Appl Math Optim (2014) 69:393–430 DOI 10.1007/s00245-013-9227-z Qualitative Phenomena for Some Classes of Quasilinear Elliptic Equations with Multiple Resonance Nikolaos S. Papageorgiou · Vicen¸ tiu D. R ˘ adulescu Published online: 5 November 2013 © Springer Science+Business Media New York 2013 Abstract We consider nonlinear nonhomogeneous Dirichlet problems driven by the sum of a p-Laplacian and a Laplacian. The hypotheses on the reaction term incor- porate problems resonant at both ±∞ and zero. We consider both cases p> 2 and 1 <p< 2 (singular case) and we prove four multiplicity theorems producing three or four nontrivial solutions. For the case p> 2 we provide precise sign information for all the solutions. Our approach uses critical point theory, truncation and comparison techniques, Morse theory and the Lyapunoff-Schmidt reduction method. Keywords Strong comparison principle · Nonlinear maximum principle · Critical group · Nodal and constant sign solutions · Resonant equations · Lyapunoff-Schmidt reduction method 1 Introduction Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω . In this paper, we study the following nonlinear nonhomogeneous Dirichlet problem − p u(z) − u(z) = f ( z, u(z) ) in Ω, u| ∂Ω = 0. (1) N.S. Papageorgiou Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece e-mail: npapg@math.ntua.gr V.D. R˘ adulescu (B ) Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania e-mail: vicentiu.radulescu@imar.ro V.D. R˘ adulescu Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania