Positive solutions of logistic equations with dependence on gradient and nonhomogeneous Kirchhoff term

Consider the equation −M(x,‖u‖)Δpu=λf(x,u,∇u)−g(x,u,∇u)−M(x,‖u‖)Δpu=λf(x,u,∇u)−g(x,u,∇u) in Ω, u=0u=0 on ∂Ω. The main aim of this paper is to prove existence results for both non-degenerate and degenerate cases of the function M. Our approach relies on the fixed point index and the cone theoretic argument.