## Tohoku Mathematical Journal

### Multiple and nodal solutions for nonlinear equations with a nonhomogeneous differential operator and concave-convex terms

#### Abstract

In this paper we consider a nonlinear parametric Dirichlet problem driven by a nonhomogeneous differential operator (special cases are the $p$-Laplacian and the $(p,q)$-differential operator) and with a reaction which has the combined effects of concave (($p-1)$-sublinear) and convex ($(p-1)$-superlinear) terms. We do not employ the usual in such cases AR-condition. Using variational methods based on critical point theory, together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small $\lambda>0$ ($\lambda$ is a parameter), the problem has at least five nontrivial smooth solutions (two positive, two negative and the fifth nodal). We also prove two auxiliary results of independent interest. The first is a strong comparison principle and the second relates Sobolev and Hölder local minimizers for $C^1$ functionals.

#### Article information

Source
Tohoku Math. J. (2), Volume 66, Number 4 (2014), 583-608.

Dates
First available in Project Euclid: 21 May 2015

https://projecteuclid.org/euclid.tmj/1432229198

Digital Object Identifier
doi:10.2748/tmj/1432229198

Mathematical Reviews number (MathSciNet)
MR3350285

Zentralblatt MATH identifier
1325.35020

#### Citation

Filippakis, Michael E.; O'Regan, Donal; Papageorgiou, Nikolaos S. Multiple and nodal solutions for nonlinear equations with a nonhomogeneous differential operator and concave-convex terms. Tohoku Math. J. (2) 66 (2014), no. 4, 583--608. doi:10.2748/tmj/1432229198. https://projecteuclid.org/euclid.tmj/1432229198

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