  In this note we consider the Liouville type theorem for a properly immersed
submanifold $M$ in a complete Riemmanian manifold $N$. 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 $q_0\in N$.
  (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.
  (ii) Let $u$ be a smooth nonnegative function on $M$ satisfying $\Delta u\geq
ku^a$ for some constant $k>0$ and $a>1$. If $|\vec{H}|\leq
C(1+dist_N(\cdot,q_0)^2)^\frac{\beta}{2}$ for some $C>0$, $0\leq\beta<1$, then
$u=0$ on $M$.
  As applications we get some nonexistence result for $p$-biharmonic
submanifolds.
