Note on semiclassical states for the Schr\"{o}dinger equation with nonautonomous nonlinearities

TL;DR

This paper investigates the existence of positive semiclassical states for a Schrödinger equation with nonautonomous nonlinearities, showing solutions under specific conditions on the potential and coefficient functions as the semiclassical parameter becomes small.

Contribution

It establishes the existence of positive solutions for the Schrödinger equation with variable potentials and nonlinearities under certain extremal conditions, extending previous results to nonautonomous cases.

Findings

Positive solutions exist for small  under specific potential and coefficient conditions.

Solutions concentrate near points where potential and coefficient functions attain extremal values.

The results apply to superlinear, subcritical nonlinearities.

Abstract

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 $\hslash>0$.