Existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

Let ( M , g ) be a noncompact, connected, orientable smooth N -dimensional Riemannian manifold without boundary. We consider the existence of solutions of problem (P) Δ g u + a u log u = 0 on M , where Δ g stands for the Laplacian in M and a > 0 .