Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 37, 2012, 579-594

EXISTENCE OF SOLUTION TO A CRITICAL EQUATION WITH VARIABLE EXPONENT

Julián Fernández Bonder, Nicolas Saintier and Analía Silva

IMAS - CONICET and Universidad de Buenos Aires, Departamento de Matemática, FCEyN
Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina; jfbonder 'at' dm.uba.ar

Universidad Nacional de General Sarmiento, Instituto de Ciencias
Juan María Gutierrez 1150 Los Polvorines, Pcia de Bs. As., Argentina; nsaintie 'at' ungs.edu.ar
and Universidad de Buenos Aires, Departamento de Matemática, FCEyN
Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina; nsaintie 'at' dm.uba.ar

IMAS - CONICET and Universidad de Buenos Aires, Departamento de Matemática, FCEyN
Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina; asilva 'at' dm.uba.ar

Abstract. In this paper we study the existence problem for the p(x)-Laplacian operator with a nonlinear critical source. We find a local condition on the exponents ensuring the existence of a nontrivial solution that shows that the Pohozaev obstruction does not holds in general in the variable exponent setting. The proof relies on the Concentration-Compactness Principle for variable exponents and the Mountain Pass Theorem.

2010 Mathematics Subject Classification: Primary 35J92, 35B33.

Key words: Sobolev embedding, variable exponents, critical exponents, concentration compactness.

Reference to this article: J. Fernández Bonder, N. Saintier and A. Silva: Existence of solution to a critical equation with variable exponent. Ann. Acad. Sci. Fenn. Math. 37 (2012), 579-594.

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doi:10.5186/aasfm.2012.3743