BOUNDEDNESS, A PRIORI ESTIMATES AND EXISTENCE OF SOLUTIONS OF ELLIPTIC SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS

Consider the semilinear elliptic system -Delta u = f(x, u, v), -Delta v = g(x,u,v), x is an element of Omega, complemented by the nonlinear boundary conditions partial derivative(v)u = (f) over tilde (y, u, v), partial derivative(v)u = (g) over tilde (y, u, v), y is an element of partial derivative Omega, where Omega is a smooth bounded domain in R-N and partial derivative(v), denotes the derivative with respect to the outer unit normal v. We show that any positive very weak solution of this problem belongs to L-infinity provided the functions f, g, (f) over tilde, (g) over tilde satisfy suitable polynomial growth conditions. In addition, all positive solutions are uniformly bounded provided the right-hand sides are bounded in L-1. We also prove that our growth conditions are optimal. Finally, we show that our results remain true for problems involving nonlocal nonlinearities and we use our a priori estimates to prove the existence of positive solutions.