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# New Super-quadratic Conditions for Asymptotically Periodic Schrödinger Equations

Published:2017-03-13
Printed: Jun 2017
• Xianhua Tang,
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R. China
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## Abstract

This paper is dedicated to studying the semilinear Schrödinger equation $$\left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N}, \\ u\in H^{1}({\mathbf{R}}^{N}), \end{array} \right.$$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results.
 Keywords: Schrödinger equation, superlinear, asymptotically periodic, ground state solutions of Nehari-Pankov type
 MSC Classifications: 35J20 - Variational methods for second-order elliptic equations 35J60 - Nonlinear elliptic equations

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