tailieunhanh - báo cáo hóa học:" Research Article Multiple Positive Solutions for a Class of Concave-Convex Semilinear Elliptic Equations in Unbounded Domains with Sign-Changing Weights"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Multiple Positive Solutions for a Class of Concave-Convex Semilinear Elliptic Equations in Unbounded Domains with Sign-Changing Weights | Hindawi Publishing Corporation Boundary Value Problems Volume 2o10 Article ID 856932 18 pages doi 2010 856932 Research Article Multiple Positive Solutions for a Class of Concave-Convex Semilinear Elliptic Equations in Unbounded Domains with Sign-Changing Weights Tsing-San Hsu Center for General Education Chang Gung University Kwei-Shan Tao-Yuan 333 Taiwan Correspondence should be addressed to Tsing-San Hsu tshsu@ Received 8 September 2010 Accepted 18 October 2010 Academic Editor Julio Rossi Copyright 2010 Tsing-San Hsu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We study the existence and multiplicity of positive solutions for the following Dirichlet equations -Am u Xafx u q-2u b x u p-2u in Q. u 0 on dQ where d 0 1 q 2 p 2 2 2N N - 2 if N 3 2 OT if N 1 2 Q is a smooth unbounded domain inJĩv a x b x satisfy suitable conditions and a x maybe change sign in Q. 1. Introduction and Main Results In this paper we deal with the existence and multiplicity of positive solutions for the following semilinear elliptic equation -Am u Xa x u q-2u b x u p-2u in Q u 0 in Q b u 0 on dQ where X 0 1 q 2 p 2 2 2N N - 2 if N 3 2 OT if N 1 2 Q is an unbounded domain and a b are measurable functions and satisfy the following conditions A1 a e C Q n Lq Q q p p - q with a max a 0 0 in Q. B1 b e C Q n LOT Q and b max b 0 0 in Q. 2 Boundary Value Problems Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied. For example Ambrosetti et al. 1 considered the following equation -Au Auq-1 up-1 in Q u 0 in Q u 0 on dQ Ex where A 0 1 q 2 p 2 . They proved that there exists Ao 0 such that Ex admits at least two positive solutions for all A e 0 Ao and has one positive solution for A A0 and no positive solution for A A0. Actually Adimurthi et al. 2 Damascelli .