Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 45043, 10 pages
doi:10.1155/JIA/2006/45043
On the nonexistence of positive solution of some singular nonlinear integral equations
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.
Abstract
We consider the singular nonlinear integral equation
u(x)=∫ℝNg(x,y,u(y))dy/|y−x|σ
for all x∈RN
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,y∈RN,u≥0, with some constants α,β,β1,γ,γ1≥0 and M>0.
We prove in an
elementary way that if
0≤α≤(N+β−γ)/(σ+γ1−β1), (1/2)(N+β+β1−γ−γ1)<σ<min{N,N+β+β1−γ−γ1},
σ+γ1−β1>0,
N≥2
the above nonlinear integral equation has no positive solution.