# American Institute of Mathematical Sciences

April  2018, 11(2): 323-344. doi: 10.3934/dcdss.2018018

## Double resonance for Robin problems with indefinite and unbounded potential

 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  December 2016 Revised  April 2017 Published  January 2018

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.

Citation: Nikolaos S. Papageorgiou, Patrick Winkert. Double resonance for Robin problems with indefinite and unbounded potential. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 323-344. doi: 10.3934/dcdss.2018018
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