Maximum principles for a class of nonlinear second-order elliptic boundary value problems in divergence form

For a class of nonlinear elliptic boundary value problems in divergence form, we construct some general elliptic inequalities for appropriate combinations of u(x) and | ￠ u|2, where u(x) are the solutions of our problems. From these inequalities, we derive, using Hopf's maximum principles, some maximum principles for the appropriate combinations of u(x) and | ￠ u|2, and we list a few examples of problems to which these maximum principles may be applied.