Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

Abstract

We consider the following Schrödinger equation $$-{\hslash }^2\Delta u+V(x)u=\Gamma (x)f(u)\mathrm{in}{\mathbb{R}}^N,$$ where $u\in H^1({\mathbb{R}}^N)$, $u>0$, $\hslash >0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $\Gamma$ attains global maximum at the same point or $V$ attains global minimum and $\Gamma$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash >0$.