Multiplicity and concentration for the nonlinear Schrödinger equation with critical frequency

Abstract:

We consider the nonlinear Schrödinger equation (E)ε² Δ v - V (x) v + | v |^{p - 1} v = 0 in R^N, 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 ∈ R^N : 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 {v<inf>k, ε</inf>}<inf>k ∈ N</inf> and {u<inf>k</inf>}<inf>k ∈ N</inf>, respectively, we show that for fixed k ∈ N and, up to rescaling v<inf>k, ε</inf>, the energy of v<inf>k, ε</inf> converges to the energy of u<inf>k</inf>. It is also shown that the solutions v<inf>k, ε</inf> 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.

Año de publicación:

2007

Keywords:

Fuente:

scopus