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

DOI: http://dx.doi.org/10.12775/TMNA.2015.067

#### Abstract

\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.

#### Keywords

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