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.
Citation
Michael E. Filippakis. Donal O'Regan. Nikolaos S. Papageorgiou. "Multiple and nodal solutions for nonlinear equations with a nonhomogeneous differential operator and concave-convex terms." Tohoku Math. J. (2) 66 (4) 583 - 608, 2014. https://doi.org/10.2748/tmj/1432229198
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