Liouville type theorems for some $(p,q)$-Laplace equations with gradient dependent reaction on Riemannian manifolds

Abstract

In this paper, we combine Bochner formula, Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $\Delta_pu+\Delta_qu+h(u,|\nabla u|^2)=0$ defined on a complete Riemannian manifold $\left(M,g\right)$, where $q\ge p>1$, $h\in C^1(\mathbb{R}\times\mathbb{R}^{+})$ and $\Delta_z u=\mathrm{div}\left(\left|\nabla u\right|^{z-2}\nabla u\right)$, with $z\in\{ p,~q\}$, is the usual $z$-Laplace operator. Under some assumptions on $p$, $q$ and $h(x,y)$, we derive concise gradient estimates for solutions to the above equation and then obtain some Liouville type theorems. In particular, we use integral estimate method to show that, if $u$ is a non-negative entire solution to $\Delta_p u +\Delta_q u=0$ ($n\le p\le q$) on a complete non-compact Riemannian manifold $M$ with non-negative Ricci curvature and $\dim M = n\ge2$, then $u$ is a trivial constant solution.