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

Michael Filippakis; Alexandru Kristály; Nikolaos S. Papageorgiou

doi:10.3934/dcds.2009.24.405

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2009, Volume 24, Issue 2: 405-440. Doi: 10.3934/dcds.2009.24.405

This issue Previous Article A semi-implicit spectral method for stochastic nonlocal phase-field models Next Article The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity

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

1.
National Technical University, Department of Mathematics, Zografou Campus, Athens, 15780
2.
Babeş-Bolyai University, Department of Economics, 400591 Cluj-Napoca
3.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received: February 29, 2008

Revised: November 30, 2008

Published: May 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J20, 35J60, 58J70.

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