Multiple clustered layer solutions for semilinear elliptic problems on Sn

We consider the following superlinear elliptic equation on S-n {epsilon(2)Delta Snu - u + f(u) = 0 in D; u > 0 in D and u = 0 on delta D, where D is a geodesic ball on Sn with geodesic radius theta(1), and Delta(Sn) is the Laplace-Beltrami operator on S-n. We prove that for any theta is an element of (pi/2, pi) and for any positive integer N >= 1, there exist at least 2N+1 solutions to the above problem for sufficiently small. Moreover, the asymptotic behavior of such solutions is also characterized. We then apply this result to the Brezis-Nirenberg problem and establish the existence of solutions which are not minimizers of the associated energy.