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