## Tohoku Mathematical Journal 2014 December SECOND SERIES VOL. 66, NO. 4

 Tohoku Math. J. 66 (2014), 583-608

Title MULTIPLE AND NODAL SOLUTIONS FOR NONLINEAR EQUATIONS WITH A NONHOMOGENEOUS DIFFERENTIAL OPERATOR AND CONCAVE-CONVEX TERMS

Author Michael E. Filippakis, Donal O'Regan and Nikolaos S. Papageorgiou

(Received April 3, 2013, revised November 25, 2013)
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.

Mathematics Subject Classification. Primary 35J20; Secondary 35J60, 35J70.

Key words and phrases. Nonlinear nonhomogeneous differential operator, nonlinear regularity theory, nonlinear maximum principle, local minimizers, strong comparison principle, constant sign solutions, nodal solutions, concave-convex nonlinearities.