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

Abstract

In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent $$displaylines{ -Delta u = lambda u - alpha u^p+ u^{2^*-1}, quad u >0 , quad hbox{in } Omega,cr u=0, quad hbox{on } partialOmega. }$$ where $Omega subset mathbb{R}^n$, $nge 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.

Keywords

• Critical Sobolev exponent
• positive solutions
• bifurcation.