Nonlinear Dirichlet problems with a crossing reaction

Shouchuan Hu; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2014.13.2749

Article Contents

2014, Volume 13, Issue 6: 2749-2766. Doi: 10.3934/cpaa.2014.13.2749

This issue Previous Article The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas Next Article

Nonlinear Dirichlet problems with a crossing reaction

Shouchuan Hu^1, and
Nikolaos S. Papageorgiou^2,

1.
College of Mathematics, Shandong Normal University, Jinan, Shandong
2.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received: December 31, 2013

Revised: May 31, 2014

Published: October 2014

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Abstract

We consider a nonlinear Dirichlet problem driven by the sum of a $p$-Laplacian ($p>2$) and a Laplacian, with a reaction that is ($p-1$)-sublinear and exhibits an asymmetric behavior near $\infty$ and $-\infty$, crossing $\hat{\lambda}_1>0$, the principal eigenvalue of $(-\Delta_p, W^{1,p}_0(\Omega))$ (crossing nonlinearity). Resonance with respect to $\hat{\lambda}_1(p)>0$ can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction $C^1$ in the $x$-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.

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Mathematics Subject Classification: 35B34, 35J20, 35J60, 58E05.

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