Abstract and Applied Analysis
Volume 2011 (2011), Article ID 526026, 13 pages
http://dx.doi.org/10.1155/2011/526026
Research Article

Multiplicity of Solutions for Nonlocal Elliptic System of ( 𝑝 , π‘ž ) -Kirchhoff Type

Bitao Cheng,1 Xian Wu,2 and Jun Liu1

1College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China
2Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China

Received 16 May 2011; Accepted 23 June 2011

Academic Editor: Josip E. PečariΔ‡

Copyright Β© 2011 Bitao Cheng et al. 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.

Abstract

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type βˆ’ [ 𝑀 1 ( ∫ Ξ© | βˆ‡ 𝑒 | 𝑝 ) ] 𝑝 βˆ’ 1 Ξ” 𝑝 𝑒 = πœ† 𝐹 𝑒 ( π‘₯ , 𝑒 , 𝑣 ) , in Ω, βˆ’ [ 𝑀 2 ( ∫ Ξ© | βˆ‡ 𝑣 | π‘ž ) ] π‘ž βˆ’ 1 Ξ” π‘ž 𝑣 = πœ† 𝐹 𝑣 ( π‘₯ , 𝑒 , 𝑣 ) , in Ω, 𝑒 = 𝑣 = 0 , on πœ• Ξ© . Under bounded condition on M and some novel and periodic condition on F, some new results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by means of Bonanno's multiple critical points theorems without the Palais-Smale condition and Ricceri's three critical points theorem, respectively.