Standing waves for a class of nonlinear Schrödinger equations with potentials in $L^\infty$

Abstract

We prove the existence of standing waves to the following family of nonlinear Schrödinger equations:

ih∂_tψ = -h²Δψ + V (x)ψ - ψ|ψ|^p-2, (t, x) ∈ R × Rⁿ

provided that $h > 0$ is small, $2 < p < 2n/(n − 2)$ when $n ≥ 3$, $2 < p < ∞$ when $n = 1, 2$ and $V (x) ∈ L^∞(R^n)$ is assumed to have a sublevel with positive and finite measure.