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

Year 2019, Volume: 3 Issue: 1, 35 - 45, 31.03.2019

Nguyen Thanh Chung

https://doi.org/10.31197/atnaa.495567

Cited By: 1

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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