A positive solution for an asymptotically linear elliptic problem on ℝⁿ autonomous at infinity

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on ℝⁿ. The main diffculties to overcome are the lack of a priori bounds for Palais-Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order to regain compactness.