Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
SEMILINEAR PROBLEMS WITH BOUNDED NONLINEAR TERM MARTIN SCHECHTER Received 17 August 2004 We solve boundary value problems for elliptic semilinear equations in which no asymptotic behavior is prescribed for the nonlinear term. 1. Introduction Many authors (beginning with Landesman and Lazer [1]) have studied resonance problems for semilinear elliptic partial differential equations of the form −∆u − λ u = f (x,u) in Ω, u = 0 on ∂Ω, (1.1) where Ω is a smooth bounded domain in Rn , λ is an eigenvalue of the linear problem −∆u = λu in Ω, u = 0 on ∂Ω, (1.2) and f (x,t) is a bounded Carath´ odory function on Ω × R. | SEMILINEAR PROBLEMS WITH BOUNDED NONLINEAR TERM MARTIN SCHECHTER Received 17 August 2004 We solve boundary value problems for elliptic semilinear equations in which no asymptotic behavior is prescribed for the nonlinear term. 1. Introduction Many authors beginning with Landesman and Lazer 1 have studied resonance problems for semilinear elliptic partial differential equations of the form -Au - Xgu f x u in Q u 0 on dQ 1.1 where Q is a smooth bounded domain in R Xg is an eigenvalue of the linear problem -Au Xu in Q u 0 on dQ 1.2 and f x t is a bounded Caratheodory function on Q X R such that f x t f x a.e. as t TO. 1.3 Sufficient conditions were given on the functions f to guarantee the existence of a solution of 1.1 . Some of the references are listed in the bibliography. They mention other authors as well. In the present paper we consider the situation in which 1.3 does not hold. In fact we do not require any knowledge of the asymptotic behavior of f x t as 1t TO. As an example we have the following. Theorem 1.1. Assume that sup F x v dx TO 1.4 veE X Q where E Xe is the eigenspace ofXe and c t F x t f x s ds. 1.5 0 Copyright 2005 Hindawi Publishing Corporation Boundary Value Problems 2005 1 2005 1-8 DOI 10.1155 BVP.2005.1 2 Semilinear problems with bounded nonlinear term Assume also that if there is a sequence uk such that Peu-k II TO 11 1 Pe uk II C 2 J F x uk dx bo 1.6 f x uk f x weakly in L2 Q where f x E Ảe and Pe is the projection onto E Ảe then b0 f 1 Bo 1.7 where Bo Jn W0 x dx W0 x supt Àf 1 Ảe t2 2F x t and u1 is the unique solution of Au Ảeu f u E xe . 1.8 Then 1.1 has at least one solution. In particular the conclusion holds if there is no sequence satisfying 1.6 . A similar result holds if 1.4 is replaced by inf F x v dx veE Ảe n TO. 1.9 In proving these results we will make use of the following theorem 2 . Theorem 1.2. Let N be a closed subspace of a Hilbert space H and let M N Assume that at least one of the subspaces M N is finite dimensional. Let G