Abstract and Applied Analysis
Volume 2010 (2010), Article ID 971268, 20 pages
doi:10.1155/2010/971268
Research Article

Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weighted 𝑝 ( 𝑥 ) -Laplacian Equations with Exponential Nonlinearities

Li Yin,1 Yunrui Guo,2 Jing Yang,1 Bibo Lu,3 and Qihu Zhang1,4

1Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China
2Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China
3School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, Henan 454000, China
4School of Mathematics and Statistics, Huazhong Normal University, Wuhan, Hubei 430079, China

Received 21 March 2010; Accepted 3 October 2010

Academic Editor: Pavel Drábek

Copyright © 2010 Li Yin 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 investigates the following 𝑝 ( 𝑥 ) -Laplacian equations with exponential nonlinearities: − Δ 𝑝 ( 𝑥 ) 𝑢 + 𝜌 ( 𝑥 ) 𝑒 𝑓 ( 𝑥 , 𝑢 ) = 0 in Ω , 𝑢 ( 𝑥 ) → + ∞ as 𝑑 ( 𝑥 , 𝜕 Ω ) → 0 , where − Δ 𝑝 ( 𝑥 ) 𝑢 = − d i v ( | ∇ 𝑢 | 𝑝 ( 𝑥 ) − 2 ∇ 𝑢 ) is called 𝑝 ( 𝑥 ) -Laplacian, 𝜌 ( 𝑥 ) ∈ 𝐶 ( Ω ) . The asymptotic behavior of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.