The generalized strauss compactness lemma and its applications on nonlinear elliptic problems

Abstract

To PROVE the existence of solutions of partial differential equations, usually it needs some compactness properties. For the bounded domain case, there is a useful one-Rellich-Kondrakov theorem. However for the unbounded domain case, we know quite a few of such results. Strauss [I I] proved a compactness lemma for the entire space RN. In this article we generalized Strauss’ result to the case of an infinite strip Q = RN x (0, l), N L 2. Several applications on elliptic equations are also given. In this section we introduce some notations, concepts, and equations which will be used later. In Section 2 we prove the compactness lemma. In Section 3 we give some applications on nonlinear elliptic equations, as one of the continuing studies of Lions [8] and Berestycki-Lions [41. Consider (0, 1) as the circle group R/Z, and Q = RN x (0, I) the locally compact abelian group with the Haar measure dxdt where dx and dt are Lebesque measure on R’ and (0, l), respectively. We therefore may consider the convolution of two functions or generalized functions on the group !A. Let f: R + R be a measurable function. The distribution function f,~: Rf R+ off is defined by, for CY > 0, f&) = Ilx E 4lf(.@l > 41