# Multiple solutions for superlinear $p$-Laplacian Neumann problems

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Aizicovici, S., Papageorgiou, N.S. and Staicu, V. Osaka J. Math. 49 (2012), 699–740 MULTIPLE SOLUTIONS FOR SUPERLINEAR p-LAPLACIAN NEUMANN PROBLEMS Dedicated to the memory of Stefan Mirica SERGIU AIZICOVICI, NIKOLAOS S. PAPAGEORGIOU and VASILE STAICU� (Received June 28, 2010, revised February 1, 2011) Abstract Our main goal is to prove the existence of multiple solutions with precise sign information for a Neumann problem driven by the p-Laplacian differential operator with a (p � 1)-superlinear term which does not satisfy the Ambrosetti–Rabinowitz condition. Using minimax methods we show that the problem has five nontrivial smooth solutions, two positive, two negative and the fifth nodal. In the semilinear case (p D 2), using Morse theory, we produce a second nodal solution (for a total of six nontrivial smooth solutions). 1. Introduction In a recent paper [2], we studied the following nonlinear Neumann problem (1.1) 8 < : �4pu(z)C �ju(z)jp�2u(z) D f (z, u(z)) in , �u �n D 0 on � , where � RN is a bounded domain with a C2 boundary � , n is the outward unit normal on � , � > 0, 2 � p < 1 and 4p stands for the p-Laplacian differential operator defined by 4p u(z) D div � kDu(z)kp�2 R N Du(z) � . Also f (z, x) is a Caratheodory function which exhibits a (p � 1)-superlinear growth near �1. More precisely, it satisfies the so-called Ambrosetti–Rabinowitz condition (AR-condition, for short), which says that there exist � > p and M > 0 such that (1.2) 0 < �F(z, x) � f (z, x)x for a.a. z 2 , all jx j � M, 2000 Mathematics Subject Classification. 35J25, 35J70, 58E05. �Partially supported by the Portuguese Foundation for Sciences and Technology (FCT) under the sabbatical leave fellowship SFRH/BSAB/794/2008. 700 S. AIZICOVICI, N.S. PAPAGEORGIOU AND V. STAICU where F(z, x) D R x0 f (z, s) ds. Integrating (1.2) we obtain the weaker condition (1.3) c0jx j� � F(z, x) for a.a. z 2 , all jx j � M , and some c0 > 0. From (1.3) we infer the much weaker co

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