Geodesic
Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent
Electronic Journal of Differential Equations, Tome 2006 (2006).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $alpha$ 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.$
Classification : 49K20, 35J65, 34B15
Keywords: critical Sobolev exponent, positive solutions, bifurcation 
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     author = {Cheng, Yuanji},
     title = {Bifurcation of positive solutions for a semilinear equation with critical {Sobolev} exponent},
     journal = {Electronic Journal of Differential Equations},
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     volume = {2006},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a20/}
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Cheng, Yuanji. Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent. Electronic Journal of Differential Equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a20/