New super-quadratic conditions for asymptotically periodic Schrödinger equation

Abstract

This paper is dedicated to studying the semilinear Schrödinger equation $\left\{\begin{array}{ll}-\Nabla u+V(x)u=f(x, u), \ \ \ \ x\in {\R}^{N},u\in H^{1}({\R}^{N}),\end{array}\right.$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C({\mathbb{R}}^N)$, $V_0(x)$ is 1-periodic in each of $x_1,x_2,\dots ,x_N$ and $\sup [\sigma (-△+V_0)\cap (-\infty ,0)]<0<\inf [\sigma (-△+V_0)\cap (0,\infty )]$, $V_1\in C({\mathbb{R}}^N)$ and ${\lim }_{|x|\to \infty }{V}_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results.

16 pages,0 figures. arXiv admin note: text overlap with arXiv:1405.2607

References in corpus (1)

• Periodic nonlinear Schrödinger equation with application to photonic crystals