Probability Theory and Related Fields

, Volume 101, Issue 2, pp 251–276 | Cite as

On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions

  • Z. Q. Chen
  • R. J. Williams
  • Z. Zhao


We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domainsD ⊂ ℝ d (d≥3), having compact boundary, with nonlinear Neumann boundary conditions on the boundary ofD. For this we use an implicit probabilistic representation, Schauder's fixed point theorem, and a recently proved Sobolev inequality forW1,2(D). Special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics.

Mathematics Subject Classification

35J65 60J65 53C21 


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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Z. Q. Chen
    • 1
  • R. J. Williams
    • 2
  • Z. Zhao
    • 3
  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Department of MathematicsUniversity of California, San DiegoLa JollaUSA
  3. 3.Department of MathematicsUniversity of MissouriColumbiaUSA

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