## Tohoku Mathematical Journal 2017 June SECOND SERIES VOL. 69, NO. 2

 Tohoku Math. J. 69 (2017), 262-286

Title ROBIN PROBLEMS WITH INDEFINITE AND UNBOUNDED POTENTIAL, RESONANT AT $-\INFTY$, SUPERLINEAR AT $+\INFTY$

Author Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu

(Received April 8, 2015, revised July 28, 2015)
Abstract. We consider a semilinear Robin problem with an indefinite and unbounded potential and a reaction which exhibits asymmetric behavior as $x\rightarrow\pm\infty$. More precisely it is sublinear near $-\infty$ with possible resonance with respect to the principal eigenvalue of the negative Robin Laplacian and it is superlinear at $+\infty$. Resonance is also allowed at zero with respect to any nonprincipal eigenvalue. We prove two multiplicity results. In the first one, we obtain two nontrivial solutions and in the second, under stronger regularity conditions on the reaction, we produce three nontrivial solutions. Our work generalizes the recent one by Recova-Rumbos (Nonlin. Anal. 112 (2015), 181--198).

Mathematics Subject Classification. Primary 35J20; Secondary 35J60, 58E05.

Key words and phrases. Indefinite and unbounded potential, Robin boundary condition, asymmetric reaction, critical groups, multiple nontrivial solutions.