Singular equations with variable exponents and concave-convex nonlinearities

Leszek Gasiński; Nikolaos S. Papageorgiou

doi:10.3934/dcdss.2022135

Article Contents

2023, Volume 16, Issue 6: 1414-1434. Doi: 10.3934/dcdss.2022135

This issue Previous Article On nonlocal Dirichlet problems with oscillating term Next Article Some recent results on singular $ p $-Laplacian systems

Singular equations with variable exponents and concave-convex nonlinearities

Leszek Gasiński^1, , and
Nikolaos S. Papageorgiou^2,

1.
Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Cracow, Poland
2.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

Corresponding author: Leszek Gasiński
Corresponding author: Leszek Gasiński

Received: February 2022

Revised: May 2022

Early access: July 2022

Published: June 2023

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Abstract

We consider a nonlinear anisotropic Dirichlet problem with a reaction that has the combined effects of three distinct nonlinearities: a parametric singular term, a parametric "concave" term (the parameter $ \lambda>0 $ is the same in both) and a nonparametric "convex" perturbation. So, the problem is a singular version of the well known "concave-convex" problem. We prove an existence and multiplicity result which is global in the parameter $ \lambda>0 $ (a bifurcation-type theorem). We also indicate some small improvements in the case of $ (p(z),q(z)) $ and $ p(z) $-equations.

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

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References

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