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Báo cáo nghiên cứu khoa học: " ON THE EXISTENCE OF SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS WITH UNBOUNDED COEFFICIENTS"

Let N be an integer ≥ 2 and D be a bounded open subset in R N . In this paper we study the following equation: | TẠP CHÍ PHÁT TRIỂN KH CN TẬP 9 SÓ 9 -2006 ON THE EXISTENCE OF SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS WITH UNBOUNDED COEFFICIENTS Bui Boi Minh Anh 1 Nguyen Minh Quan 1 Tran Tuan Anh 2 Vo Dang Khoa 3 1 State University of New York at Buffalo USA 2 Georgia Institute of Technology Atlanta Georgia USA 3 University of Medicine and Pharmacy Hochiminh City Vietnam Manuscript Received on March 20th 2006 Manuscript Revised October 2 d. 2006 ABSTRACT Using the topological degree of class S introduced by F. E. Browder in 1 and 2 we extend some results of the papers 3 and 4 to the case of Banach spaces with locally bounded conditions. 1. INTRODUCTION Let N be an integer 2 and D be a bounded open subset in RN. In this paper study the following equation _N d Ẻ7É-ai x du Vu - Ẽgi x u lr go x u a x 0 L i i dxi J The p - Laplace equation -Apu f x u 0 is a special case of 1.1 . If p 2 ổu ai x Vu dx then 1.1 has the form Vx e D we 1.1 and Au E gi x u u go x u a x 0. The problem 1.2 has been solved in 4 Theorem 3.1 p.514 by using topological degree for operators of class B . However that method doesn t work when id-2 du p 2 and ai x Vu Vu -. The one we use here can solve the problem 1.2 for all ỔXi p 1. Moreover our result is also stronger than Theorem 11 in 3 p.357 where the authors prove the existence result for the Dirichlet problem Ku f x u t lul dD 0 with the condition 10 that the function b is in Lp D but not in Lc D . 1.2 the in D. 2. TOPOLOGICAL DEGREE OF CLASS S In this section we recall the class S introduced by Browder see 1 2 . Definition 2.1. Let D be a bounded open set of a reflexive Banach space X and f be a mapping from D into the dual space X of X. We say f is of class S if f has the following properties Trang 17 Science Technology Development Vol 9 No.9- 2006 i f xn converges weakly to f x if xn n converges strongly to x in D i.e. f is a demicontinuous mapping on D. ii xn converges strongly to x if xn converges weakly to x in D and lnmupn. f xn xn -x 0. Definition