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arXiv · 1404.0771

Existence of nontrivial solutions for periodic Schrodinger equations with new nonlinearities

Shaowei Chen·Dawei Zhang
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Abstract

We study the Schr\"{o}dinger equation: \begin{eqnarray} - \Delta u+V(x)u+f(x,u)=0,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{eqnarray} where $V$ is periodic and $f$ is periodic in the $x$-variables, $0$ is in a gap of the spectrum of the operator $-\Delta+V$. We prove that under some new assumptions for $f$, this equation has a nontrivial solution. Our assumptions for the nonlinearity $f$ are very weak and greatly different from the known assumptions in the literature.

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Shaowei Chen, Dawei Zhang. 2014-04-03. Existence of nontrivial solutions for periodic Schrodinger equations with new nonlinearities. https://arxiv.org/abs/1404.0771

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