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 $\hat{I}{”}_{p}u+\hat{I}{”}_{q}u+h(u,|\nabla u{|}^2)=0$ 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 $\hat{I}{”}_{z}u=\mathrm{div}\left({\left|\nabla u\right|}^{z-2}\nabla u\right)$, 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 $\hat{I}{”}_{p}u+\hat{I}{”}_{q}u=0$ ($n\le p\le q$) on a complete non-compact Riemannian manifold $M$ with non-negative Ricci curvature and $\dim M=n\ge 2$, then $u$ is a trivial constant solution.