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On an Anisotropic Logistic Equation

Author

Listed:
  • 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
    Department of Mathematics, University of Craiova, 200585 Craiova, Romania)

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 ) ) ).

Suggested Citation

  • Leszek Gasiński & Nikolaos S. Papageorgiou, 2024. "On an Anisotropic Logistic Equation," Mathematics, MDPI, vol. 12(9), pages 1-13, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:9:p:1280-:d:1381401
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