Multiplicity and concentration for the nonlinear Schrödinger equation with critical frequency

We consider the nonlinear Schrödinger equation equation(E)ε2Δv−V(x)v+|v|p−1v=0in RN, and the limit problem equation(L)Δu+|u|p−1u=0in Ω, with boundary condition u=0u=0 on ∂Ω∂Ω, where Ω=int{x∈RN:V(x)=infV=0} is assumed to be non-empty, connected and smooth. We prove the existence of an infinite number of solutions for (E)(E) and (L)(L) sharing the topology of their level sets, as seen from the Ljusternik–Schnirelman scheme. Denoting their solutions as {vk,ε}k∈N{vk,ε}k∈N and {uk}k∈N{uk}k∈N, respectively, we show that for fixed k∈Nk∈N and, up to rescaling vk,εvk,ε, the energy of vk,εvk,ε converges to the energy of ukuk. It is also shown that the solutions vk,εvk,ε for (E)(E) concentrate exponentially around ΩΩ and that, up to rescaling and up to a subsequence, they converge to a solution of (L)(L).