Summary
We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domainsD ⊂ ℝd(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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Research supported in part by NSF Grants DMS 8657483 and GER 9023335
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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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DOI: https://doi.org/10.1007/BF01375828