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 ut=△pu+cuσ,where c is a nonnegative constant and σ is a negative constant. Moreover, by the gradient estimate, we can get the following Liouville theorem for the elliptic equation (⋆)△pu+cuσ=0.Let Mn 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 uσ−1≥θ (θ is a positive constant). Then in the region |∇u|>0 and p≥2nn+1, then u can only be the constant solutions to Eq. (⋆). At last, we give the corresponding Harnack inequality for positive solutions to equation ut=△pu+cuσ.