Nonlinear Elliptic Problems

with Aa having at most polynomial growth, by applying a general theorem concerning nonlinear functional equations in reflexive Banach spaces. Our result in [ l] extended and generalized results announced earlier by M. I. Visik [6; 7; 8] and obtained by more concrete analytic arguments. Visik's detailed account of his results which has just appeared in [9] has one feature which goes beyond the framework of methods developed in [ l ] , namely that the hypotheses of strong ellipticity or monotonicity which are assumed involve only the variation of Aa with respect to the highest order derivatives and not the lower order derivatives of u. It is our object in the present note to announce some results which constitute an extension of our preceding methods to cover this point. The detailed proof of these results will appear in [4].