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We present a unifled approach to study large positive solutions (i.e., u(x) ! 1 as x ! @›) of the equation ¢u + hu ¡ kˆ(u) = ¡f in an arbitrary domain ›. We assume ˆ(u) is convex and grows su-ciently fast as u ! 1. Equations of this type arise in geometry (Yamabe problem, two dimensional curvature equation), probability (superdifiusion). We prove that both existence and uniqueness are local properties of points of the boundary @›, i.e., they depend only on properties of › in arbitrary small neighborhood of each boundary point. We also flnd several new necessary and su-cient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries.

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