Strongly singular problems in exterior domains

Abstract

We prove existence and uniqueness of a weak solution of the singular problem{−Δpu=K(x)uδ in Ωe,u>0 in Ωe,u=0on ∂Ω,u(x)→0as|x|→∞, where Ω⊂RN(N>2) is a simply connected, bounded domain containing the origin with smooth boundary, Ωe=RN∖Ω‾ is the exterior domain, 1<p<N, K(x) is an appropriately decaying weight function and δ≥1. Additionally, we prove existence results and discuss decay rates of the solutions near ∂Ω and at infinity when 1uδ is replaced with f(u)uδ or 1uδ+λf(u) for some positive function f and bifurcation parameter λ>0. We use approximation scheme as well as sub- and supersolution method to prove our results. Finally, we establish that solutions corresponding to the nonlinearity 1uδ+λf(u) approach the unique solution corresponding to the singular term 1uδ as λ→0+.