  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.
