New super-quadratic conditions for asymptotically periodic Schr\"odinger equation

TL;DR

This paper introduces new super-quadratic conditions for the asymptotically periodic Schrödinger equation, broadening the class of nonlinearities for which solutions can be established, especially under weaker assumptions than previous results.

Contribution

It proposes novel super-quadratic conditions that are less restrictive than existing ones, enabling the analysis of solutions to asymptotically periodic Schrödinger equations.

Findings

Established weaker super-quadratic conditions for Schrödinger equations.

Extended the existence theory to broader classes of nonlinearities.

Provided analytical framework for asymptotically periodic potentials.

Abstract

This paper is dedicated to studying the semilinear Schr\"odinger 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 $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\R^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]<0<\inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\R^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results.