In this paper, the second order nonlinear elliptic differential equations ∑i,j=1n∂∂xj[ai,j(x,y)∂∂xjy]+q(x)f(y)=e(x)are considered in an exterior Ω ⊂ Rn, where q(x) is allowed to change sign. Some sufficient conditions for any solutions y(x) of (E) to be satisfied |x|→∞liminf|y(x)|=0 are obtained. Particularly, these results improve the previous results for second order ordinary differential equations.