On an Anisotropic Logistic Equation

Leszek Gasiński () and Nikolaos S. Papageorgiou
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Leszek Gasiński: Department of Mathematics, University of the National Education Commission, Krakow, Podchorazych 2, 30-084 Krakow, Poland
Nikolaos S. Papageorgiou: Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece

Mathematics, 2024, vol. 12, issue 9, 1-13

Abstract: We consider a nonlinear Dirichlet problem driven by the ( p ( z ) , q ) -Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ . If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution u λ * and establish the monotonicity of the map λ ⟼ u λ * and its asymptotic behaviour as the parameter λ decreases to the critical value λ ^ 1 ( q ) > 0 (the principal eigenvalue of ( − Δ q , W 0 1 , q ( Ω ) ) ).

Keywords: anisotropic operator; equidiffusive logistic reaction; uniqueness; minimal positive solution; anisotropic regularity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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