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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Multiple nontrivial solutions for nonlinear eigenvalue problems


Authors: D. Motreanu, V. V. Motreanu and N. S. Papageorgiou
Journal: Proc. Amer. Math. Soc. 135 (2007), 3649-3658
MSC (2000): Primary 35J60; Secondary 35J70
Published electronically: August 7, 2007
MathSciNet review: 2336581
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Abstract: In this paper we study a nonlinear eigenvalue problem driven by the $ p$-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.


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Additional Information

D. Motreanu
Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France
Email: motreanu@univ-perp.fr

V. V. Motreanu
Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France
Email: viorica@univ-perp.fr

N. S. Papageorgiou
Affiliation: Department of Mathematics, National Technical University, Athens 15780, Greece
Email: npapg@math.ntua.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08927-7
PII: S 0002-9939(07)08927-7
Keywords: Nonlinear eigenvalue problem, $p$-Laplacian, principal and second eigenvalue, upper-lower solution, truncation, strong maximum principle, second deformation lemma
Received by editor(s): August 1, 2006
Received by editor(s) in revised form: September 6, 2006
Published electronically: August 7, 2007
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society