On an elliptic problem with boundary blow-up and a singular weight: the radial case

Abstract

In this work we consider the non-autonomous problem Δu = a(x)um in the unit ball B ⊂ RN, with the boundary condition u|∂B = +∞, and m > 0. Assuming that a is a continuous radial function with a(x) ˜ C0 dist(x, ∂B)−γ as dist(x, ∂B) → 0, for some C0 > 0, γ > 0, we completely determine the issues of existence, multiplicity and behaviour near the boundary for radial positive solutions, in terms of the values of m and γ. The case 0 < m ≤ 1, as well as estimates for solutions to the linear problem m = 1, are a significant part of our results.