tailieunhanh - Báo cáo nghiên cứu khoa học: " EXISTENCE OF SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS WITH SINGULAR CONDITIONS"

In this paper, we study the existence of generalized solution for a class of singular elliptic equation: −diva ( x, u ( x ) , ∇u ( x ) ) + f ( x, u ( x ) , ∇u ( x ) ) = 0 . Using the Galerkin approximation in [2, 10] and test functions introduced by Drabek, Kufner, Nicolosi in [5], we extend some results about elliptic equations in [2, 3, 4, 6, 10]. | TẠP CHÍ PHÁT TRIỂN KH CN TẬP 9 SÓ 9 -2006 EXISTENCE OF SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS WITH SINGULAR CONDITIONS Chung Nhan Phu Tran Tan Quoc University of Natural Sciences VNU-HCM Manuscript Received on March 24th 2006 Manuscript Revised October 2nd 2006 ABSTRACT In this paper we study the existence of generalized solution for a class of singular elliptic equation -diva x u x Vu x f x u x Vu x 0. Using the Galerkin approximation in 2 10 and test functions introduced by Drabek Kufner Nicolosi in 5 we extend some results about elliptic equations in 2 3 4 6 10 . The aim of this paper is to prove the existence of generalized solutions in Wqp Q for the quasilinear elliptic equations -diva x u x Vu x f x u x Vu x 0 . proving the existence of u e W0 p Q such that J a x u x Vu x Vọdx J f x u x Vu x ọdx 0 Vọe C Q Q Q where Q is a bounded domain in n N 2 with smooth boundary p e 1 N and a Qx X N N f Qx X N satisfy the following conditions Each ai x n z is a Caratheodory function that is measurable in x for any fixed z n z e N 1 and continuous in z for almost all fixedx e Q ai x n z c1 x n lzlp-1 k1 x vi 1 N a x n z -a x n z z-z 0 a x n z z z p . x eQ Vne vz z e N z z . where c1 e Lc Q c1 0 k1 e Lp Q a e 0 p -1 X 0. and f Qx X N is a Caratheodory function satisfying f x n z c2 x n P lz Y k2 x f x n z n -c3 x - b n q - d z r where c2 is a positive function in Lc Q c3 is a positive function in L Q k2 e Lp Q Np and r q e 0 p b d are positive constants Y e 0 p -1 Pe 0 p -1 with p T . Trang 27 Science Technology Development Vol 9 2006 Because c1 c2 e Lc Q we cannot define operator on the whole space Wqp Q . Therefore we cannot use the property of S operator as usual. To overcome this difficulty in every Qn we find solution un e W0 p Qn of the equation -diva x u x Vu x f x u x Vu x 0 where Qn is an increasing sequence of open subsets of Q with smooth boundaries such that Qn is contained in Qn 1 and Q U 1 Qn. In this .