On the Nonexistence of Positive Solution of Some Singular Nonlinear Integral Equations

We consider the singular nonlinear integral equation u(x)= ∫RN g(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 0, N ≥ 2, the above nonlinear integral equation has no positive solution.