Boundary Value Problems
Volume 2010 (2010), Article ID 607453, 17 pages
doi:10.1155/2010/607453
Research Article

Existence and Nonexistence of Positive Solutions for Singular 𝑝 -Laplacian Equation in ℝ 𝑁

Caisheng Chen, Zhenqi Wang, and Fengping Wang

Department of Mathematics, Hohai University, Nanjing, Jiangsu 210098, China

Received 15 August 2010; Accepted 10 December 2010

Academic Editor: Zhitao Zhang

Copyright © 2010 Caisheng Chen 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

We study the existence and nonexistence of solutions for the singular quasilinear problem − d i v ( | 𝑥 | − 𝑎 𝑝 | ∇ 𝑢 | 𝑝 − 2 ∇ 𝑢 ) = ℎ ( 𝑥 ) 𝑓 ( 𝑢 ) + 𝜆 𝐻 ( 𝑥 ) 𝑔 ( 𝑢 ) , 𝑥 ∈ ℝ 𝑁 , 𝑢 ( 𝑥 ) > 0 , 𝑥 ∈ ℝ 𝑁 , l i m | 𝑥 | → ∞ 𝑢 ( 𝑥 ) = 0 , where 1 < 𝑝 < 𝑁 , 0 ≤ 𝑎 < ( 𝑁 − 𝑝 ) / 𝑝 and 𝑓 ( 𝑢 ) a n d 𝑔 ( 𝑢 ) behave like 𝑢 𝑚 and 𝑢 𝑛 with 0 < 𝑚 ≤ 𝑝 − 1 < 𝑛 at the origin. We obtain the existence by the upper and lower solution method and the nonexistence by the test function method.