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

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domains D ⊂ ℝ^d(d≥3), having compact boundary, with nonlinear Neumann boundary conditions on the boundary of D. For this we use an implicit probabilistic representation, Schauder's fixed point theorem, and a recently proved Sobolev inequality for W^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.