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.