S IDENTITY AND THE MOVING PLANE PROCEDUREWalter

Positive solutions of a class of nonlinear elliptic partial diierential equations are shown to be symmetric by means of the moving plane argument coupled with Spectral Theory results and Picone's Identity. The method adapts easily to situations where the moving plane procedure gives rise to variational problems with positive eigenfunctions. 0. Introduction Consider the problem: ?u = p(x)g(u) in (1) u = 0 on @ where is a cylinder in R n : = (?1; 1) 0 with 0 a domain (= bounded, open, connected set) of R n?1. Are all C 2 () positive solutions symmetric in x 1 ? If the boundary of 0 is reasonably smooth, then under suitable conditions on p; g the Nirenberg showed that the result is still true for general nonlinear equations if, for example, u 2 W 2;n `oc (() \ C() with no regularity assumed on @; while Dancer, 11], dealt with the case of u 2 H 1;2 0 (() \ L 1 ((). It is the purpose of this paper to discuss the symmetry of positive solutions under somewhat weaker conditions than those that to the best of our knowledge have been applied earlier. We avoid direct use of pointwise considerations near @ by employing related arguments from Spectral Theory and Picone's Identity. In this way, we are able in particular to bypass various \corner Lemma" and Maximum Principle procedures. Most of the paper is devoted to the problem ?u = f(u) in (2) u = 0 on @