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June  2009, 24(2): 405-440. doi: 10.3934/dcds.2009.24.405

Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation

Michael Filippakis 1, , Alexandru Kristály 2, and  Nikolaos S. Papageorgiou 3,

1. 

National Technical University, Department of Mathematics, Zografou Campus, Athens, 15780, Greece
2. 

Babeş-Bolyai University, Department of Economics, 400591 Cluj-Napoca
3. 

Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received  March 2008 Revised  December 2008 Published  March 2009

We consider nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential depending on a parameter $\lambda\ >\ 0$. The main result guarantees the existence of two positive, two negative and a nodal (sign-changing) solution for the studied problem whenever $\lambda\ >\ 0$ belongs to a small interval (0, λ*) and $p$ ≥ 2. We do not impose any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by using variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass theorem, and a Brézis-Nirenberg type result for C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.
Keywords: upper-lower solutions., multiple solutions, nonsmooth critical point theory, positive solution, $p$-Laplacian, nodal solution.
Mathematics Subject Classification: Primary: 35J20, 35J60, 58J7.
Citation: Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405
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