The authors prove a theorem for the existence of a solution of the nonlinear elliptic equation \(-\Delta u+2=R(x)\ell\) 4, \(u\in S\) 2 under some conditions on R(x) but not symmetry. This is the first existence result where R(x) is not symmetric. The proof of the existence uses a mass center analysis technique and a continuous flow in H 1(S 2) controlled by \(\nabla R\).