Existence of saddle-type solutions for a class of quasilinear problems in R^2

Abstract

The main goal of the present paper is to prove the existence of saddle-type solutions for the following class of quasilinear problems $$ -\Delta_{\Phi}u + V'(u)=0\quad \text{in }\mathbb{R}^2, $$% where $$ \Delta_{\Phi}u=\text{div}(\phi(|\nabla u|)\nabla u), $$% $\Phi\colon \mathbb{R}\rightarrow [0,+\infty)$ is an N-function and the potential $V$ satisfies some technical condition and we have as an example $ V(t)=\Phi(|t^2-1|)$.