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 Ω⊂\RNΩ⊂\R�, N≥3�≥3, is a smooth bounded domain, M(t)=a+btκ�(�)=�+���, a,κ>0�,�>0, b≥0�≥0, λ� is a positive parameter, Δ2p(x)u=Δ(|Δu|p(x)−2Δu)Δ�(�)2�=Δ(|Δ�|�(�)−2Δ�) is the operator of fourth order called the p(x)�(�)-biharmonic operator, Δp(x)u=div(|∇u|p(x)−2∇u)Δ�(�)�=div⁡(|∇�|�(�)−2∇�) is the p(x)�(�)-Laplacian, p:¯¯¯¯Ω→\R�:Ω¯→\R is a log-H\"{o}lder continuous function and f:¯¯¯¯Ω×\R→\R�:Ω¯×\R→\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.