We consider the linear elliptic equation $-/Delta u=V(x)u$ in  a punctured domain or an exterior domain in $/R^N$ with $N/ge 3$. Making use of the Kelvin transform and a nonexistence result of a Hardy potential problem in a bounded domain, we establish the nonexistence of positive  supersolution if the potential $V$ satisfies some suitable condition. Some applications are also obtained for optimal nonexistence of positive supersolution to some linear and nonlinear elliptic equations as well as elliptic systems. This is based on joint works with H.Y.Chen, R.Peng, and A.Quaas.

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