## Topological Methods in Nonlinear Analysis

### Multiplicity theorems for resonant and superlinear nonhomogeneous elliptic equations

#### Abstract

We consider nonlinear elliptic equations driven by the sum of a $p$-Laplacian ($p> 2$) and a Laplacian. We consider two distinct cases. In the first one, the reaction $f(z,\cdot)$ is $(p-1)$-linear near $\pm\infty$ and resonant with respect to a nonprincipal variational eigenvalue of $(-\Delta_{p},W_{0}^{1,p}(\Omega))$. We prove a multiplicity theorem producing three nontrivial solutions. In the second case, the reaction $f(z,\cdot)$ is $(p-1)$-superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. We prove two multiplicity theorems. In the first main result we produce six nontrivial solutions all with sign information and in the second theorem we have five nontrivial solutions. Our approach uses variational methods combined with the Morse theory, truncation methods, and comparison techniques.

#### Article information

Source
Topol. Methods Nonlinear Anal. Volume 48, Number 1 (2016), 283-320.

Dates
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.tmna/1475266381

Digital Object Identifier
doi:10.12775/TMNA.2016.048

Zentralblatt MATH identifier
1368.35088

#### Citation

Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D. Multiplicity theorems for resonant and superlinear nonhomogeneous elliptic equations. Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 283--320. doi:10.12775/TMNA.2016.048. https://projecteuclid.org/euclid.tmna/1475266381

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