The existence of positive solution to some asymptotically linear elliptic equations in exterior domains
Gongbao Li, Gao-Feng Zheng · Revista Matemática Iberoamericana · 2006
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 . Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution.