MULTIPLE SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

We consider the elliptic problem with nonlinear boundary conditions: -Delta u + bu = f(x, u) in Omega, -partial derivative(nu)u = vertical bar u vertical bar(q-1)u - g(u) on partial derivative Omega, where Omega is a bounded domain in R-n. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since Lq+1(partial derivative Omega) subset of H-1(Omega) does not hold for large q, the standard variational method can not be applied directly. To overcome this difficulty, we use approximation methods and uniform a priori estimates for solutions of approximate equations.