Existence of Solutions for Semilinear Integro-differential Equations of p-Kirchhoff Type

Abstract

In our research we will study the existence of weak solutions to the problem $$ -[M(\|u\|^{p}_{1,p})]^{p-1}\Delta_{p} u = f(x,u)+\int_{\Omega}k(x,y)H(u)dy \quad \mbox{in }\Omega,$$ \noindent with zero Dirichlet boundary condition on a bounded smooth domain of $\mathbb{R}^{n} $, $ $ $1<p<N$; $M$,$f$,$k$ and $H$ are given functions. By means of the Galerkin method and using of the Brouwer Fixed Point theorem we get our results. The uniqueness of a weak solution is also considered.