Perturbations of nonlinear eigenvalue problems

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

doi:10.3934/cpaa.2019068

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2019, Volume 18, Issue 3: 1403-1431. Doi: 10.3934/cpaa.2019068

This issue Previous Article A remark on norm inflation for nonlinear Schrödinger equations Next Article Hilbert transforms along double variable fractional monomials

Perturbations of nonlinear eigenvalue problems

1.
National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
2.
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
3.
Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
4.
Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
5.
Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia

* Corresponding author
* Corresponding author

Received: June 30, 2018

Revised: June 30, 2018

Published: May 2019

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Abstract

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $λ$ varies. We also show that there exists a minimal positive solution $\overline{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$. Special attention is given to the particular case of the $p$-Laplacian.

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

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