In this paper, we are concerned with the asymptotically linear elliptic problem $-\Delta u+ \lambda_{0}u=f(u), u\in H_{0}^{1}(\Omega ) $ in an exterior domain $\Omega= \mathbb{R}^{N}\setminus\overline{\mathcal{O}} \left( N\geqslant 3\right) $ with $\mathcal{O}$ a smooth bounded and star-shaped open set, and $\lim_{t\rightarrow +\infty }\frac{ f(t)}{t}=l$, $0
