Sharp volume tests and entropy thresholds for quasilinear elliptic inequalities with gradient reactions

We investigate positive weak solutions of -Δₘ u≥ uᵖ|∇ u|q on complete noncompact Riemannian manifolds without curvature, volume doubling, or Poincaré assumptions. For the superhomogeneous mixed reaction we derive a curvature-free integral volume test by combining two first-order estimates with a recurrence for the mass of a fixed density. The same approach organizes the pure-gradient and capacity criteria and yields sharp finite hierarchies of iterated-logarithmic growth bounds, with counterexamples on smooth model manifolds. In the subhomogeneous region, vanishing lower growth of log V(R)/Rm-q forces the Liouville property. On the homogeneous line $p+q=m-1$, we obtain the optimal strict lower-volume-entropy threshold (p+1)((m-1)/p)p/(p+1) for $p>0$ (and $1$ for $p=0$), together with examples at equality. The existing six-region classification is not claimed as new; the advances concern refined volume hypotheses and their model sharpness. The tests remain valid for smooth weighted measures and for locally bounded positive Sobolev solutions. Sharpness is understood within the stated model families; no characterization from volume alone on an individual manifold is asserted.