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Assume that the sectional curvature $K^N$ of $N$ satisfies $K^N\\geq-L(1+dist_N(\\cdot,q_0)^2)^\\frac{\\alpha}{2}$ for some $L>0, 2>\\alpha\\geq 0$ and $q_0\\in N$.\n  (i) If $\\Delta|\\vec{H}|^{2p-2}\\geq k|\\vec{H}|^{2p}$($p>1$) for some constant $k>0$, then we prove that $M$ is minimal.\n  (ii) Let $u$ be a smooth nonnegative function on $M$ satisfying $\\Delta u\\geq ku^a$ for some constant $k>0$ and $a>1$. 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