Quasilinear Elliptic Hamilton-Jacobi Equations on Complete Manifolds

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 445-449.Let $(M^n,g)$ be a $n$-dimensional complete and non-compact Riemannian manifold with Ricci tensor $Ricc_g$ and sectional curvature $Sec_g$. Assume $Ricc_g\geq (1-n)B^2$ and $scal_g(x)=o(dist^2(x,a))$ for some $a\in M$ if $p>2$. Then for $q>p-1\geq 1$, any $C^1$ solution of (E) $-\Gd_pu+\abs{\nabla u}^q=0$ on $M$ satisfies $\abs{\nabla u(x)}\leq c_{n,p,q}B^{\frac{1}{q+1-p}}$ for some constant $c_{n,p,q}>0$. As a consequence there exists $c_{n,p}>0$ such that any positive $p$-harmonic function $v$ on $M$ satisfies $v(a)e^{-c_{n,p}B\dist (x,a)}\leq v(x)\leq v(a)e^{c_{d,p}B\dist (x,a)}$ for any $x\in M$