Nonlinear systems of elliptic equations with natural growth conditions and sign conditions

In the theory of nonlinear systems of partial differential equations, with a nonlinear term depending on the gradient having natural growth, which means, for instance, a quadratic growth for a function expected to be in H1, it is essential to look for solutions which are bounded. However, there are natural cases in which bounded solutions are out of reach. This paper revisits previous works in this subject, with some improvement in the proofs, and some extension of the results. The case of systems which will naturally include the case of a single equation is considered. An alternative proof which does not seem to carry over to systems is also proposed.