Positive radial symmetric solutions to an exterior elliptic Robin boundary-value problem and application

Abstract

We study the existence, uniqueness, and behavior of positive radial symmetric solutions of the exterior boundary-value problem − Δ u = H ˆ ( x ) f ( u ) in Ω ≔ R n ∖ B ¯ ( 0 , r 0 ) , ∂ u ∂ n + a u = γ on ∂ Ω , and u ( x ) → 0 as | x | → ∞ . When a > 0 , the Robin boundary condition has a “wrong” sign, and this makes the analysis of the problem interesting and non-trivial. We determine a sharp condition on the coefficient a for the existence of positive spherically symmetric solutions of the boundary-value problem. The important model case f ( u ) = ( u + 1 ) − 7 , a = 1 / ( 2 r 0 ) , and γ = − 1 / ( 2 r 0 ) , is related to the initial-data problem in general relativity. For this model case, as an example of application of our results, we obtain the existence and uniqueness of the conformal factor corresponding to the Hamiltonian constraint equation in the case of a single black hole in the radial symmetric regime.