Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - February 2006 - Volume 49, Issue 01  

Research Article

MULTIPLE POSITIVE SOLUTIONS FOR A CRITICAL GROWTH PROBLEM WITH HARDY POTENTIAL

Pigong Hana1

a1 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhong Guancun East Road, Beijing 100080, China (pghan@amss.ac.cn)

Abstract

In this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:

$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$

where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.

(Received December 04 2004)

Keywords

  • Primary 35J60;
  • 35B33;
  • positive solution;
  • subsolution and supersolution;
  • Palais–Smale condition;
  • critical Sobolev exponent