Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli

We consider a semilinear elliptic equation, Delta u + u(p) = 0 on Omega(R)={x is an element of R-n\R - 1 < \x\ < R + 1} with zero Dirichlet boundary condition, where 1 < p infinity for n = 2, 1<p<(n + 2)/(n - 2) for n > 2. We prove that, when the space dimension n is three, the number of nonequivalent nonradial positive solutions of the equation goes to infinity as R --> infinity. The same result has been known for n = 2 and n greater than or equal to 4; in those cases, the result was obtained by showing that the minimal energy solutions in various symmetry classes have different energy levels. As we will show in this paper, this is not true if n = 3. This makes the case n = 3 highly exceptional, and explains why past attempts failed in this case. In this paper we will prove the above result by considering local-rather than global-minimizers in some Symmetry classes. (C) 1997 Academic Press.