Positive Radial Solutions of a Nonlinear Boundary Value Problem

In this work we study the following quasilinear elliptic equation: |x|a?u -udiv= 0 (a(|x|) + g(u))? = |x|ßup in ? on ?? where a is a positive continuous function, g is a nonnegative and nondecreasing continuous function, ? = BR, is the ball of radius R > 0 centered at the origin in RN, 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. © 2018 American Institute of Mathematical Sciences. All rights reserved.