# Nontrivial Solutions for Schrödinger Equation with Local Super-Quadratic Conditions

Research paper by **Xianhua Tang, Xiaoyan Lin; Jianshe Yu**

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

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.