Positive radial solutions to classes of singular problems on the exterior domain of a ball

Abstract

We study positive radial solutions to singular boundary value problems of the form:{−Δu=λK(|x|)f(u)uα,in Ω,∂u∂η+c˜(u)u=0,|x|=r0,u(x)→0,|x|→∞, where Δu:=div(∇u) is the Laplacian operator of u, Ω={x∈RN‖x|>r0>0,N>2}, λ>0, K∈C([r0,∞),(0,∞)) is such that K(s)≤1sN+βˆ for s≫1 for some βˆ>1, α<min⁡{1,βˆN−2} and ∂u∂η is the outward normal derivative of u on |x|=r0. Here, f∈C1([0,∞),R) is such that f(s)s1+α→0 as s→∞, and c˜∈C([0,∞),(0,∞)). We analyse the cases when (a) f(0)>0 and (b) f(0)<0. We discuss existence, non-existence, multiplicity and uniqueness results. We prove our existence results by the method of sub and supersolutions.