Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 135, pp. 1-8.

Title: Bifurcation of positive solutions for a semilinear equation
       with critical Sobolev exponent

Author: Yuanji Cheng (Malmo Univ., Sweden)
	 
Abstract: 
 In this note we consider bifurcation of positive solutions to the
 semilinear elliptic boundary-value problem with critical Sobolev
 exponent
 $$\displaylines{
 -\Delta u = \lambda u - \alpha u^p+ u^{2^*-1}, \quad u >0 ,
 \quad \hbox{in }  \Omega,\cr
 u=0,  \quad \hbox{on } \partial\Omega.
 }$$
 where $\Omega \subset \mathbb{R}^n$, $n\ge 3 $ is a bounded
 $C^2$-domain  $\lambda>\lambda_1$, $1<p < 2^* -1= \frac{n+2}{n-2} $
 and $\alpha >0$ is a bifurcation parameter.
 Brezis and Nirenberg [2] showed that a lower order (non-negative)
 perturbation can contribute to regain the compactness and whence
 yields existence of solutions. We study the equation with an
 indefinite perturbation and prove a bifurcation result of two
 solutions for this equation.
 
Submitted August 12, 2005. Published October 25, 2006.
Math Subject Classifications: 49K20, 35J65, 34B15.
Key Words: Critical Sobolev exponent; positive solutions; bifurcation.