It is a well-known fact, documented by striking examples, that the solutions of sufficiently nonlinear elliptic differential equations behave quite differently from the solutions of linear elliptic differential equations. Directing one's attention to the study of certain properties of the solutions of certain classes of differential equations, one actually can define measures of nonlinearity, conditions which, when fulfilled, guarantee the validity of the properties under consideration. This has been done by a number of authors in various ways and in regard to different properties of interest-solvability of Dirichiet's problem, possibility of isolated singularities, existence of nonlinear entire solutions, validity of certain a priori estimates, etc.-see for instance S. Bernstein [1], L. Bers [3], R. Finn [4], [5], [6], [7], [8], D. Gilbarg [9], H. Jenkins [10], [1], [12], H. Jenkins and J. Serrin [13], J. Leray [14], Johannes and Joachim Nitsche [16], [17], J. Serrin [18], [19]. Of course, in most cases these measures only lead to sufficient conditions. The first such discussion was carried out by S. Bernstein in 1912 (see [1, especially pp. 455-469]) for quasi-linear elliptic differential equations