tailieunhanh - Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 429813, 18 pages

Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 429813, 18 pages doi: Research Article Superlinear Singular Problems on the Half Line ˚ ´ ˇ Irena Rachunkova and Jan Tomecek Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palack´ University, tˇ . 17. listopadu 12, 771 46 Olomouc, Czech Republic y r Correspondence should be addressed to Irena Rachunkov´ , rachunko@ a ˚ Received 19 October 2010; Accepted 7 December 2010 Academic Editor: Zhitao Zhang Copyright q 2010 I. Rachunkov´ and J. Tomeˇ ek. This is an open access article distributed under a c ˚ the Creative Commons Attribution. | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 429813 18 pages doi 2010 429813 Research Article Superlinear Singular Problems on the Half Line Irena Rachunkova and Jan Tomecek Department of Mathematical Analysis and Applications of Mathematics Faculty of Science Palacky University tr. 17. listopadu 12 771 46 Olomouc Czech Republic Correspondence should be addressed to Irena Rachunkova rachunko@ Received 19 October 2010 Accepted 7 December 2010 Academic Editor Zhitao Zhang Copyright 2010 I. Rachunkova and J. Tomecek. 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. The paper studies the singular differential equation p f u p f f u which has a singularity at t 0. Here the existence of strictly increasing solutions satisfying sup u f t e 0 to L 0 is proved under the assumption that f has two zeros 0 and L and a superlinear behaviour near -to. The problem generalizes some models arising in hydrodynamics or in the nonlinear field theory. 1. Introduction Let us consider the problem paw p f f Ù u 0 0 m to L where L is a positive real parameter. Definition . Let c 0. A function u e C1 0 c n C2 0 c satisfying on 0 c is called a solution of on 0 c . Definition . Let u be a solution of on 0 c for each c 0. Then u is called a solution of on 0 to . If u moreover fulfils conditions it is called a solution of problem . Definition . A strictly increasing solution of problem is called a homoclinic solution. 2 Boundary Value Problems In this paper we are interested in the existence of strictly increasing solutions and in particular of homoclinic solutions. In what follows we assume f e Liploc R f 0 f L 0 f x 0 for x e 0 L there exists B 0 such that f x 0 for x e b 0 F b FL where F x - f z dz 0 p e C 0 to n