Existence of Prescribed${L}^{2}$-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: $I_{{\rho }^2}=\inf \{(1/2)\int {ℝ}^3{|\nabla u|}^{2}dx+(1/4)\iint {ℝ}^3({|u(x)|}^2{|u(y)|}^2/|x-y|)dxdy$− $(1/p)\int {ℝ}^3{|u|}^{p}dx:u\in B_{\rho }\}$can be achieved for $p\in (\mathrm{2,3})$and $\rho >\mathrm{0}$small, where $B_{\rho }:=\{u\in H^{\mathrm{1}}({ℝ}^{\mathrm{3}}):{\parallel u\parallel }_{\mathrm{2}}=\rho \}$. Moreover, the properties of $I_{{\rho }^{\mathrm{2}}}/{\rho }^{\mathrm{2}}$and the associated Lagrange multiplier ${\lambda }_{\rho }$are also given if $p\in (\mathrm{2,8}/\mathrm{3}]$.