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On p-Laplace equations with concave terms and asymmetric perturbations

Published online by Cambridge University Press:  11 February 2011

D. Motreanu
Affiliation:
Department of Mathematics, University of Perpignan, 66860 Perpignan, France (motreanu@univ-perp.fr)
V. V. Motreanu
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer Sheva, Israel (motreanu@bgu.ac.il)
N. S. Papageorgiou
Affiliation:
Department of Mathematics, National Technical University, Athens 15780, Greece (npapg@math.ntua.gr)

Abstract

We consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a concave term and a nonlinear perturbation, which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)-superlinear on ℝ+ and (p − 1)-(sub)linear on ℝ. Using variational methods based on the critical point theory together with truncation techniques, Ekeland's variational principle, Morse theory and the lower-and-upper-solutions approach, we show that the problem has at least four non-trivial smooth solutions. Also, we provide precise information about the sign of these solutions: two are positive, one is negative and one is nodal (sign changing).

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2011

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