Existence of three solutions for a Navier boundary value problem involving the (p(x),q(x))-biharmonic

In this paper, we study (p(x),q(x))-biharmonic systems with Navier boundary condition on a bounded domain and obtain three solutions under appropriate hypotheses. The technical approach is mainly based on the general three critical points theorem obtained by Ricceri.

In this paper, we consider the Navier boundary value problem involving the (p(x),q(x))-biharmonic systems

where \lambda ,\mu \in [0,+\mathrm{\infty }), \mathrm{\Omega }\subset R^N (N\ge 1) is a nonempty bounded open set with a boundary ∂ Ω of class C^1, F,G:\mathrm{\Omega }\times R\times R\to R are functions such that F(\cdot ,s,t), G(\cdot ,s,t) are measurable in Ω for all (s,t)\in R\times R and F(x,\cdot ,\cdot ) is C^1 in R\times R for a.e. x\in \mathrm{\Omega }, F_i denotes the partial derivative of F with respect to i, i=u,v, so does G_i. And p,q\in C(\overline{\mathrm{\Omega }}), , . Moreover,

is the critical exponent just as in many papers. Obviously, , for all x\in \mathrm{\Omega }.

In what follows, E denotes the Cartesian product of two Sobolev spaces W^{2,p(x)}(\mathrm{\Omega })\cap W_0^{1,p(x)}(\mathrm{\Omega }) and W^{2,q(x)}(\mathrm{\Omega })\cap W_0^{1,q(x)}(\mathrm{\Omega }), i.e., E=(W^{2,p(x)}(\mathrm{\Omega })\cap W_0^{1,p(x)}(\mathrm{\Omega }))\times (W^{2,q(x)}(\mathrm{\Omega })\cap W_0^{1,q(x)}(\mathrm{\Omega })), and X denotes the Sobolev space W^{2,p(x)}(\mathrm{\Omega })\cap W_0^{1,p(x)}(\mathrm{\Omega }).

In recent years, the study of differential equations and variational problems with p(x)-growth conditions has been an interesting topic resulting from nonlinear electrorheological fluids (see [1]) and elastic mechanics (see [2]).

Some authors considered elliptic systems (see [3–16]) which have been used in a wide range of applications. Existence and multiplicity results for elliptic systems involving variational structure have been extensively investigated.

In [3], Boccardo and Figueiredo studied the following boundary value problem involving the (p,q)-Laplacian:

where p and q are real numbers larger than 1.

In [4], applying the fibering method established by Pohozaev, Bozhkova and Mitidieri, the authors studied the existence of multiple solutions for quasilinear system involving a pair of (p,q)-Laplacian operators.

In [5], Chun Li and Chun-Lei Tang ensured the existence of three solutions for the problem

where p>N, q>N and F satisfies suitable assumptions.

In [6], Afrouzi and Heidarkhani studied the existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the (p_1,\dots ,p_n)-Laplacian. In [7], Jing-Jing Liu and Xia-Yang Shi proved the existence of multiple solutions for a quasilinear system involving a pair of (p(x),q(x))-Laplacian operators. In [8], Bin Ge and Ji-Hong Shen obtained multiple solutions for a class of differential inclusion systems involving the (p(x),q(x))-Laplacian.

Recently, Lin Li and Chun-Lei Tang (see [9]) considered the Navier boundary value problem involving the (p,q)-biharmonic systems

where p>max\{1,\frac{N}{2}\}, q>max\{1,\frac{N}{2}\}, and F, G satisfy suitable assumptions.

The main result of this paper is the following theorem.

Theorem 1.1 Suppose that there exist two positive constants C, d and two functions \gamma (x),\beta (x)\in C(\overline{\mathrm{\Omega }}) with , such that

(j1) F(x,s,t)\ge 0 for a.e. x\in \mathrm{\Omega } and all (s,t)\in [0,d]\times [0,d];

(j2) \mathrm{\exists }{p}_1(x),q_1(x)\in C(\overline{\mathrm{\Omega }}) and , such that

(j3) |F(x,s,t)|\le C(1+|s{|}^{\gamma (x)}+|t{|}^{\beta (x)}) for a.e. x\in \mathrm{\Omega } and all (s,t)\in R\times R;

(j4) F(x,0,0)=0 for a.e. x\in \mathrm{\Omega }. Then there exist an open interval \mathrm{\Lambda }\subseteq [0,+\mathrm{\infty }) and a positive real number r with the following property: for each \lambda \in \mathrm{\Lambda } and each function G:\mathrm{\Omega }\times R\times R\mapsto R, measurable in Ω, C^1 in R\times R and satisfying

where p_2,q_2\in C(\overline{\mathrm{\Omega }}) and , for all x\in \overline{\mathrm{\Omega }}, there exists \delta >0 such that, for each \mu \in [0,\delta ], problem (P) has at least three weak solutions whose norms in (W^{2,p(x)}(\mathrm{\Omega })\cap W_0^{1,p(x)}(\mathrm{\Omega }))\times (W^{2,q(x)}(\mathrm{\Omega })\cap W_0^{1,q(x)}(\mathrm{\Omega })) are less than r.

The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge about the Lebesgue and Sobolev spaces with variable exponents, and present Ricceri’s three-critical-points theorem. In Section 3, we prove the main result.

Assume that Ω is a bounded domain of R^N (N\ge 1) with a smooth boundary ∂ Ω. Let

For p\in L_{+}^{\mathrm{\infty }}(\mathrm{\Omega }), set

For p\in L_{+}^{\mathrm{\infty }}(\mathrm{\Omega }), define

with the norm

and

endowed with the norm

We denote by W_0^{1,p(x)}(\mathrm{\Omega }) the closure of C_0^{\mathrm{\infty }}(\mathrm{\Omega }) in W^{1,p(x)}(\mathrm{\Omega }).

For the basic properties of the spaces L^{p(x)}(\mathrm{\Omega }), W^{1,p(x)}(\mathrm{\Omega }) and W_0^{1,p(x)}(\mathrm{\Omega }), please refer to [17–20]. Now we recite some known results which will be used later.

Proposition 2.1 (see [17])

1. (i)

   The spaces L^{p(x)}(\mathrm{\Omega }), W^{1,p(x)}(\mathrm{\Omega }) and W_0^{1,p(x)}(\mathrm{\Omega }) are separable and reflexive Banach spaces;

2. (ii)

   If q\in C_{+}(\overline{\mathrm{\Omega }}) and for any x\in \overline{\mathrm{\Omega }}, then the imbedding from W^{1,p(x)}(\mathrm{\Omega }) to L^{q(x)}(\mathrm{\Omega }) is compact and continuous;

3. (iii)

   There is a constant C>0 such that |u{|}_{p(x)}\le C|\mathrm{\nabla }u{|}_{p(x)} \mathrm{\forall }u\in W_0^{1,p(x)}(\mathrm{\Omega }).

By (iii) of Proposition 2.1, we know that |\mathrm{\nabla }u{|}_{p(x)} and \parallel u\parallel are equivalent norms on W_0^{1,p(x)}(\mathrm{\Omega }). We use |\mathrm{\nabla }u{|}_{p(x)} to replace \parallel u\parallel in the following discussion.

Proposition 2.2 (see [18])

Set \rho (u)={\int }_{\mathrm{\Omega }}|u{|}^{p(x)}dx for u,u_k\in L^{p(x)}(\mathrm{\Omega }), we obtain

1. (1)

   For u\ne 0, |u{|}_{p(x)}=\lambda \iff \rho (\frac{u}{\lambda })=1;

2. (2)

   ;

3. (3)

   If |u{|}_{p(x)}>1, then |u{|}_{p(x)}^{p^{-}}\le \rho (u)\le |u{|}_{p(x)}^{p^{+}};

4. (4)

   If , then |u{|}_{p(x)}^{p^{+}}\le \rho (u)\le |u{|}_{p(x)}^{p^{-}};

5. (5)

   {lim}_{k\to \mathrm{\infty }}|u_k{|}_{p(x)}=0\iff {lim}_{k\to \mathrm{\infty }}\rho (u_k)=0;

6. (6)

   |u_k{|}_{p(x)}\to \mathrm{\infty }\iff \rho (u_k)\to \mathrm{\infty }.

In this paper, the space E is endowed with the following equivalent norm:

where

Similar to Proposition 2.2, we obtain the following.

Proposition 2.3 Let \varphi (u)={\int }_{\mathrm{\Omega }}|\mathrm{\Delta }u{|}^{p(x)}dx for u,u_k\in W^{2,p(x)}(\mathrm{\Omega }), we obtain:

1. (1)

   For u\ne 0, \parallel u\parallel =\lambda \iff \varphi (\frac{u}{\lambda })=1;

2. (2)

   ;

3. (3)

   If \parallel u\parallel >1, then {\parallel u\parallel }^{p^{-}}\le \varphi (u)\le {\parallel u\parallel }^{p^{+}};

4. (4)

   If , then {\parallel u\parallel }^{p^{+}}\le \varphi (u)\le {\parallel u\parallel }^{p^{-}};

5. (5)

   {lim}_{k\to \mathrm{\infty }}\parallel u_k\parallel =0\iff {lim}_{k\to \mathrm{\infty }}\varphi (u_k)=0;

6. (6)

   \parallel u_k\parallel \to \mathrm{\infty }\iff \varphi (u_k)\to \mathrm{\infty }.

Let G(u)={\int }_{\mathrm{\Omega }}\frac{1}{p(x)}|\mathrm{\Delta }u{|}^{p(x)}dx, u\in X and denote L=G^{\prime }:X\to X^{\ast }, then

Proposition 2.4 (see [17])

1. (i)

   L:X\to X^{\ast } is a continuous, bounded and strictly monotone operator;

2. (ii)

   L is a mapping of type (S_{+}), i.e., if u_n\to u in X and \overline{{lim}_{n\to \mathrm{\infty }}}((L(u_n)-L(u),u_n-u))\le 0, then u_n\to u in X;

3. (iii)

   L:X\to X^{\ast } is a homeomorphism.

Proposition 2.5 (see [21])

Let X be a separable and reflexive real Banach space; I\subseteq R; let \mathrm{\Phi }:X\to R be a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X^{\ast }; J:X\to R is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. In addition, Φ is bounded on each bounded subset of X. Assume that

for all \lambda \in I\subseteq [0,\mathrm{\infty }[, and that there exists \rho \in R such that

Then there exist a nonempty open set A\subseteq I and a positive real number r with the following property: for every \lambda \in A and every C^1 functional \mathrm{\Psi }:X\to R with a compact derivative, there exists \delta >0 such that, for each \mu \in [0,\delta ], the equation

has at least three solutions in X whose norms are less than r.

Proposition 2.6 (see [22])

Let X be a nonempty set, and let Φ, J be two real functionals on X. Assume that there are r>0 and x_0,x_1\in X such that

Then, for each ρ satisfying

one has

Definition 3.1 A weak solution of problem (P) is any (u,v)\in E such that

for all \mathrm{\forall }(\xi ,\eta )\in E. We define the corresponding energy functional of problem (P) as

where

Then H(u,v) is a C^1 functional and the critical points of it are weak solutions of problem (P).

Proof of Theorem 1.1 Let Φ, J, Ψ as above. So, for each u,v,\xi ,\eta \in E, one has

Therefore, the weak solutions of problem (P) are exactly the solutions of the equation

In view of Proposition 2.4 (or [17] for details), certainly, Φ is a continuously Gâteaux differentiable and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on E^{\ast }. Moreover, J and Ψ are continuously Gâteaux differentiable functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X under our assumptions.

By Proposition 2.3, set G(u)={\int }_{\mathrm{\Omega }}\frac{1}{p(x)}|\mathrm{\Delta }u{|}^{p(x)}dx just as before, then for \parallel u\parallel \ge 1,

for ,

Actually, for , set C_0\ge \frac{1}{p^{+}}{\parallel u\parallel }^{p^{-}}-\frac{1}{p^{+}}{\parallel u\parallel }^{p^{+}}\ge 0, then we can obtain

Hence we have

So, there exists a constant C_1\ge 0 such that

holds for any (u,v)\in E.

holds for any (u,v)\in E, where constants C_2\ge 0, C_3\ge 0. Here, by using conditions (j3) and (ii) of Proposition 2.1, combining the two inequalities above, we can obtain

Due to , , we get

Then assumption (2.1) of Proposition 2.5 is fulfilled.

Next, we derive that assumption (2.2) is also fulfilled. It is easy to verify the conditions of Proposition 2.6. Let (u_0,v_0)=(0,0), we can easily have

Then there exist \gamma >0 and (u_1,v_1)\in E such that \mathrm{\Phi }(u_1,v_1)>\gamma and (2.3) is satisfied.

There is a point x^0\in \mathrm{\Omega } since it is a nonempty bounded open set. Let r_2>r_1>0, put

where B(x,r) is the open ball in R^N of radius r centered at x,

Let (u_1(x),v_1(x))=(w(x),w(x)), then by (j1) we can derive that

From (j2), \mathrm{\exists }\eta \in [0,1], C_1>0 such that

By (j3), there are nine positive real numbers M_i (i=1,2,\dots ,9) according to |s|, |t| larger or smaller than η and 1. For example, when |s|>1, some

Set M=max\{C_1,M_1,\dots ,M_9\}, then

Hence, fix γ such that . And for , by the Sobolev embedding theorem (X\to L^{p_1^{-}}(\mathrm{\Omega }) is continuous), there exist suitable positive constants C_2 and C_3 such that

Since p_1^{-}>p^{+}, q_1^{-}>q^{+}, we have

We choose w(x)\in X as above such that -J(w,w)>0. Fix {\gamma }_0 such that . Then we divide the proof into two cases.

1. (i)

   For , by (3.2) we have

   \begin{array}{rl}\mathrm{\Phi }(u_1,v_1)& =\mathrm{\Phi }(w,w)\\ ={\int }_{\mathrm{\Omega }}(\frac{1}{p(x)}|\mathrm{\Delta }w{|}^{p(x)}+\frac{1}{q(x)}|\mathrm{\Delta }w{|}^{q(x)})dx\\ \ge min\{\frac{1}{p^{+}},\frac{1}{q^{+}}\}{\int }_{\mathrm{\Omega }}(|\mathrm{\Delta }w{|}^{p(x)}+|\mathrm{\Delta }w{|}^{q(x)})dx\\ \ge min\{\frac{1}{p^{+}},\frac{1}{q^{+}}\}({\parallel w\parallel }^{p^{+}}+{\parallel w\parallel }^{q^{+}})\\ \ge {\gamma }_0>\gamma .\end{array}

By (3.3), we obtain

1. (ii)

   For \parallel w\parallel \ge 1, from (3.1) we get

   \begin{array}{rl}\mathrm{\Phi }(u_1,v_1)& =\mathrm{\Phi }(w,w)\\ ={\int }_{\mathrm{\Omega }}(\frac{1}{p(x)}|\mathrm{\Delta }w{|}^{p(x)}+\frac{1}{q(x)}|\mathrm{\Delta }w{|}^{q(x)})dx\\ \ge min\{\frac{1}{p^{+}},\frac{1}{q^{+}}\}{\int }_{\mathrm{\Omega }}(|\mathrm{\Delta }w{|}^{p(x)}+|\mathrm{\Delta }w{|}^{q(x)})dx\\ \ge min\{\frac{1}{p^{+}},\frac{1}{q^{+}}\}({\parallel w\parallel }^{p^{-}}+{\parallel w\parallel }^{q^{-}})\\ \ge {\gamma }_0>\gamma .\end{array}

From (3.3), we have

For any (u,v)\in {\mathrm{\Phi }}^{-1}((-\mathrm{\infty },\gamma ]), we can obtain , i.e.,

Then we can have

So,

This inequality implies

i.e.,

Therefore we have

So, we can get that

Then

that is,

Hence we can find \gamma >0, u_1=v_1=w and \mathrm{\Phi }(w,w)\le \gamma satisfying (2.3). Also, we can find ρ satisfying

Put I=[0,\mathrm{\infty }), moreover, \mathrm{\Phi }(u,v), -J(u,v) fulfil the assumption of Proposition 2.6. So, applying Proposition 2.6, we can easily get that (2.2) is fulfilled.

Thus, Φ, J and Ψ fulfil all the assumptions of Proposition 2.5, and our conclusion follows from Proposition 2.5. □

Remark Applying Theorem 2.1 in [23] to the proof of Theorem 1.1, an upper bound of the interval of parameters λ, for which (P) has at least three weak solutions, is obtained. To be precise, in the conclusion of Theorem 1.1, one has

for each h>1 and (u_1,v_1) as in the proof of Theorem 1.1 (namely, u_1=v_1=w).

The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This work was supported by the Scientific Research Project of Guangxi Education Department (No. 201204LX672).

Affiliations

1. School of Information & Statistics, Guangxi University of Financial and Economics, Nanning, Guangxi, 530003, China

   • Feng-Li Huang

2. Department of Construction and Information Engineering, Guangxi Modern Vocational Technology College, Hechi, Guangxi, 547000, China

   • Guang-Sheng Chen

3. School of Mathematics and Statistics, Xuchang University, Xuchang, Henan, 461000, China

   • Yu-Qi Niu

4. School of Civil Engineering, Hebei University of Technology, Tianjin, 300130, China

   • Ti Song

Corresponding author

Correspondence to Guang-Sheng Chen.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.

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