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 $\lambda>\lambda_1$, $10$ 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.