Quasilinear elliptic Hamilton–Jacobi equations on complete manifolds

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Let (Mn,g) be an n-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor Riccg and sectional curvature Secg. Assume Riccg⩾(1−n)B2, and either p>2 and Secg(x)=o(dist2(x,a)) when dist2(x,a)→∞ for a∈M, or 1<p<2 and Secg(x)⩽0. If q>p−1>0, any C1 solution of (E) −Δpu+|∇u|q=0 on M satisfies |∇u(x)|⩽cn,p,qB1q+1−p for some constant cn,p,q>0. As a consequence, there exists cn,p>0 such that any positive p-harmonic function v on M satisfies v(a)e−cn,pBdist(x,a)⩽v(x)⩽v(a)ecn,pBdist(x,a) for any (a,x)∈M×M.