Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

Bartosz Bieganowski, Jarosław Mederski

Published 2018-08-28Version 1

We consider the following Schr\"{o}dinger equation $$ - \hslash ^2 \Delta u + V(x)u = \Gamma(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ 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 $\hslash>0$.