Existence and multiplicity of radially symmetric solutions for nonlinear Schrödinger equations

Abstract

In this paper we study the following nonlinear Schrödinger equations in $\mathbb{R}^N$: $$-\Delta u + V(x)u= g(u),\quad u \in H^1(\mathbb{R}^N), $$where $N \ge 2$, $V \in C^1(\mathbb{R}^N,\mathbb{R})$ and $g \in C(\mathbb{R},\mathbb{R})$. For a wide class of nonlinearities, which satisfy the Berestycki-Lions type condition, we show the existence and multiplicity of radially symmetric solutions. We use a new deformation argument under a new version of the Palais-Smale condition.