Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

A concave—convex elliptic problem involving the fractional Laplacian

C. Brändlea1, E. Coloradoa1, A. de Pabloa1 and U. Sáncheza1

a1 Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain (eduardo.colorado@uc3m.es)

Abstract

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave—convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

(Received January 25 2011)

(Accepted November 02 2011)