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Wang's multiplicity result for superlinear $ (p,q)$-equations without the Ambrosetti-Rabinowitz condition


Authors: Dimitri Mugnai and Nikolaos S. Papageorgiou
Journal: Trans. Amer. Math. Soc. 366 (2014), 4919-4937
MSC (2010): Primary 35J20; Secondary 35J60, 35J92, 58E05
DOI: https://doi.org/10.1090/S0002-9947-2013-06124-7
Published electronically: October 28, 2013
MathSciNet review: 3217704
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Abstract: We consider a nonlinear elliptic equation driven by the sum of a $ p$-Laplacian and a $ q$-Laplacian, where $ 1<q\leq 2\leq p<\infty $ with a $ (p-1)$-superlinear Carathéodory reaction term which doesn't satisfy the usual Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the problem has at least three nontrivial solutions; among them one is positive and one is negative.


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Additional Information

Dimitri Mugnai
Affiliation: Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
Email: mugnai@dmi.unipg.it

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

DOI: https://doi.org/10.1090/S0002-9947-2013-06124-7
Keywords: Superlinear reaction, Ambrosetti--Rabinowitz condition, strong deformation retract, nonlinear regularity, Morse relation, critical groups
Received by editor(s): August 6, 2012
Received by editor(s) in revised form: January 23, 2013
Published electronically: October 28, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.