manuscripta math. 69, 43 - 70 (1990) manuscripta mathematica 9 Sprlnger-Verlag 1990 Solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary conditions* MYRIAM COMTE AND MARIETTE C. KNAAP 1 We consider the problem -Au=[u[~'-xu+Au in {2 with ~=0 on al'l, where ~ is a bounded domain in R N, p=(Nq-2)/(N--2) is the critical Sobolev exponent,, the outward pointing normal and A a constant. Our main result is that if f~ is a ball in R N, then for every AER the problem admits infinitely many solutions. Next we prove that for every bounded domain G in R 3, symmetric with respect to a plane, there exists a constant p>0 such that for every A<p. this problem has at least one non-trivial solution. 1. Introduction In this paper we consider the problem -Au = [u[p-lu+Au in fl (1.1) (I) a___uu _ an - 0 on aft, (1.2) where 12 is a bounded domain of class C 2 in R N, N _> 3 and n is the * This work was supported by the Paris VI-Leiden exchange program 1 Supported by the Netherlands organisation for scientific research NWO, under num- ber 611-306-016. 43