Infinitely many bound states for some nonlinear scalar field equations

Abstract

In this paper we consider the problem \(-\Delta u + a(x)u = \vert u\vert ^{p-2}u\) in \(\mathbb{R}^N\), where p > 2 and \(p 2. Assuming that the potential a(x) is a regular function such that \(\liminf_{\vert x\vert\rightarrow + \infty} a(x) = a_\infty > 0\) and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.