Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic PDE's

We prove pointwise gradient bounds for entire solutions of pde's of the form u(x) = ψ(x, u(x), u(x)), where â"' is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.