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

Abstract

This paper is concerned with the existence and multiplicity of solutions of a Dirichlet problem for p(.)- Kirchhoff-type equation $left{begin{matrix} M(int_{Omega }frac{left | triangledownu right |^{p(x)}}{p(x)})(-Delta _{p(x)}u)=f(x,u), in nonfrenchspacingOmega , u=0,nonfrenchspacingonpartialOmega. end{matrix}right.$ 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.