Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum

Abstract

We study the following nonlinear Schrodinger equation\begin{equation*}\begin{cases} -\Delta u + V(x) u = g(x,u) & \hbox{for } x\in\R^N,\\ u(x)\to 0 & \hbox{as } |x|\to\infty,\end{cases}\end{equation*}where $V\colon \R^N\to\R$ and $g\colon \R^N\times\R\to\R$ are periodic in $x$. We assume that $0$ is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical term g satisfies a Nehari type monotonicity condition. We employ a Nehari manifold type technique in a strongly indefitnite setting and obtain the existence of a ground state solution. Moreover, we get infinitely many geometrically distinct solutions provided that $g$ is odd.

Locations

• Topological Methods in Nonlinear Analysis - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF