EXISTENCE OF WEAK SOLUTIONS FOR A SEMILINEAR PROBLEM WITH A NONLINEAR BOUNDARY CONDITION

This paper shows conditions for the existence of weak solutions of the problem {-Delta u = lambda(1)u + f(x, u) - h (x) in Omega, partial derivative u/partial derivative n = g(x, u) on partial derivative Omega, where Omega is a bounded domain in R-N (N >= 1) with smooth boundary, partial derivative u/partial derivative n denotes the derivative of u with respect to the outer normal n, f : Omega x R -> R and g : partial derivative Omega x R -> R are bounded Caratheodory functions, h is an element of L-2 ( Omega) and lambda(1) > 0 is the principal eigenvalue of -Delta on Omega with zero Dirichlet boundary conditions. Our method is based on the minimum principle.