The existence of positive solution to some asymptotically linear elliptic equations in exterior domains

Abstract

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 $\Omega ={\mathbb{R}}^N\setminus \overline{\mathcal{O}}\left(N⩾3\right)$ with $\mathcal{O}$ a smooth bounded and star-shaped open set, and ${\lim }_{t\to +\infty }\frac{f(t)}{t}=l$, $0<l<+\infty$. Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution.