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

In this paper, we are concerned with the asymptotically linear elliptic problem -?u + ?0u = f(u), u I H01(O) in an exterior domain O = RnO (N = 3) with O a smooth bounded and star-shaped open set, and limt?+8 f(t)/t = l, 0 < l < +8. Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution