Tohoku Mathematical Journal

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

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

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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

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

First available in Project Euclid: 21 May 2015

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Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations

Nonlinear nonhomogeneous differential operator nonlinear regularity theory nonlinear maximum principle local minimizers strong comparison principle constant sign solutions nodal solutions concave-convex nonlinearities


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.

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