Robin problems with indefinite linear part and competition phenomena

Nikolaos S. Papageorgiou; Vicenšiu D. Rădulescu; Dušan D. Repovš

doi:10.3934/cpaa.2017063

Article Contents

2017, Volume 16, Issue 4: 1293-1314. Doi: 10.3934/cpaa.2017063

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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
^* Corresponding author: Vicenţiu D. Rădulescu

Received: July 2016

Revised: February 2017

Published: May 2017

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Abstract

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 $\lambda \mapsto \tilde{u}_\lambda$.

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Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35J92.

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