Liouville theorem for p-Laplacian Lichnerowicz equation on compact manifolds

Abstract

Abstract In this paper, we obtain the gradient estimates for positive solutions to the following p -Laplacian Lichnerowicz equation u t = △ p u + c u σ , 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 ( ⋆ ) △ p u + c u σ = 0 . Let M n be a Riemannian manifold of dimension n with R i c ( M ) ≥ − K for some K ≥ 0 . Suppose that u is a positive solution to Eq. ( ⋆ ) with u σ − 1 ≥ θ ( θ is a positive constant). Then in the region | ∇ u | > 0 and p ≥ 2 n 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 = △ p u + c u σ .