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 <min{N ,N + β + β1 − γ − γ1}, σ + γ1 − β1 > 0, N ≥ 2, the above nonlinear integral equation has no positive solution.


Introduction
We consider the nonexistence of positive solutions of the following singular nonlinear integral equation 1) where b N = 2((N − 1)ω N+1 ) −1 with ω N+1 being the area of unit sphere in R N+1 , N ≥ 2, σ is a given positive constant with 0 < σ < N, and g : R 2N × R + → R is given continuous function satisfying the following. There exist the constants α,β,β 1 ,γ,γ 1 ≥ 0 and M > 0 such that 2 On the nonexistence of positive solutions (1.4) of which the boundary value u(x) = v(x,0) together with some auxiliary conditions will be a solution of the equation In [3] the authors have studied a problem (1.3), (1.4) for N = 2 with the Laplace equation (1.3) having the axial symmetry u rr + 1 r u r + u zz = 0 ∀r > 0, ∀z > 0, (1.6) and with the nonlinear boundary condition of the form −u z (r,0) = I 0 exp − r 2 /r 2 0 + u α (r,0) ∀r > 0, (1.7) where I 0 , r 0 , α are given positive constants. The problem (1.6), (1.7) is the stationary case of the problem associated with ignition by radiation. In the case of 0 < α ≤ 2 the authors in [3] have proved that the following nonlinear integral equation (1.8) associated to the problem (1.6), (1.7) has no positive solution. Afterwards, this result has been extended in [8] to the general nonlinear boundary condition −u z (r,0) = g r,u(r,0) ∀r > 0. (1.9) In [7] the problem (1.3), (1.4) is considered for N = 2 and for a function g continuous, nondecreasing and bounded below by the power function of order α with respect to the third variable and it is proved that for 0 < α ≤ 2 such a problem has no positive solution.

The theorem of nonexistence of positive solution
Without loss of generality, we can suppose that b N = 1 with a change of the constant M in the assumption (1.2) of g. We rewrite the integral equation (1.1): Then we have the main result as follows.

the integral equation (2.1) has no continuous positive solution.
Remark 2.2. The result of theorem is stronger than that in [1,7]. Indeed, corresponding to the same equation (1.5), the following assumptions which were made in [1,7] are not needed here.
(G 1 ) g(y,u) is nondecreasing with respect to variable u, that is, exists and is positive.
First, we need the following lemma.
where ω N is the area of unit sphere in R N .
The proof of lemma can be found in [9].
Theorem is proved completely.