Integral Volume Criteria for Semilinear and Quasilinear Liouville Theorems on Complete Manifolds

Let $M$ be a connected complete noncompact Riemannian manifold without boundary, and let $V_o(R)$ denote its ball volume. We prove Liouville theorems without any curvature assumption. For $p>1$ and $q>p-1$, divergence of ${\int }^{\mathrm{\infty }}{R}^{pq\mathrm{/}(p-1)-1}{V}_o(R{)}^{-(q-p+1)\mathrm{/}(p-1)}\text{dd}R$ forces every nonnegative Sobolev weak supersolution of $-{\mathrm{\Delta }}_{p}u\ge u^q$ to vanish. For $q>\max \{1,p-1\}$, divergence of ${\int }^{\mathrm{\infty }}{R}^{p-1}{V}_o(R{)}^{-(q-p+1)}\text{dd}R$ forces every nonnegative global distributional energy supersolution of $u_t-{\mathrm{\Delta }}_{p}u\ge u^q$ to vanish. Consequently, the corresponding Cauchy problem has no global solution in this class with a nonzero nonnegative local $L^1$ initial trace.