Existence of PrescribedL2-Norm Solutions for a Class of Schrödinger-Poisson Equation

摘要

By using the standard scaling arguments, we show that the infimum of the following minimization problem:Iρ2=inf{(1/2)∫ℝ3|∇u|2dx+(1/4)∬ℝ3(|u(x)|2|u(y)|2/|x-y|)dx dy−  (1/p)∫ℝ3|u|pdx:u∈Bρ}can be achieved forp∈(2,3)andρ>0small, whereBρ:={u∈H1(ℝ3):∥u∥2=ρ}. Moreover, the properties ofIρ2/ρ2and the associated Lagrange multiplierλρare also given ifp∈(2,8/3].