Abstract

We consider the nonlinear Schrödinger equation (E)ε2 Δ v - V (x) v + | v |p - 1 v = 0 in RN, and the limit problem (L)Δ u + | u |p - 1 u = 0 in Ω, with boundary condition u = 0 on ∂ Ω, where Ω = int {x ∈ RN : V (x) = inf V = 0} is assumed to be non-empty, connected and smooth. We prove the existence of an infinite number of solutions for (E) and (L) sharing the topology of their level sets, as seen from the Ljusternik-Schnirelman scheme. Denoting their solutions as {vk, ε}k ∈ N and {uk}k ∈ N, respectively, we show that for fixed k ∈ N and, up to rescaling vk, ε, the energy of vk, ε converges to the energy of uk. It is also shown that the solutions vk, ε for (E) concentrate exponentially around Ω and that, up to rescaling and up to a subsequence, they converge to a solution of (L). © 2005 Elsevier Ltd. All rights reserved.