3].

Lemma 2.1

Let \(1<s<\frac{N}{2}\). Then for \(u \in W^{1,s}_{0}(\Omega )\cap W^{2,s}(\Omega )\), one has

$$\begin{aligned} \int _\Omega \frac{{|u(x){|^s}}}{{|x{|^{2s}}}}dx \le \frac{1}{{\mathcal {H}}_s}\int _\Omega | \Delta u(x){|^s}dx, \end{aligned}$$

where \({\mathcal {H}}_s:= (\frac{N(s-1)(N-2s)}{s^2})^s\).

Remark 2.3

Suppose \(1<s_i<\frac{N}{2}\) and \(p_i \in C(\Omega )\) be as in (2.1), for \(i=1,\ldots ,n\). Then there exists \(\alpha\) such that

$$\begin{aligned} \int _\Omega \frac{|u(x)|^{s_i}}{|x|^{2s_i}}dx \le \frac{\alpha }{{\mathcal {H}}_s}|\Delta u|^{s_i}_{p_i(x)}, \end{aligned}$$

for \(u \in W^{1,p_i(x)}_{0}(\Omega )\cap W^{2,p_i(x)}(\Omega )\), where

$$\begin{aligned} {\mathcal {H}}_s=\min \left\{ {\mathcal {H}}_{s_i}=\left( \frac{N(s_i-1)(N-2s_i)}{s_i^2}\right) ^{s_i}, i=1,\ldots ,n\right\} . \end{aligned}$$

Proof

Since for each \(i=1,\ldots , n\), \(s_i<p_i(x)\) a.e. in \(\Omega\), so by (2.5), we have

$$\begin{aligned} W^{1,p_i(x)}_{0}(\Omega )\cap W^{2,p_i(x)}(\Omega )\hookrightarrow W^{1,s_i}_{0}(\Omega )\cap W^{2,s_i}(\Omega ); \end{aligned}$$

moreover, there exist constants \(\alpha _{s_i}\) such that

$$\begin{aligned} |\Delta u|_{s_i}\le \alpha _{s_i} |\Delta u|_{p_i(x)}. \end{aligned}$$

By using classical Hardy–Rellich inequality, we have

$$\begin{aligned} \int _\Omega \frac{|u(x)|^{s_i}}{|x|^{2s_i}}dx \le \frac{1}{{\mathcal {H}}_{s_i}}\int _\Omega | \Delta u(x)|^{s_i}dx, \end{aligned}$$

for \(u \in W^{1,s_i}_{0}(\Omega )\cap W^{2,s_i}(\Omega )\). Then we gain

$$\begin{aligned} \int _\Omega \frac{|u(x)|^{s_i}}{|x|^{2s_i}}dx \le \frac{\alpha ^{s_i}_{s_i}}{{\mathcal {H}}_{s_i}}|\Delta u|^{s_i}_{p_i(x)}. \end{aligned}$$

Put \(\alpha =\max \{\alpha ^{s_i}_{s_i}, i=1,\ldots ,n\}\) and \({\mathcal {H}}_s=\min \{{\mathcal {H}}_{s_i}=(\frac{N(s_i-1)(N-2s_i)}{s_i^2})^{s_i}, i=1,\ldots ,n\}\), this completes the proof. \(\square\)

In what follows, we set

$$\begin{aligned} X:=\prod _{i=1}^n\big ( W^{2,p_i(x)}(\Omega )\cap W_0^{1,p_i(x)}(\Omega )\big ), \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert u\Vert =\Vert (u_1,\ldots ,u_n)\Vert =\sum _{i=1}^n|\Delta u_i|_{p_i(x)} \end{aligned}$$

for \(u=(u_1,\ldots ,u_n)\in X\). From Remark 2.1, we conclude that the embedding

$$\begin{aligned} X\hookrightarrow C^0({\overline{\Omega }})\times \cdots \times C^0({\overline{\Omega }}) \end{aligned}$$

is compact and if we put

$$\begin{aligned} K:=\max _{1\le i\le n} k_i, \end{aligned}$$

where \(k_i, 1\le i\le n\) are as in relation (2.6), it is clear that \(K>0\) and one has

$$\begin{aligned} | u_i |_{\infty }\leqslant K |\Delta u_i|_{p_i(x)}\qquad i=1,\ldots ,n. \end{aligned}$$
(2.7)

We mean by a weak solution to the problem (1.1) is as follows:

Definition 2.1

We say that \(u=(u_1,\ldots ,u_n)\in X\) is a weak solution of problem (1.1) if

$$\begin{aligned} {\sum _{i=1}^n\int _{\Omega }|\Delta u_i|^{p_i(x)-2}\Delta u_i \Delta v_i dx}&+\sum _{i=1}^n\int _{\Omega }b_i(x)\frac{|u_i|^{s_i-2}u_i v_i}{|x|^{2s_i}}dx \\ {}&-\lambda \sum _{i=1}^n\int _{\Omega }F_{u_i}(x,u_1,\ldots , u_n)v_idx=0 \end{aligned}$$

for every \(v=(v_1,\ldots ,v_n)\in X\).

We introduce the functional \(\Phi :X\longrightarrow {\mathbb {R}}\) as follows

$$\begin{aligned} \Phi (u_1,\ldots ,u_n):=\sum _{i=1}^n \int _{\Omega }\frac{1}{p_i(x)}\vert \Delta u_i \vert ^{p_i(x)} dx+\sum _{i=1}^n\int _{\Omega }b_i(x)\frac{|u_i(x)|^{s_i}}{s_i|x|^{2s_i}}dx. \end{aligned}$$

Remark 2.4

There exists positive constant \({\hat{C}}\) such that

$$\begin{aligned} \frac{1}{p_i^+} |\Delta u_i|_{p_i(x)}^{\check{p_i}}\le \Phi (u_1,\ldots ,u_n)\le {\hat{C}} \sum _{i=1}^n \Big (|\Delta u_i|_{p_i(x)}^{\hat{p_i}}+|\Delta u_i|^{s_i}_{p_i(x)}\Big ) \end{aligned}$$

for each \(1\le i\le n\) and \(u=(u_1,\ldots ,u_n)\in X\).

Proof

From (2.2) and Remark 2.3, for every \(1\le i\le n\), one has the following estimate

$$\begin{aligned} \frac{1}{p_i^+} |\Delta u_i|_{p_i(x)}^{\check{p_i}}&\le \frac{1}{p_i^+} \int _\Omega |\Delta u_i|^{p_i(x)}dx\\ {}&\le \sum _{i=1}^n \frac{1}{p_i^+}\int _\Omega |\Delta u_i|^{p_i(x)}dx\\ {}&\le \Phi (u_1,\ldots ,u_n)\\ {}&=\sum _{i=1}^n \int _{\Omega }\frac{1}{p_i(x)}\vert \Delta u_i \vert ^{p_i(x)} dx+\sum _{i=1}^n\int _{\Omega }b_i(x)\frac{|u_i(x)|^{s_i}}{s_i|x|^{2s_i}}dx\\&\le \sum _{i=1}^n \frac{1}{p^-}\int _\Omega (|\Delta u_i|^{p_i(x)})dx+\frac{\alpha }{{\mathcal {H}}_{s}}\sum _{i=1}^n \frac{|b_i|_\infty }{s_i}|\Delta u_i|^{s_i}_{p_i(x)}\\ {}&\le {\hat{C}} \sum _{i=1}^n \Big (|\Delta u_i|_{p_i(x)}^{\hat{p_i}}+|\Delta u_i|^{s_i}_{p_i(x)}\Big ), \end{aligned}$$

where \(p^{-}:=\min \{p_i^-:i=1,\ldots ,n\}\). Now, it is enough to set

$$\begin{aligned} {\hat{C}} =\max _{1\le i\le n}\left\{ \frac{1}{p^-},\frac{\alpha }{{\mathcal {H}}_{s}}\frac{|b_i|_{\infty }}{s_i}\right\} . \end{aligned}$$

\(\square\)

Remark 2.5

Due to the Remark 2.4\(\Phi\) is coercive.

Proof

Let \(u=(u_1,\ldots ,u_n)\in X\) and \(\Vert u\Vert \rightarrow \infty\). By definition of \(\Vert \cdot \Vert\), there exists \(1\le i_0\le n\) such that \(|\Delta u_{i_0}|_{p_{i_0}(x)}\rightarrow \infty\). Then Remark 2.4 result in \(\Phi (u)\rightarrow \infty\). \(\square\)

Furthermore, \(\Phi\) is sequentially weakly lower semicontinuous and it is known that \(\Phi\) is continuously Gâteaux differentiable functional whose derivative given by

$$\begin{aligned} \Phi '(u_1,\ldots ,u_n)(v_1,\ldots ,v_n)= {\sum _{i=1}^n\int _{\Omega }\Big ( |\Delta u_i|^{p_i(x)-2}\Delta u_i \Delta v_i +} b_i(x) \dfrac{\vert u_i\vert ^{s_i-2}u_iv_i}{|x|^{2s_i}}\Big )dx, \end{aligned}$$

for each \((v_1,\ldots ,v_n)\in X\).

Now imagine that the function

$$\begin{aligned} F:\Omega \times {\mathbb {R}}^n\rightarrow {\mathbb {R}} \end{aligned}$$

is a measurable function with respect to \(x\in \Omega\) for each \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) and is \(C^1\) with respect to \((t_1, \ldots , t_n)\in {\mathbb {R}}^n\) for a.e. \(x\in \Omega\); \(F_{u_i}\) denotes the partial derivative of F with respect to \(u_i\). We define \(\Psi :{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) with

$$\begin{aligned} \Psi (u_1,\ldots ,u_n):=\int _{\Omega }F(x,u_1,\ldots ,u_n)dx. \end{aligned}$$

The functional \(\Psi\) is well defined, continuously Gâteaux differentiable with compact derivative, whose Gâteaux derivative at point \(u=(u_1,\ldots ,u_n)\in X\) is as follows

$$\begin{aligned} \Psi '(u_1,\ldots ,u_n)(v_1,\ldots ,v_n)=\sum _{i=1}^n\int _{\Omega }F_{u_i}(x, u_1(x),\ldots ,u_n(x))v_i(x) dx \end{aligned}$$

for every \((v_1,\ldots ,v_n)\in X\). Define

$$\begin{aligned} I_{\lambda }(u)=\Phi (u)-\lambda \Psi (u) \end{aligned}$$

for each \(u=(u_1,\ldots ,u_n)\); if \(I_{\lambda }^{\prime }(u)=0\) we have

$$\begin{aligned} {\sum _{i=1}^n\int _{\Omega }|\Delta u_i|^{p_i(x)-2}\Delta u_i \Delta v_i dx}&+\sum _{i=1}^n\int _{\Omega }b_i(x)\frac{|u_i|^{s_i-2}u_i v_i}{|x|^{2s_i}}dx \\ {}&=\lambda \sum _{i=1}^n\int _{\Omega }F_{u_i}(x,u_1,\ldots , u_n)v_idx, \end{aligned}$$

then the critical points of \(I_{\lambda }\) are exactly the weak solutions of the problem (1.1). We set

$$\begin{aligned} \delta (x)= sup\left\{ {\delta >0: B(x,\delta )\subseteq \Omega }\right\} , \end{aligned}$$

and we define

$$\begin{aligned} R:=sup_{x\in \Omega }\delta (x). \end{aligned}$$
(2.8)

Obviously, there exists \(x^0=(x^0_1,\ldots ,x^0_N)\in \Omega\) such that

$$\begin{aligned} B(x^0,R)\subseteq \Omega . \end{aligned}$$

Now, we are ready to sketch the main result of the paper.

3 Main approach

We formulate our main result as follows.

Theorem 3.1

Assume that \(F:{\overline{\Omega }}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) satisfies the following conditions

(F1):

\(F(x,0,\ldots ,0)=0\), for a.e. \(x\in \Omega\);

(F2):

There exist \(\eta \in L^1(\Omega )\) and n positive continuous functions \(\gamma _i\), \(1\le i\le n\), with \(\gamma _i(x)<p_i(x)\) a.e in \(\Omega\) such that

$$\begin{aligned} 0\le F(x,u_1,\ldots ,u_n)\le \eta (x) \left( 1+\sum _{i=1}^n \vert u_i \vert ^{\gamma _i(x)}\right) ; \end{aligned}$$
(F3):

There exist \(r>0\), \(\delta >0\) and \(1\le i_*\le n\) such that

$$\begin{aligned} \frac{1}{p_{i_*}^+}\left( \dfrac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{\check{p_{i_*}}}m\left( R^N-\left( \frac{R}{2}\right) ^N\right) >r, \end{aligned}$$

where \(m:=\frac{\pi ^{\frac{N}{2}}}{\frac{N}{2}\Gamma (\frac{N}{2})}\) is the measure of unit ball of \({\mathbb {R}}^{N}\) and \(\Gamma\) is the Gamma function.

Such that

$$\begin{aligned} G_{r}<H_\delta , \end{aligned}$$
(3.1)

where

$$\begin{aligned} G_{r}:=\frac{|\eta |_1}{r}\left( 1+ \sum _{i=1}^nK^{\hat{\gamma _i}} \left( p_i^+r\right) ^\frac{\hat{\gamma _i}}{\check{p_i}}\right) , \end{aligned}$$

and

$$\begin{aligned} {H_\delta :=\dfrac{\sum _{i=1}^n inf_{x\in \Omega } F(x,\delta ,\ldots ,\delta )}{{\hat{C}}\sum _{i=1}^n\left( \left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{\hat{p_i}}+\left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{s_i}\right) (2^N-1)}.} \end{aligned}$$

Then for each

$$\begin{aligned} \lambda \in \Lambda _{r,\delta }:=\left( \frac{1}{H_\delta }, \frac{1}{G_{r}} \right) \end{aligned}$$

the problem (1.1) admits at least three distinct weak solutions in X.

Proof

Our aim is to apply Theorem 1.1. According to previous section, the space

$$\begin{aligned} X=\prod _{i=1}^n\big ( W^{2,p_i(x)}(\Omega )\cap W_0^{1,p_i(x)}(\Omega )\big ) \end{aligned}$$

and the functionals \(\Phi ,\Psi :X\rightarrow {\mathbb {R}}\) defined as above satisfy the regularity assumptions of Theorem 1.1. By condition (F1) and definition of \(\Phi , \Psi\), it is clear that

$$\begin{aligned} \inf _{x\in X} \Phi =\Phi (0)=\Psi (0)=0. \end{aligned}$$

Now Fix \(\delta >0\) and R defined as in (2.8). By w, we consider the function of the space \(W^{2,p_i(x)}(\Omega )\cap W_0^{1,p_i(x)}(\Omega ), 1\le i\le n\), defined by

$$\begin{aligned} w(x):= \left\{ \begin{array}{ll} 0&{}\quad x\in \Omega {\setminus } B(x^0,R), \\ \delta &{}\quad x\in B\left( x^0,\frac{R}{2}\right) , \\ \frac{\delta }{R^2-\left( \frac{R}{2}\right) ^2} \left( R^2-\sum _{i=1}^N\left( x_i-x_i^0\right) ^2\right) &{}\quad x\in B(x^0,R){\setminus } B\left( x^0,\frac{R}{2}\right) , \end{array}\right. \end{aligned}$$
(3.2)

where \(x=(x_1,\ldots ,x_N)\in \Omega\). Then

$$\begin{aligned} \sum _{i=1}^N\dfrac{\partial ^2 w}{\partial x_i^2}(x)= \left\{ \begin{array}{ll} 0&{}\quad x\in \left( \Omega {\setminus } B(x^0,R)\right) \cup B\left( x^0,\frac{R}{2}\right) , \\ -\dfrac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}&{}\quad x\in B(x_0,R){\setminus } B\left( x^0,\frac{R}{2}\right) . \end{array}\right. \end{aligned}$$

So, by Remark 2.4, for \(1\le i_*\le n\), one has

$$\begin{aligned}&\frac{1}{p_{i_*}^+}\left( \dfrac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{\check{p_{i_*}}}m\left( R^N-\left( \frac{R}{2}\right) ^N\right) \\ {}&\quad <\Phi (w,\ldots , w)\\ {}&\quad \le { {\hat{C}}\sum _{i=1}^n \left( \left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{\hat{p_i}}+\left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{s_i}\right) m\left( R^N-\left( \frac{R}{2}\right) ^N\right) ,} \end{aligned}$$

then by assumption (F3), we gain \(\Phi (w,\ldots ,w)>r\). On the other hand, we have

$$\begin{aligned} \Psi (w,\ldots , w)&\ge \sum _{i=1}^n\int _{B(x_0,\frac{R}{2})}F(x,w, \ldots ,w)dx \\ {}&\ge \sum _{i=1}^n inf_{x\in \Omega } F(x,\delta ,\ldots ,\delta ) m\left( \frac{R}{2}\right) ^N, \end{aligned}$$

where m is the measure of unit ball of \({\mathbb {R}}^{N}\) and so,

$$\begin{aligned} \nonumber \dfrac{\Psi (w,\ldots ,w)}{\Phi (w,\ldots ,w)}&\ge \dfrac{\sum _{i=1}^n inf_{x\in \Omega } F(x,\delta ,\ldots ,\delta )m\left( \frac{R}{2}\right) ^N}{\hat{C }\sum _{i=1}^n \left( \left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{\hat{p_i}}+\left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{s_i}\right) m\left. \left( R^N-\left( \frac{R}{2}\right) ^N\right) \right) }\\ \nonumber \\ {}&=\dfrac{\sum _{i=1}^n inf_{x\in \Omega } F(x,\delta ,\ldots ,\delta )}{{\hat{C}}\sum _{i=1}^n \left( \left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{\hat{p_i}}+\left( \frac{2\delta N}{R^2-\left( \frac{R}{2}\right) ^2}\right) ^{s_i}\right) (2^N-1)} =H_\delta . \end{aligned}$$
(3.3)

Now, let \(u=(u_1,\ldots ,u_n)\in \Phi ^{-1}(-\infty ,r)\), from Remark 2.4, for each \(i=1,\ldots ,n\) we have

$$\begin{aligned} |\Delta u|_{p_i(x)} \le \big (p_i^+\Phi (u_1,\ldots ,u_n)\big )^\frac{1}{\check{p_i}}\le \left( p_i^+r\right) ^\frac{1}{\check{p_i}}. \end{aligned}$$
(3.4)

Then for every \(u=(u_1,\ldots , u_n)\in \Phi ^{-1}(-\infty ,r)\), from condition (F2), Hölder inequality and (2.6), we gain

$$\begin{aligned} \int _{\Omega }F(x,u_1,\ldots ,u_n)dx&\le \int _{\Omega } sup_{u\in \Phi ^{-1}(-\infty ,r)}F(x,u_1,\ldots ,u_n)dx\\ {}&\le \int _{\Omega }\eta (x) \left( 1+\sum _{i=1}^n \vert u_i \vert ^{\gamma _i(x)}\right) dx\\ {}&\le |\eta |_1\left( 1+\sum _{i=1}^n |u_i|^{\hat{\gamma _i}}_\infty \right) \\ {}&\le |\eta |_1\left( 1+ \sum _{i=1}^n K^{\hat{\gamma _i}}|\Delta u_i|_{p_i(x)}^{\hat{\gamma _i}}\right) . \end{aligned}$$

So,

$$\begin{aligned} \nonumber \frac{1}{r}sup_{u\in \Phi ^{-1}(-\infty ,r)}\Psi (u)&=\frac{1}{r}sup_{u\in \Phi ^{-1}(-\infty ,r)}\int _{\Omega }F(x,u_1,\ldots ,u_n)dx\\&\le \frac{|\eta |_1}{r}\left( 1+\sum _{i=1}^nK^{\hat{\gamma _i}} \left( p_i^+r\right) ^\frac{\hat{\gamma _i}}{\check{p_i}}\right) =G_r. \end{aligned}$$
(3.5)

From assumption (3.1), relations (3.3) and (3.5), one has

$$\begin{aligned} \dfrac{1}{r} sup_{u\in \Phi ^{-1}(-\infty ,r)}\Psi (u_i)<\dfrac{\Psi (w,\ldots ,w)}{\Phi (w,\ldots ,w)}. \end{aligned}$$

Therefore, the assumption (i) of Theorem 1.1 is satisfied.

Now, we prove that the functional \(I_{\lambda }\) for all \(\lambda >0\) is coercive.

With the same arguments as used before, we have

$$\begin{aligned} \Psi (u) =\int _{\Omega }F(x,u_1,\ldots ,u_n)dx \le |\eta |_1\left( 1+\sum _{i=1}^n K^{\hat{\gamma _i}}|\Delta u_i|_{p_i(x)}^{\hat{\gamma _i}}\right) . \end{aligned}$$

The last inequality and Remark 2.4 lead to

$$\begin{aligned} I_{\lambda }(u)\ge \frac{1}{p_i^+}|\Delta u_i|_{p_i(x)}^{\check{p_i}}-\lambda |\eta |_1\left( 1+ \sum _{i=1}^n K^{\hat{\gamma _i}} |\Delta u_i|_{p_i(x)}^{\hat{\gamma _i}}\right) , \end{aligned}$$

for each \(i=1,\ldots ,n\). Now, suppose that \(u\in X\) and \(\Vert u\Vert \rightarrow \infty\). So, there exists \(1\le i_0\le n\) such that \(|\Delta u_{i_0}|_{p_{i_0}(x)}\rightarrow \infty\). Since \(\gamma _{i_0}(x)<p_{i_0}(x)\) a.e. in \(\Omega\), so \(I_{\lambda }\) is coercive.

Then condition (ii) holds. So for each

$$\begin{aligned} \Lambda _{\delta ,r}:=\left( \frac{1}{H_\delta }, \frac{1}{G_{r}} \right) \subseteq \left( \dfrac{\Phi (w,\ldots ,w)}{\Psi (w,\ldots ,w)},\dfrac{r}{sup_{u\in \Phi ^{-1}(-\infty ,r)}\Psi (u_i)}\right) , \end{aligned}$$

Theorem 1.1 ensures that for each \(\lambda \in \Lambda _{r,\delta }\), the functional \(I_{\lambda }\) admits at least three critical points in X that are weak solutions of the problem (1.1). \(\square\)

Remark 3.1

An interesting problem is to probe the existence and multiplicity of solutions of this system under the Steklov boundary conditions [8] or in the Heisenberg Sobolev spaces and Orlicz Sobolev spaces. Interested reader can see details of these spaces in [5, 6, 12,13,14,15,16, 20] and references therein.