Multiplicity and concentration for the nonlinear Schrödinger equation with critical frequency (Q860728)

The authors consider the nonlinear Schrödinger equation \(\epsilon{^{2}}\Delta v-V(x)v+| v| {^{p-1}}v=0\) in \({\mathbb{R}}{^{n}}\) with \(\epsilon>0,\) \(n\geq 3,\) and \(p\in(1,(n+2)/(n-2)),\) where \(V\) is a nonnegative, continuous potential that becomes infinite as \(| x| \to \infty.\) By applying the Ljusternik-Schnirelman theory, it is shown that there are infinitely many solutions to this equation and such solutions decay exponentially as \(| x| \to \infty.\) It is also shown that, as \(\epsilon\to 0,\) such solutions satisfy \(\Delta u+| u| {^{p-1}}u=0\) in a bounded, connected set \(\Omega\) with \(u=0\) on the boundary of \(\Omega.\) A similar problem has been analyzed by \textit{Y. Ding} and \textit{A. Szulkin} [in: T. Cazenave et al. (eds.), Contributions to nonlinear analysis, Birkhäuser, Basel, pp. 221--231 (2006; Zbl 1131.35332)].