LARGE SOLUTIONS FOR YAMABE AND SIMILAR PROBLEMS ON DOMAINS IN RIEMANNIAN MANIFOLDS

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Abstract

We present a unified approach to study large positive solutions (i.e., u(x) -> infinity as x -> partial derivative Omega) of the equation Delta u + hu - k psi(u) = -f in an arbitrary domain Omega. We assume psi(u) is convex and grows sufficiently fast as u -> infinity. 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 partial derivative Omega; i.e., they depend only on properties of 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.

Original languageEnglish
Pages (from-to)5131-5178
Number of pages48
JournalTransactions of the American Mathematical Society
Volume363
Issue number10
Publication statusPublished - Oct 2011

Keywords

  • Elliptic equations
  • large solutions
  • NONLINEAR ELLIPTIC-EQUATIONS
  • COMPLETE CONFORMAL METRICS
  • NEGATIVE SCALAR CURVATURE
  • LIPSCHITZ-DOMAINS
  • POSITIVE SOLUTIONS
  • DIFFERENTIAL-EQUATIONS
  • POTENTIAL-THEORY
  • PRESCRIBED SINGULARITIES
  • BOUNDARY
  • UNIQUENESS

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