A Moser/Bernstein type theorem in a Lie group with a left invariant metric under a gradient decay condition

We say that a PDE in a Riemannian manifold $M$ is geometric if,$\$whenever $u$ is a solution of the PDE on a domain $\Omega$ of $M$, the composition $u_{\phi}:=u\circ\phi$ is also solution on $\phi^{-1}\left( \Omega\right)$, for any isometry $\phi$ of $M.$ We prove that if $u\in C^{1}\left( \mathbb{H}^{n}\right)$ is a solution of a geometric PDE satisfying the comparison principle, where $\mathbb{H}^{n}$ is the hyperbolic space of constant sectional curvature $-1,$ $n\geq2,$ and if $$\limsup_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla u\right\Vert \right) =0,$$ where $S_{R}$ is a geodesic sphere of $\mathbb{H}^{n}$ centered at fixed point $o\in\mathbb{H}^{n}$ with radius $R,$ then $u$ is constant. Moreover, given $C>0,$ there is a bounded non-constant harmonic function $v\in C^{\infty }\left( \mathbb{H}^{n}\right)$ such that $$\lim_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla v\right\Vert \right) =C.$$ The first part of the above result is a consequence of a more general theorem proved in the paper which asserts that if $G$ is a non compact Lie group with a left invariant metric, $u\in C^{1}\left( G\right)$ a solution of a left invariant PDE (that is, if $v$ is a solution of the PDE on a domain $\Omega$ of $G$, the composition $v_{g}:=v\circ L_{g}$ of $v$ with a left translation $L_{g}:G\rightarrow G,$ $L_{g}\left( h\right) =gh,$ is also solution on $L_{g}^{-1}\left( \Omega\right)$ for any $g\in G),$ the PDE satisfies the comparison principle and% $$\limsup_{R\rightarrow\infty}\left( \sup_{g\in B_{R}}\left\Vert \operatorname*{Ad}\nolimits_{g}\right\Vert \sup_{S_{R}}\left\Vert \nabla u\right\Vert \right) =0,$$ where $\operatorname*{Ad}\nolimits_{g}:\mathfrak{g}\rightarrow\mathfrak{g}$ is the adjoint map of $G$ and $\mathfrak{g}$ the Lie algebra of $G,$ then $u$ is constant.