Soluções normalizadas para equações de Schrödinger de massa supercrítica com potencial

Abstract

In this work, we are interested in proving the existence of normalized solutions to the Schrödinger equation -∆u+V(x)u+λu=|u|^(p-2)u, in R^N, in the supercritical mass case and subcritical Sobolev case, 2+4/N < p < 2^* ≡ 2N/(N-2)^+. The existence of a solution (u, λ) ϵ H^1(R^N) x R^N with a prescribed norm will be ensured under various technical conditions on the potential V:R^N → R. Firstly, we will prove the existence of a solution where V is positive and vanishing at infinity. Next, we will prove the existence of a solution if V is a negative potential. The respective solutions will be obtained as a critical point of functionals constrained to the sphere S_ρ in L^2(R^N) and λ will be a Lagrange multiplier. The results will be proved using Variational Methods.