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 $$ \left\{\begin{array}{l} -a\left(x,\int_{\Omega}g(u)\,dx \right)\Delta u =f(u), \quad x \in \Omega \\ u=0, \hspace{2 cm} x \in \partial \Omega, \end{array}\right. \leqno{(P)} $$ where $\Omega \subset \mathbb{R}^N \, ( N \geq 2)$ 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.