Variational problems on the Sphere

By various variational approaches, existence of infinitely many weak solutions for the following eigenvalue problem (S lambda) -Delta(h)w + alpha(sigma)w = lambda K (sigma) f (w), sigma is an element of S-d, w is an element of H-alpha(2) (S-d) on the unit sphere Sd are established for certain eigenvalues lambda > 0, depending on oscillating properties of f either at infinity or at zero. Here a, K are sufficiently smooth and positive maps on S-d. These multiplicity results can be applied to solve Emden-Fowler equations in the Euclidean case.