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Journal of Convex Analysis 24 (2017), No. 3, 769--793
Copyright Heldermann Verlag 2017

Asymmetric, Noncoercive, Superlinear (p,2)-Equations

Nikolaos S. Papageorgiou
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

Vicentiu D. Radulescu
Institute of Mathematics, Romanian Academy of Sciences, P. O. Box 1-764, 014700 Bucharest, Romania


We examine a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a $p$-Laplacian $(p\geq 2)$ and a Laplacian (a $(p,2)$-equation). The reaction term is asymmetric and it is superlinear in the positive direction and sublinear in the negative direction. The superlinearity is not expressed using the Ambrosetti-Rabinowitz condition, while the asymptotic behavior as $x\rightarrow-\infty$ permits resonance with respect to any nonprincipal eigenvalue of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. Using variational methods based on the critical point theory and Morse theory (critical groups), we prove a multiplicity theorem producing three nontrivial solutions.

Keywords: (p,2)-equation, asymmetric reaction, superlinear growth, multiple solutions, nonlinear regularity, critical groups.

MSC: 35J20, 35J60, 58E05

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