Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case

Abstract

In present paper, we prove the existence of solutions (λ,u)∈R×H1(RN) to the following Schrödinger equation{−Δu(x)+V(x)u(x)+λu(x)=g(u(x))inRN0≤u(x)∈H1(RN),N≥3 satisfying the normalization constraint ∫RNu2dx=a. We treat the so-called mass super-critical case here. Under an explicit smallness assumption on V with lim|x|→∞⁡V(x)=supx∈RN⁡V(x) and some Ambrosetti-Rabinowitz type conditions on g, we can prove the existence of ground state normalized solutions for prescribed mass a>0. Furthermore, we emphasize that the mountain pass characterization of a minimizing solution of the probleminf⁡{∫[12|∇u|2+12V(x)u2−G(u)]dx:‖u‖L2(RN)2=a,P[u]=0}, where G(s)=∫0sg(τ)dτ andP[u]=∫[|∇u|2−12〈∇V(x),x〉u2−N(12g(u)u−G(u))]dx.