Existence of positive solutions for nonlinear elliptic equations with convection terms

We prove the existence of positive solutions for the equation −∑i=1N∂∂xiai(x,u,∇u)=f(x,u,∇u)inΩ under the Dirichlet boundary condition, where the essential point is the dependence of the terms of the elliptic equation on the solution u and its gradient ∇u. We develop an approach based on approximate solutions and on a new strong maximum principle.