Double resonance for Robin problems with indefinite and unbounded potential

Nikolaos S. Papageorgiou; Patrick Winkert

doi:10.3934/dcdss.2018018

Article Contents

2018, Volume 11, Issue 2: 323-344. Doi: 10.3934/dcdss.2018018

This issue Previous Article Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method Next Article Ambrosetti-Prodi type result to a Neumann problem via a topological approach

Double resonance for Robin problems with indefinite and unbounded potential

Nikolaos S. Papageorgiou^1, and
Patrick Winkert^2,

1.
National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
2.
Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

Received: November 30, 2016

Revised: March 31, 2017

Published: April 2018

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Abstract

We study a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential term. The nonlinearity $f(x, s)$ is a Carathéodory function which is asymptotically linear as $ s\to ± ∞$ and resonant. In fact we assume double resonance with respect to any nonprincipal, nonnegative spectral interval $ \left[ \hat{λ}_k, \hat{λ}_{k+1}\right]$. Applying variational tools along with suitable truncation and perturbation techniques as well as Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of constant sign.

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

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