Dirichlet $(p,q)$-equations at resonance

Leszek Gasiński; Nikolaos S. Papageorgiou

doi:10.3934/dcds.2014.34.2037

Article Contents

2014, Volume 34, Issue 5: 2037-2060. Doi: 10.3934/dcds.2014.34.2037

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Dirichlet $(p,q)$-equations at resonance

Leszek Gasiński^1, and
Nikolaos S. Papageorgiou^2,

1.
Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków
2.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received: January 2013

Revised: July 2013

Published: May 2014

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Abstract

We consider a parametric nonlinear Dirichlet equation driven by the sum of a $p$-Laplacian and a $q$-Laplacian ($1 < q < p < +\infty$, $p ≥ 2$) and with a Carathéodory reaction which at $\pm\infty$ is resonant with respect to the principal eigenvalue $\widehat{\lambda}_1(p) > 0$ of $(-\Delta_p, W^{1,p}_0(\Omega))$. Using critical point theory, truncation and comparison techniques and critical groups (Morse theory), we show that for all small values of the parameter $\lambda>0$, the problem has at least five nontrivial solutions, four of constant sign (two positive and two negative) and the fifth nodal (sign-changing).

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Mathematics Subject Classification: 35J20, 35J60, 35J92, 58E05.

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