The Role Of Domain Shape On The Asymptotically Linear Elliptic Problem

In this paper, we discuss the role of domain shape on the existence and mul-tiplicity of positive solutions to some asymptotically linear elliptic problems. Inthe first part, we are concerned with some asymptotically linear elliptic equa-tions on bounded domains. We are interested in the role of domain topologyon the number of positive solutions to this problem. In the second part, westudy the existence of positive solutions to some asymptotically linear ellipticequations on exterior domain.In the first chapter, we are concerned with the elliptic problem with asymp-totically linear term at infinitywhereΩ(?) R~N (N≥3) is a smooth bounded domain. We obtain that under ourassumptions the problem has at least catΩ+1 distinct positive solutions whenλis large enough, where catΩis the Ljusternik-Schnirelmann category of (?)relative to (?) itself.In the second chapter, we are concerned with the asymptotically linearelliptic problem in an exteriorwhereΩ= R~N\(?) (N≥3) and (?) is a smooth bounded and star-shaped openset, and lira (?) f(t)/t= l,0＜l＜+∞Using a precise deformation lemma andalgebraic topology argument, we prove under our assumptions that the problempossesses at least one positive solution.