# American Institute of Mathematical Sciences

July  2017, 16(4): 1293-1314. doi: 10.3934/cpaa.2017063

## Robin problems with indefinite linear part and competition phenomena

 1 Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece 2 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13,200585 Craiova, Romania 4 Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva ploščad 16, SI-1000 Ljubljana, Slovenia

* Corresponding author: Vicenţiu D. Rădulescu

Received  July 2016 Revised  February 2017 Published  April 2017

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.

Citation: Nikolaos S. Papageorgiou, Vicenšiu D. Rădulescu, Dušan D. Repovš. Robin problems with indefinite linear part and competition phenomena. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1293-1314. doi: 10.3934/cpaa.2017063
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