Existence and multiplicity of solutions for p(.)-Kirchhoff-type equations

Abstract

his paper is concerned with the existence and multiplicity of solutions of a Dirichlet problem for $p(.)$-Kirchhoff-type equation% \begin{equation*} \left\{ \begin{array}{c} M\left( \int_{\Omega }\frac{\left\vert \nabla u\right\vert ^{p(x)}}{p(x)}% dx\right) \left( -\Delta _{p(x)}u\right) =f(x,u),\text{ in }\Omega , \\ u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on }\partial \Omega .% \end{array}% \right. \end{equation*}% Using the mountain pass theorem, fountain theorem, dual fountain theorem and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on $f$ and $M$, we obtain results on existence and multiplicity of solutions.