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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: EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF SUPERLINEAR p-LAPLACIAN EQUATIONS | EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF SUPERLINEAR p-LAPLACIAN EQUATIONS JUAN WANG AND CHUN-LEI TANG Received 16 May 2006 Revised 5 July 2006 Accepted 6 July 2006 By a variant version of mountain pass theorem the existence and multiplicity of solutions are obtained for a class of superlinear p-Laplacian equations -Apu f x u . In this paper we suppose neither f satisfies the superquadratic condition in Ambrosetti-Rabinowitz sense nor f x t t p-1 is nondecreasing with respect to 1t . Copyright 2006 J. Wang and C.-L. Tang. 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 and main results In this paper we consider the following superlinear p-Laplacian equations with Dirichlet boundary value condition -ApU f x u x e u u 0 x e du 1.1 where ApU is the p-Laplacian operator ApU div Vu p-2 Vu withp 1 u is abounded domain in RN N 1 with smooth boundary du f e C u X R R is subcritical in t that is there is a q e p np n - p when N p q e p ro when N p such that f x t n lim 0 1.2 t-A t q-1 v 7 uniformly in a.e. x e u. We are interested in the case that f is superlinear in t at infinity that is f x t t _ z lim TO. 1.3 f LA t p Ambrosetti and Rabinowitz have got the solutions of problem 1.1 by a very famous mountain pass theorem in 1 . But they supposed the well-known AR condition holds Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 47275 Pages 1-12 DOI 10.1155 BVP 2006 47275 2 A class of superlinear p-Laplacian equations that is for some p p M 0 0 pF x s f x s s V s M x e Q. AR It is easy to see that the AR implies 1.3 . This AR condition usually plays a very important role in verifying that the functional corresponding to problem has a MountainPass geometry and shows that a related PS c sequence is bounded see 1 2 5 12 . But there are always many functions .
