Strongly singular problems in exterior domains

We prove existence and uniqueness of a weak solution of the singular problem {-Delta(p)u = K(x)/u(delta) in Omega(e) u > 0 in Omega(e), u = 0 on partial derivative Omega, u(x) -> 0 as |x| -> infinity, where Omega subset of R-N (N > 2) is a simply connected, bounded domain containing the origin with smooth boundary, Omega(e) = R-N Omega is the exterior domain, 1 < p < N, K(x) is an appropriately decaying weight function and delta >= 1. Additionally, we prove existence results and discuss decay rates of the solutions near partial differential and at infinity when 1/(u)delta is replaced with f(u)/u(delta) 1/u(delta) or + lambda f (u) for some positive function f and bifurcation parameter lambda > 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 1/u(delta) 1 +lambda f(u) approach the unique solution corresponding to the singular term 1/u(delta) as lambda -> 0(+). (c) 2021 Elsevier Inc. All rights reserved.