Existence of solution for a class of nonlocal problem via dynamical methods

Abstract

In this paper we use the dynamical methods to establish the existence of nontrivial solution for a class of nonlocal problem of the type $$ \left\{\begin{array}{l} -a\left(x,\int_Ωg(u)\,dx \right)Δu =f(u), \quad x \in Ω\\ u=0, \hspace{2 cm} x \in \partial Ω, \end{array}\right. \leqno{(P)} $$ where $\Omega \subset {\mathbb{R}}^N(N\ge 2)$ is a smooth bounded domain and $a:\overline{\Omega }\times \mathbb{R}\to \mathbb{R}$ and $g,f:\mathbb{R}\to \mathbb{R}$ are $C^1$-functions that satisfy some technical conditions.