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

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 for p∈(2,3) and ρ>0 small, where Bρ:={u∈H1(ℝ3):∥u∥2=ρ}. Moreover, the properties of Iρ2/ρ2 and the associated Lagrange multiplier λρ are also given if p∈(2,8/3]