Osaka Journal of Mathematics

Multiple solutions for superlinear $p$-Laplacian Neumann problems

Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu

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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 ($p = 2$), using Morse theory, we produce a second nodal solution (for a total of six nontrivial smooth solutions).

Article information

Osaka J. Math., Volume 49, Number 3 (2012), 699-740.

First available in Project Euclid: 15 October 2012

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Primary: 35J25: Boundary value problems for second-order elliptic equations 35J70: Degenerate elliptic equations 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Aizicovici, Sergiu; Papageorgiou, Nikolaos S.; Staicu, Vasile. Multiple solutions for superlinear $p$-Laplacian Neumann problems. Osaka J. Math. 49 (2012), no. 3, 699--740.

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