Existence of entire solutions for semilinear elliptic problems on ${\mathbb R}^{N}$

Abstract

In this paper, we consider the existence of positive and negative entire solutions of semilinear elliptic problem $$ -\Delta u + u = g(x,u), \quad u \in H^{1}({\mathbb R}^{N})\tag{P} $$ where $N \geq 2$ and $g:{\mathbb R}^{N} \times {\mathbb R }\to {\mathbb R}$ is a continuous function with superlinear growth and $g(x,0) = 0$ on ${\mathbb R}^{N} $.