Hokkaido Mathematical Journal

PRINARI Francesca, VISCIGLIA Nicola,

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

Hokkaido Mathematical Journal, 37 (2008) pp.611-625

Fulltext

PDF

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 × R^n$ 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.

MSC(Primary)35J60
MSC(Secondary)35B20, 47J30
Uncontrolled Keywordsstanding waves, minimization problems, compact perturbations.