Nontrivial Solutions for Schrödinger Equation with Local Super-Quadratic Conditions

Research paper by Xianhua Tang, Xiaoyan Lin; Jianshe Yu

Indexed on: 01 May '18Published on: 02 Apr '18Published in: Journal of Dynamics and Differential Equations


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.