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.