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 (


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)