Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 45043, 10 pages

On the nonexistence of positive solution of some singular nonlinear integral equations

Nguyen Thanh Long

Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University - Ho Chi Minh City, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Vietnam

Received 15 February 2004; Accepted 24 August 2004

Copyright © 2006 Nguyen Thanh Long. 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.


We consider the singular nonlinear integral equation u(x)=Ng(x,y,u(y))dy/|yx|σ for all xRN where σ is a given positive constant and the given function g(x,y,u) is continuous and g(x,y,u)M|x|β1|y|β(1+|x|)γ1(1+|y|)γuα for all x,yRN,u0, with some constants α,β,β1,γ,γ10 and M>0. We prove in an elementary way that if 0α(N+βγ)/(σ+γ1β1), (1/2)(N+β+β1γγ1)<σ<min{N,N+β+β1γγ1}, σ+γ1β1>0, N2 the above nonlinear integral equation has no positive solution.