Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

Abstract

We consider the following Schrödinger equation −ℏ2Δu+V(x)u=Γ(x)f(u)inRN,where u∈H1(RN), u>0, ℏ>0 and f is superlinear and subcritical nonlinear term. We show that if V attains local minimum and Γ attains global maximum at the same point or V attains global minimum and Γ attains local maximum at the same point, then there exists a positive solution for sufficiently small ℏ>0.