Existence of solution to a critical trace equation with variable exponent

Article type: Research Article

Authors: Fernández Bonder, Julián^{a; b; *} | Saintier, Nicolas^{c; d} | Silva, Analía^{a; b}

Affiliations: [a] IMAS, CONICET, Universidad de Buenos Aires, Buenos Aires, Argentina | [b] Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina. E-mails: jfbonder@dm.uba.ar, asilva@dm.uba.ar | [c] Instituto de Ciencias, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina | [d] Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina. E-mails: nsaintie@dm.uba.ar, nsaintie@ungs.edu.ar

Correspondence: [*] Corresponding author: Julián Fernández Bonder, IMAS, CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428), Buenos Aires, Argentina. E-mail: jfbonder@dm.uba.ar; URL: http://mate.dm.uba.ar/~jfbonder.

Abstract: In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the p(x)-Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration–compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain pass theorem.

Keywords: Sobolev embedding, variable exponents, critical exponents, concentration compactness

DOI: 10.3233/ASY-151289

Journal: Asymptotic Analysis, vol. 93, no. 1-2, pp. 161-185, 2015