Existence of solutions for sublinear equations on exterior domains

Abstract

In this article we consider the radial solutions of ∆u + K(r)ƒ(u) = 0 on the exterior of the ball of radius R > 0, B_R, centered at the origin in ℝ^N with u = 0 on ∂B_R and lim_r→∞ u(r) = 0 where N > 2, ƒ is odd with ƒ < 0 on (0, β), ƒ > 0 on (β, ∞), ƒ(u) ~ u^p with 0 < p < 1 for large u and K(r) ~ r^-α with (N+2)-p(N-2)/2 ≤ α < N - p(N - 2) for large r. We prove existence of n solutions - one with exactly n zeros on [R, ∞) - if R > 0 is sufficiently small. If R > 0 is sufficiently large then there are no solutions with lim_r→∞ u(r) = 0.