Rigidity results for the $p$-Laplace type equations on compact Riemannian manifolds

In this paper, we obtain two rigidity results for $p$-Laplace type equation and $n$-Laplace equation with exponential nonlinearity on $n$-dimensional compact Riemannian manifolds by using of nonlinear flow and the carr'e du champ methods, respectively, where rigidity means that the PDE has only constant solution when a parameter is in a certain range. Moreover, an interpolation inequality is derived as an application.