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# 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: http://dx.doi.org/10.4310/DPDE.2015.v12.n1.a2

#### Authors

#### 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.

Published 19 March 2015