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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Sub-super solutions for (p-q) Laplacian systems | Haghaiegh and Afrouzi Boundary Value Problems 2011 2011 52 http www.boundaryvalueproblems.eom content 2011 1 52 RESEARCH o Boundary Value Problems a SpringerOpen Journal Open Access Sub-super solutions for p-q Laplacian systems Somayeh Haghaiegh1 and Ghasem Alizadeh Afrouzi2 Correspondence Haghaieghi_ch86@yahoo.com department of Mathematics Science and Research Branch Islamic Azad University Tehran Iran Full list of author information is available at the end of the article Abstract In this work we consider the system Apu A g x a u f v in AqV A g x b v h u in u V 0 on d Z where Q is a bounded region in RN with smooth boundary 30 Ap is the p-Laplacian operator defined by Apu div Vu p-2Vu p q 1 and g x is a C1 sign-changing the weight function that maybe negative near the boundary. f h a b are C1 nondecreasing functions satisfying a 0 0 b 0 0. Using the method of sub-super solutions we prove the existence of weak solution. 1 Content In this paper we study the existence of positive weak solution for the following system A.pu A g x a u f v in AqV A g x b v h u in u V 0 on dfà 1 where o is a bounded region in RN with smooth boundary do Ap is the p-Laplacian operator defined by Apu div Vu p 2 Vu p q 1 and g x is a C1 sign-changing the weight function that maybe negative near the boundary. f h a b are C1 non-decreasing functions satisfying a 0 0 b 0 0. This paper is motivated by results in 1-5 . We shall show the system 1 with signchanging weight functions has at least one solution. 2 Preliminaries In this article we use the following hypotheses 1 Al . f M h s q 1j as s VM 0 A2 lim f s lim h s as s . A3 lima r limSq T 0 as s . Let lp lq be the first eigenvalue of -Ap -Aq with Dirichlet boundary conditions and ộp ộq be the corresponding positive eigenfunctions with ộp ộq 1. Springer 2011 Haghaiegh and Afrouzi licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http creativecommons.org licenses by 2.0 .
