On an Anisotropic Logistic Equation

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 lambda. 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(lambda)* and establish the monotonicity of the map lambda bar right arrow u(lambda)* and its asymptotic behaviour as the parameter lambda decreases to the critical value (lambda(1)) over cap (q) > 0 (the principal eigenvalue of (-Delta(q),W-0(1,q)(Omega))).