Topological Methods in Nonlinear Analysis

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

Jarosław Mederski

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We study the following nonlinear Schrödinger equation \begin{equation*} \begin{cases} -\Delta u + V(x) u = g(x,u) & \hbox{for } x\in\mathbb R^N,\\ u(x)\to 0 & \hbox{as } |x|\to\infty, \end{cases} \end{equation*} where $V\colon \mathbb R^N\to\mathbb R$ and $g\colon \mathbb R^N\times\mathbb R\to\mathbb R$ 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.

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Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 755-771.

First available in Project Euclid: 21 March 2016

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Mederski, Jarosław. Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 755--771. doi:10.12775/TMNA.2015.067.

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