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On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions
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  • Published: June 1995

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

  • Z. Q. Chen1,
  • R. J. Williams2 &
  • Z. Zhao3Ā 

Probability Theory and Related Fields volumeĀ 101,Ā pages 251–276 (1995)Cite this article

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Summary

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domainsD āŠ‚ ā„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 forW 1,2(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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, 14853-7901, Ithaca, NY, USA

    Z. Q. Chen

  2. Department of Mathematics, University of California, San Diego, 92093-0112, La Jolla, CA, USA

    R. J. Williams

  3. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    Z. Zhao

Authors
  1. Z. Q. Chen
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  2. R. J. Williams
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  3. Z. Zhao
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Additional information

Research supported in part by NSF Grants DMS 8657483 and GER 9023335

This article was processed by the authors using the\(LaT_E X\) style filepljourlm from Springer-Verlag.

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Chen, Z.Q., Williams, R.J. & Zhao, Z. On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions. Probab. Th. Rel. Fields 101, 251–276 (1995). https://doi.org/10.1007/BF01375828

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  • Received: 14 January 1994

  • Revised: 14 June 1994

  • Issue Date: June 1995

  • DOI: https://doi.org/10.1007/BF01375828

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Mathematics Subject Classification

  • 35J65
  • 60J65
  • 53C21
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