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

Abstract

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].

Multiple normalized solutions of Chern–Simons–Schrödinger system

“…[1,2,[6][7][8]13,17,20,22,26]. In [7,8,13], for p in some ranges and c > 0, by using the concentration-compactness method of Lions [18,19], the authors obtained the minimizers oñ…”

Multiplicity of normalized solutions for a class of nonlinear Schrödinger–Poisson–Slater equations

“…The above normalized problem associated to (E λ ), has been studied in the literature [8,9,10,11,15,16,26]. In the cited references, the existence and non-existence of normalized solutions to (E λ ) are established, depending strongly on the value of p ∈ (2, 6) and of the parameter c > 0.…”