Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 723-738

REGULARITY OF STABLE SOLUTIONS TO QUASILINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MODELS

Rodrigo G. Clemente and João Marcos do Ó

Rural Federal University of Pernambuco, Department of Mathematics
52171-900, Recife, Pernambuco, Brazil; rodrigo.clemente 'at' ufrpe.br

Brasília University, Department of Mathematics
70910-900, Brasília, DF, Brazil; jmbo 'at' pq.cnpq.br

Abstract. We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equations involving the p-Laplace Beltrami operator with zero Dirichlet boundary conditions.

2010 Mathematics Subject Classification: Primary 35B35, 35D10, 35J62, 35J70, 35J75.

Key words: Nonlinear PDE of elliptic type, p-Laplacian, singular non-linearity, semi-stable solutions, extremal solutions, regularity.

Reference to this article: R. G. Clemente and J. M. do Ó: Regularity of stable solutions to quasilinear elliptic equations on Riemannian models. Ann. Acad. Sci. Fenn. Math. 44 (2019), 723-738.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4448