Liouville theorem for p-Laplacian Lichnerowicz equation on compact manifolds

In this paper, we obtain the gradient estimates for positive solutions to the following p-Laplacian Lichnerowicz equation u(t) = Delta(p)u + cu(sigma) , where c is a nonnegative constant and sigma is a negative constant. Moreover, by the gradient estimate, we, can get the following Liouville theorem for the elliptic equation Delta(p)u + cu(sigma) = 0. Let M-n be a Riemannian manifold of dimension n with Ric(M) >= -K for some K >= 0. Suppose that u is a positive solution to Eq. (*) with u(sigma-1) >= theta (theta is a positive constant). Then in the region |del u| > 0 and p >= 2n/n+1, then u can only be the constant solutions to Eq. (*). At last, we give the corresponding Harnack inequality for positive solutions to equation u(t) = Delta(p)u + cu sigma. (C) 2017 Elsevier B.V. All rights reserved.