MULTIPLE POSITIVE SOLUTIONS AND BIFURCATION FOR AN EQUATION RELATED TO CHOQUARD’S EQUATION

Proceedings of the Edinburgh Mathematical Society

Tassilo Küpper¹ 

[1] Zhengjie Zhang

关键词: multiple solutions;    Choquard’s equation;    bifurcation;   

DOI  :  10.1017/S0013091502000779

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

In this paper we study the existence of multiple positive solutions and the bifurcation problem for the following equation:$$ -Delta u+u=iggl(int_{mathbb{R}^3}frac{|u(y)|^2}{|x-y|},mathrm{d}yiggr)u+mu f(x),quad xinmathbb{R}^3, $$where $f(x)in H^{-1}(mathbb{R}^3)$, $f(x)geq0$, $f(x)otequiv0$. We show that there are positive constants $mu^{*}$ and $mu^{**}$ such that the above equation possesses at least two positive solutions for $muin(0,mu^{*})$, and no positive solution for $mu>mu^{**}$. Furthermore, we prove that $mu=mu^{*}$ is a bifurcation point for the equation under study.AMS 2000 Mathematics subject classification: Primary 35J60; 35J70

【 授权许可】

Unknown