Positive solutions for the p-Laplacian with dependence on the gradient

We prove a result of existence of positive solutions for the p-Laplacian problem −Δpu = ω(x)f(u, |∇u|) with Dirichlet boundary condition in a bounded domain Ω subset of RN, where ω is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behaviour is assumed on f, but simple geometric assumptions in a neighbourhood of the first eigenvalue of the p-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder fixed point theorem and this result is used to construct an ordered pair of sub- and super-solution, also valid for nonlinearities which are super-linear at both the origin and +∞, which is a remarkable fact. We apply our method to the p-growth problem −Δpu = λu(x)q−1(1 + |∇u(x)|p) (1 < q < p) in Ω with Dirichlet boundary conditions and give examples of super-linear nonlinearities which are also handled by our method