EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO ELLIPTIC PROBLEMS WITH SUBLINEAR MIXED BOUNDARY CONDITIONS∗

In this work we consider a class of semilinear elliptic problems with nonlinear bound-ary conditions of mixed type. Under some monotonicity properties of the nonlinear-ities involved, we show that positive solutions are unique, and that their existence is characterized by the sign of some associated eigenvalues. One of the most important contributions of this work relies in the fact that we deal with boundary conditions of the form ∂u/∂ν = g(x, u) on Γ and u = 0 on Γ′, where ν is the outward unit normal to Γ while Γ,Γ ′ are open, Γ ∩ Γ ′ = ∅, ∂Ω = Γ ∪ Γ′, but Γ,Γ ′ need not be disjoint. 1. Introduction an