Normalized concentrating solutions to nonlinear elliptic problems

Abstract

We prove the existence of solutions (λ,v)∈R×H1(Ω) of the elliptic problem{−Δv+(V(x)+λ)v=vp in Ω,v>0,∫Ωv2dx=ρ. Any v solving such problem (for some λ) is called a normalized solution, where the normalization is settled in L2(Ω). Here Ω is either the whole space RN or a bounded smooth domain of RN, in which case we assume V≡0 and homogeneous Dirichlet or Neumann boundary conditions. Moreover, 1<p<N+2N−2 if N≥3 and p>1 if N=1,2. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of Ω as the prescribed mass ρ is either small (when p<1+4N) or large (when p>1+4N) or it approaches some critical threshold (when p=1+4N).