This paper mainly discusses regional shape Existence of solutions for asymptotically linear elliptic problems and the impact of multiple solutions . The first part of the article discusses asymptotically linear elliptic problem on a bounded domain , and to study the impact of the regional topology on the number of equations . The second part of the study asymptotically linear elliptic equation on the Outland existence . In the first chapter , we consider the asymptotic linear elliptic problems which Ω ( ? ) The R ~~ N ( N ≥ 3 ) is a smooth bounded domain , and ( ? ) F (t ) / t = l , 0 < l <∞. We prove that under certain conditions, as long as λ is sufficiently large , the problem ( 1) there exists at least catΩ 1 Positive Solutions . Where , catΩ is (? ) With respect to the number of itself Ljusternik-Schnirelmann domains . In the second chapter , we consider the problem of asymptotically linear elliptic Outland where Ω = R ~ N \\ ( ? ) ( N ≥ 3 ) , ( ? ) Is a bounded star-shaped region , and ( ?) f (t) / t = l, 0 |