Translator Disclaimer
2013 Existence of Prescribed L 2 -Norm Solutions for a Class of Schrödinger-Poisson Equation
Yisheng Huang, Zeng Liu, Yuanze Wu
Abstr. Appl. Anal. 2013(SI61): 1-11 (2013). DOI: 10.1155/2013/398164

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 $p\in (\mathrm{2,8}/\mathrm{3}]$.

Citation

Download Citation

Yisheng Huang. Zeng Liu. Yuanze Wu. "Existence of Prescribed L 2 -Norm Solutions for a Class of Schrödinger-Poisson Equation." Abstr. Appl. Anal. 2013 (SI61) 1 - 11, 2013. https://doi.org/10.1155/2013/398164

Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 1298.35193
MathSciNet: MR3090286
Digital Object Identifier: 10.1155/2013/398164

Rights: Copyright © 2013 Hindawi

JOURNAL ARTICLE
11 PAGES


SHARE
Vol.2013 • No. SI61 • 2013
Back to Top