Solutions to a nonlinear Schr\"odinger equation with periodic potential and zero on the boundary of the spectrum

Jarosław Mederski

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arXiv · 1308.4320

Solutions to a nonlinear Schr\"odinger equation with periodic potential and zero on the boundary of the spectrum

Jarosław Mederski

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Abstract

We study the following nonlinear Schr\"odinger equation $$-\Delta u + V(x) u = g(x,u),$$ where V and g 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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Jarosław Mederski. 2013-08-20. Solutions to a nonlinear Schr\"odinger equation with periodic potential and zero on the boundary of the spectrum. https://arxiv.org/abs/1308.4320

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