Nontrivial Solutions for Schrödinger Equation with Local Super-Quadratic Conditions

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Abstract

This paper is dedicated to studying the semilinear Schrödinger equation
$$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\ u\in H^{1}({\mathbb {R}}^{N}), \end{array}\right. \end{aligned}$$
where \(V\in \mathcal {C}(\mathbb {R}^N, \mathbb {R})\) is sign-changing and either periodic or coercive and \(f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})\) is subcritical and local super-linear (i.e. allowed to be super-linear at some \(x\in \mathbb {R}^N\) and asymptotically linear at other \(x\in \mathbb {R}^N\)). Instead of the common condition that \(\lim _{|t|\rightarrow \infty }\frac{\int _{0}^{t} f(x, s)\mathrm {d}s}{t^2}=\infty \) uniformly in \(x\in \mathbb {R}^N\), we use a local super-quadratic condition \(\lim _{|t|\rightarrow \infty }\frac{\int _{0}^{t} f(x,s)\mathrm {d}s}{t^2}=\infty \) a.e. \(x\in G\) for some domain \(G\subset \mathbb {R}^N\) to show the existence of nontrivial solutions for the above problem.

Keywords

Schrödinger equation Superlinear Asymptotically linear Local super-quadratic conditions 

Mathematics Subject Classification

35J20 35J60 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China
  2. 2.Department of MathematicsHuaihua CollegeHuaihuaPeople’s Republic of China
  3. 3.School of Mathematics and Information SciencesGuangzhou UniversityGuangzhouPeople’s Republic of China

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