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 Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 865408 11 pages doi 2009 865408 Research Article Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems Kuan-Ju Chen Department of Applied Science Naval Academy . Box 90175 Zuoying Kaohsiung 8 303 Taiwan Correspondence should be addressed to Kuan-Ju Chen kuanju@ Received 16 December 2008 Accepted 6 July 2009 Recommended by Wenming Zou We proved a multiplicity result for strongly indefinite semilinear elliptic systems -Am u 1 1 x a v p-2v in RN -Av v 1 1 x b u q-2u in RN where a and b are positive numbers which are in the range we shall specify later. Copyright 2009 Kuan-Ju Chen. 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. 1. Introduction In this paper we shall study the existence of multiple solutions of the semilinear elliptic systems -Am u 1 v p 2v in RN I1 x i 1 . . _xr -Av v - - b u q 2u in RN where a and b are positive numbers which are in the range we shall specify later. Let us consider that the exponents p q 2 are below the critical hyperbola 1 1 1 N-2 for N 3 p q N 2 Boundary Value Problems so one of p and q could be larger than 2N N - 2 for that matter the quadratic part of the energy functional T 1 1 1 1 a I u v Vu Vv uv dx - - I fl v pdx - - -- b u qdx has to be redefined and we then need fractional Sobolev spaces. Hence the energy functional I is strongly indefinite and we shall use the generalized critical point theorem of Benci 1 in a version due to Heinz 2 to find critical points of I . And there is a lack of compactness due to the fact that we are working in RN. In 3 Yang shows that under some assumptions on the functions f and g there exist infinitely many solutions of the semilinear elliptic systems -Au u g x v in RN 1 4 -Av v f x ù in R. We shall
