Electron. J. Diff. Eqns., Vol. 2006(2006), No. 135, pp. 1-8.

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

Yuanji Cheng

Abstract:
In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent
$$\displaylines{ -\Delta u = \lambda u - \alpha u^p+ u^{2^*-1}, \quad u greater than 0 , \quad \hbox{in } \Omega,\cr u=0, \quad \hbox{on } \partial\Omega. }$$
where $\Omega \subset \mathbb{R}^n$, $n\ge 3 $ is a bounded $C^2$-domain $\lambda greater than \lambda_1$, 1 less than p less than $2^* -1= \frac{n+2}{n-2}$ and $\alpha$ greater than 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.

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Yuanji Cheng
School of Technology and Society
Malmo University
SE-205 06 Malmo, Sweden
email: yuanji.cheng@ts.mah.se

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