Normalized concentrating solutions to nonlinear elliptic problems

We prove the existence of solutions (λ, v) ∈ R × H^1(\Omega) of the elliptic problem -\Delta v + (V (x) + λ)v= v^p in \Omega v > 0, \|v\|_{L^2(\R^N}=\rho Any v solving such problem (for some λ) is called a normalized solution, where the normalization is settled in L^2(\R^N). Here \Omega is either the whole space R^N or a bounded smooth domain of R^N, in which case we assume V ≡ 0 and homogeneous Dirichlet or Neumann boundary conditions. Moreover, 1 < p < (N+2)/(N−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 as the prescribed mass ρ is either small (whenp < 1 + 4/N ) or large (when p > 1 + 4/N ) or it approaches some critical threshold (when p= 1 + 4/N ).