Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms

This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term -Delta u = f (x, u, del u) on Omega restricted by the boundary condition u vertical bar(partial derivative Omega) = 0, where Omega is a bounded domain in R-N with sufficiently smooth boundary partial derivative Omega, N >= 2, and f : (Omega) over bar x R x R-N -> R. is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity f (x, xi, eta) when vertical bar(xi, eta)vertical bar is small or large enough.