Normalized concentrating solutions to nonlinear elliptic problems

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