Double-tower solutions for higher-order prescribed curvature problem

We consider the following higher-order prescribed curvature problem on S-N: D-m (u) over tilde = (K) over tilde (y)(u) over tilde (m)*(-1) on S-N, (u) over tilde > 0 in S-N, where (K) over tilde (y) > 0 is a radial function, m* = 2N/N-2m, and D-m is the 2m-order differential operator given by D-m = Pi(m)(i=1)(-Delta g + 1/4(N - 2i)(N + 2i-2)), where g = g(S)N is the Riemannian metric. We prove the existence of infinitely many double-tower type solutions, which are invariant under some non-trivial sub-groups of O(3), and their energy can be made arbitrarily large.