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 On Multiple Solutions of Concave and Convex Nonlinearities in Elliptic Equation on RN | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 147308 19 pages doi 2009 147308 Research Article On Multiple Solutions of Concave and Convex Nonlinearities in Elliptic Equation on RN Kuan-Ju Chen Department of Applied Science Naval Academy 90175 Zuoying Taiwan Correspondence should be addressed to Kuan-Ju Chen kuanju@ Received 18 February 2009 Accepted 28 May 2009 Recommended by Martin Schechter We consider the existence of multiple solutions of the elliptic equation on RN with concave and convex nonlinearities. 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 First we look for positive solutions of the following problem -Aw u a xfup-1 1b x uq-1 in RN u 0 in RN u e H1 rn where 1 0 is a real parameter 1 p 2 q 2 2N N - 2 N 3. We will impose some assumptions on a x and b x . Assume a1 a x 0 a x e La a-1 RN n L RN where 1 a 2 p b1 b x e C Rn b x b x- 0 as x x b x b x- for all x e RN Such problems occur in various branches of mathematical physics and population dynamics and sublinear analogues or superlinear analogues of problem have been considered by many authors in recent years see 1-4 . Little information is known about the combination of sublinear and superlinear case of problem . In 5 6 they deal with the analogue of problem when RN is replaced by a bounded domain Q. For the RN case the existence of positive solutions for problem was proved by few people. 2 Boundary Value Problems In the present paper we discuss the Nehari manifold and examine carefully the connection between the Nehari manifold and the fibrering maps then using arguments similar to those used in 7 we will prove the existence of the two positive solutions by using Ekeland s Variational Principle 8 . In 5 Ambrosetti et al.