Mathematics, Vol. 12, Pages 1280: On an Anisotropic Logistic Equation

Mathematics, Vol. 12, Pages 1280: On an Anisotropic Logistic Equation

Mathematics doi: 10.3390/math12091280

Authors: Leszek Gasiński Nikolaos S. Papageorgiou

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 uλ* and establish the monotonicity of the map λ⟼uλ* and its asymptotic behaviour as the parameter λ decreases to the critical value λ^1(q)>0 (the principal eigenvalue of (−Δq,W01,q(Ω))).