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A concave—convex elliptic problem involving the fractional Laplacian

Published online by Cambridge University Press:  30 January 2013

C. Brändle
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain (eduardo.colorado@uc3m.es)
E. Colorado
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain (eduardo.colorado@uc3m.es)
A. de Pablo
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain (eduardo.colorado@uc3m.es)
U. Sánchez
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain (eduardo.colorado@uc3m.es)
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2013
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