On the existence of solutions for a class of fourth order elliptic equations of Kirchhoff type with variable exponent

Abstract

In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent

$$

\left\{\begin{array}{ll}

\Delta^2_{p(x)}u-M\left(\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx\right)\Delta_{p(x)} u  = \lambda f(x,u) \quad \text{ in }\Omega,\\

u=\Delta u = 0 \quad \text{ on } \partial\Omega, 

\end{array}\right.

$$

where $\Omega \subset \R^N$, $N \geq 3$, is a smooth bounded domain, $M(t)=a+bt^\kappa$, $a, \kappa>0$, $b \geq 0$, $\lambda$ is a positive parameter, $\Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u)$ is the operator of fourth order called the $p(x)$-biharmonic operator, $\Delta_{p(x)}u = \operatorname{div} \left(|\nabla u|^{p(x)-2}\nabla u\right)$ is the $p(x)$-Laplacian, $p:\overline\Omega \to \R$ is a log-H\"{o}lder continuous function and $f: \overline\Omega\times \R\to \R$ is a continuous function satisfying some certain conditions. Using Ekeland's variational principle combined with variational techniques, an existence result is established in an appropriate function space.

Keywords

• Fourth order elliptic equations,Kirchhoff type problems,Variable exponents,Ekeland's variational principle

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