Boundary Value Problems
Volume 2010 (2010), Article ID 429813, 18 pages
doi:10.1155/2010/429813
Research Article

Superlinear Singular Problems on the Half Line

Irena Rachůnková and Jan Tomeček

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, tř . 17. listopadu 12, 771 46 Olomouc, Czech Republic

Received 19 October 2010; Accepted 7 December 2010

Academic Editor: Zhitao Zhang

Copyright © 2010 Irena Rachůnková and Jan Tomeček. 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

The paper studies the singular differential equation ( 𝑝 ( 𝑡 ) 𝑢 ) = 𝑝 ( 𝑡 ) 𝑓 ( 𝑢 ) , which has a singularity at 𝑡 = 0 . Here the existence of strictly increasing solutions satisfying s u p { | 𝑢 ( 𝑡 ) | ∶ 𝑡 ∈ [ 0 , ∞ ) } ≥ 𝐿 > 0 is proved under the assumption that 𝑓 has two zeros 0 and 𝐿 and a superlinear behaviour near − ∞ . The problem generalizes some models arising in hydrodynamics or in the nonlinear field theory.