tailieunhanh - Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 524862, 21 pages

Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 524862, 21 pages doi: Research Article Multiple Positive Solutions of Semilinear Elliptic Problems in Exterior Domains Tsing-San Hsu and Huei-Li Lin Department of Natural Sciences, Center for General Education, Chang Gung University, Tao-Yuan 333, Taiwan Correspondence should be addressed to Huei-Li Lin, hlin@ Received 30 July 2010; Accepted 30 November 2010 Academic Editor: Wenming Z. Zou Copyright q 2010 . Hsu and . Lin. 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. | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 524862 21 pages doi 2010 524862 Research Article Multiple Positive Solutions of Semilinear Elliptic Problems in Exterior Domains Tsing-San Hsu and Huei-Li Lin Department of Natural Sciences Center for General Education Chang Gung University Tao-Yuan 333 Taiwan Correspondence should be addressed to Huei-Li Lin hlin@ Received 30 July 2010 Accepted 30 November 2010 Academic Editor Wenming Z. Zou Copyright 2010 . Hsu and . Lin. 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. Assume that q is a positive continuous function in RN and satisfies the suitable conditions. We prove that the Dirichlet problem -Au u q z u p-2u admits at least three positive solutions in an exterior domain. 1. Introduction For N 3 and 2 p 2 2N N - 2 we consider the semilinear elliptic equations - Au u q z u p 2u in Q u e Hj Q - Au u q u p-2u in Q u e H0 Q where Q is an unbounded domain RN. Let q be a positive continuous function in RN and satisfy lim q z qm 0 q z q . z q1 2 Boundary Value Problems Associated with and we define the functional a b b J and J for u e H0 Q. a u ị ỘVu 2 u2 dz u H1 b u J q z updz b tu J q-siu dz J u 1 a u - 1 b u 2 p J u 1 a ù - 1 b u 2p where u max u 0 0. By Rabinowitz 1 Proposition the functionals a b b J and J are of C2. It is well known that admits infinitely many solutions in a bounded domain. Because of the lack of compactness it is difficult to deal with this problem in an unbounded domain. Lions 2 3 proved that if q z qx 0 then has a positive ground state solution in V. Bahri and Li 4 proved that there is at least one positive solution of in RN when lim z q z qx 0 and q z q x - Cexp -ổ z for Ỗ 2. Zhu 5 has studied the multiplicity of solutions of in v as