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 boundary conditions of mixed type. Under some monotonicity properties of the nonlinearities 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 on the fact that we deal with boundary conditions of the form partial derivative u/partial derivative nu = g(x, u) on Gamma and u = 0 on Gamma, where nu is the outward unit normal to Gamma while Gamma, Gamma are open, Gamma boolean AND Gamma = empty set, partial derivative Omega = (Gamma) over bar boolean OR (Gamma) over bar, but (Gamma) over bar, (Gamma) over bar need not be disjoint.