This paper is concerned with strongly nonlinear (and possibly degenerate) elliptic partial differential equations in unbounded regions. To broaden the class of problems for which solutions exist, the equation and boundary conditions are expressed by use of set-valued functions; this involves no technical complications. The concept of "solution" is so formulated that existence is needed only in bounded regions. Uniform boundedness is fi'rst established, and compactness of support is then deduced by a comparison argument, similar to that in recent work of Brezis, but simpler in detail. The central problems here are not associated with the comparison argument, but with the nonlinearities. Our hypotheses are given only when Igrad u I is small, so that the minimal surface operator (for example) is just as tractable as the Laplacian. Further nonlinearity is allowed by the use of the Bernstein-Serrin condition on the quadratic form, and by a suitably generalized version of the Meyers-Serrin concept of essential dimension. Although the boundary can have corners, we allow nonlinear boundary conditions of mixed type. Counterexamples show that certain seemingly ad hoc distinctions are in fact necessary to the truth of the theorems. 1. Historical resume'. It is well known that certain singular variational problems on [0, oo) have solutions with compact support. The history of this subject need not detain us; we mention only that it starts in 1952, or earlier, and continues active to the present time. In the most interesting cases the minimizing function is a spline function with infinitely many maxima and minima, the Euler equation is of the fourth order, and the maximum principles which underlie the more familiar methods of comparison and estimation do not apply. The situation is entirely different when the Euler equation is of the second order. In that case many comparison methods are available, and the compactness can be deduced by exhibiting a super solution and subsolution, both of which themselves have compact support. This method was first used by Brezis [1]. It applies with equal ease to ordinary and partial differential equations, and gives powerful results with little calculation. Received by the editors May 14, 1974. AMS (MOS) subject classifications (1970). Primary 35BXX, 35B05, 35B40, 35B45, 35JXX, 35J25. (1)This research was supported in part under NSF Grant No. GP-33580X. Copyright @ 1976, American Mathematical Society 133 This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 06:24:03 UTC All use subject to http://about.jstor.org/terms