Large solutions for Yamabe and similar problems on domains in Riemannian manifolds

Abstract

We present a unified approach to study large positive solutions (i.e., u ( x ) → ∞ u(x)\to \infty as x → ∂ Ω x\to \partial \Omega ) of the equation Δ u + h u − k ψ ( u ) = − f \Delta u+hu-k\psi (u)=-f in an arbitrary domain Ω \Omega . We assume ψ ( u ) \psi (u) is convex and grows sufficiently fast as u → ∞ u\to \infty . Equations of this type arise in geometry (Yamabe problem, two dimensional curvature equation) and probability (superdiffusion). We prove that both existence and uniqueness are local properties of points of the boundary ∂ Ω \partial \Omega ; i.e., they depend only on properties of Ω \Omega in arbitrarily small neighborhoods of each boundary point. We also find several new necessary and sufficient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries.