Dynamics of Partial Differential Equations

Volume 12 (2015)

Number 1

Nodal and multiple solutions for nonlinear elliptic equations involving a reaction with zeros

Pages: 13 – 42

DOI: https://dx.doi.org/10.4310/DPDE.2015.v12.n1.a2

Authors

Leszek Gasinski (Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland)

Nikolaos S. Papageorgiou (Department of Mathematics, Zografou Campus, National Technical University, Athens, Greece)

Abstract

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a Carathéodory reaction. Imposing control on the growth of the reaction only near zero and assuming that it has two constant sign $z$-dependent zeros, we prove two multiplicity theorem producing three nontrivial smooth solutions, one positive, the second negative and the third nodal. Then for the particular case of $(p, 2)$-equations and assuming that the reaction is $(p - 1)$-superlinear near $\pm \infty$, without satisfying the Ambrosetti-Rabinowitz condition, we show that the problem has at least six nontrivial smooth solutions.

Keywords

local minimizers, critical groups, constant sign and nodal solutions, truncations, multiplicity theorems, extremal solutions

2010 Mathematics Subject Classification

Primary 35J20. Secondary 35J60, 35J92, 58E05.

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Published 19 March 2015