The existence of positive solution to some asymptotically linear elliptic equations in exterior domains (Q879630)

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