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Let fl be a bounded domain in Iw" with n 23. We are concerned with the problem of existence of a function u satisfying the nonlinear elliptic equationwhere A is a real constant. The exponent p = ( n + 2 ) / ( n -2) is critical from the viewpoint of Sobolev embedding. Indeed solutions of (0.1) correspond to critical points of the functional where F(x, u ) = f," f (x, t ) dt. Note that p + 1 = 2n/(n -2) is the limiting Sobolev exponent for the embedding HA(fl) c Lp+'(R). Since this embedding is not compact, the functional CP does not satisfy the (PS) condition. Hence there are serious difficulties when trying to find critical points by standard variational methods. In fact, there is a sharp contrast between the case p < ( n + 2 ) / ( n -2) for which the Sobolev embedding is compact, and the case p = (n + 2 ) / ( n -2).Many existence results for problem (0.1) are known when p < (n + 2 ) / ( n -2) (see the review article by P. L. Lions [20] and the abundant list of references in [20]). On the other hand, a well-known nonexistence result of Pohozaev [24]

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We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.

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We prove an inequality (Lemmas 1.1') which has been applied by one of the authors and by J. Moser in their papers in this issue. The inequality expresses that a function, which in every subcube C of a cube C, can be approximated in the L1 mean by a constant ac with an error independent of C, differs then also in the L" mean from a, in C by an error of the same order of magnitude. More precisely, the measure of the set of points in C, where the function differs from a, by more than an amount u decreases exponentially as a increases.I n Section 2 we apply Lemma 1' to derive a result of Weiss and Zygmund [3], and in Section 3 we present an extension of Lemma 1'. LEMMA 1. Let ~( x ) be a n integrable function defined in a finite cube C, in n-dimensional space; x = (xl, --+, XJ. Assume that there is a constant K such that for every parallel subcube C , and some constant a,, the inequality holds. Here dx denotes element of volume and m ( C ) i s the Lebesgue measure of C. Then, if p(u) is the measure of the set of points where /u-aco~ > 0, we have (2) p(u) 5 Be"+m(C,) for u > 0, where B, b are constants depending only on n.