Positive Radial Solutions of a Nonlinear Boundary Value Problem

In this work we study the following quasilinear elliptic equation: |x|^a?u ^–_u^div_{= 0} (a(|x|) + g(u))^? = |x^|ßup in ? on ?? where a is a positive continuous function, g is a nonnegative and nondecreasing continuous function, ? = B_R, is the ball of radius R > 0 centered at the origin in R^N, N = 3 and, the constants a, ß ? R, ? ? (0, 1) and p > 1. We derive a new Liouville type result for a kind of”broken equation”. This result together with blow-up techniques, a priori estimates and a fixed-point result of Krasnosel’skii, allow us to ensure the existence of a positive radial solution. In this paper we also obtain a non–existence result, proven through a variation of the Pohozaev identity.