Trong bài báo này chúng ta nghiên cứu sự tồn tại của các giải pháp yếu của vấn đề Dirichlet cho một hệ thống phương trình elliptic semilinear trên một miền không bị chặn trong Rn. | 33 4 2005 381-389 V í e t mi ai m J o u r mi ai l of MATHEMATICS ê VAST 2005 On a System of Semilinear Elliptic Equations on an Unbounded Domain Hoang Quoc Toan Faculty of Math. Mech. and Inform. Vietnam National University 334 Nguyen Trai Hanoi Vietnam Received May 12 2004 Revised August 28 2005 Abstract. In this paper we study the existence of weak solutions of the Dirichlet problem for a system of semilinear elliptic equations on an unbounded domain in R . The proof is based on a fixed point theorem in Banach spaces. 1. Introduction . .R .R. . . . R . -. R 382 o 2. Preliminaries and Notations X. 1 ---. .-. .X X Proposition . . . is a Hilbert space which is dense in and the embedding of . into is continuous and compact. 2 2 2 383 . x- . . I . Proposition that satisfies the hypothesis. and . Then for any in there exists a unique solution of the following problem .in .as. Furthermore if. .in . .. . . . . . . . . Definition. A pair. . is called a weak solution of the problem. .if . . It is seen that if. then the weak a classical solution of the problem. 3. Existence of Weak Solutions for the Dirichlet Problem
