# Hokkaido Mathematical Journal

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

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### 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 35B20, 47J30 standing waves, minimization problems, compact perturbations.