Non-Nehari manifold method for asymptotically periodic Schrödinger equation

Abstract

We consider the semilinear Schrödinger equation $$\{\begin{array}{l}-△u+V(x)u=f(x,u),x\in {\mathbb{R}}^N,u\in H^1({\mathbb{R}}^N),\end{array}$$ where $f$ is a superlinear, subcritical nonlinearity. We mainly study 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 ${\lim }_{|x|\to \infty }{V}_1(x)=0$. Inspired by previous work of Li et al. \cite{LWZ}, Pankov \cite{Pa} and Szulkin and Weth \cite{Sz}, we develop a more direct approach to generalize the main result in \cite{Sz} by removing the "strictly increasing" condition in the Nehari type assumption on $f(x,t)/|t|$. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold ${\mathcal{N}}^0$ by using the diagonal method.

19 pages,0 figures, SCIENCE CHINA Mathematics 2014

References in corpus (1)

• Periodic nonlinear Schrödinger equation with application to photonic crystals

Cited by in corpus (4)

• Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials
• New super-quadratic conditions for asymptotically periodic Schrödinger equation
• Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems
• Existence of ground state solution and concentration of maxima for a class of indefinite variational problems