Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 08, pp. 1-10.

Title: Estimates on potential functions and boundary behavior 
       of positive solutions for sublinear Dirichlet problems

Authors: Ramzi Alsaedi (King Abdulaziz Univ., Rabigh, Saudi Arabia)
         Habib Maagli  (King Abdulaziz Univ., Rabigh, Saudi Arabia)
         Noureddine Zeddini (King Abdulaziz Univ., Rabigh, Saudi Arabia)

Abstract:
 We give  global estimates on some potential of functions in a bounded domain
 of the Euclidean space ${\mathbb{R}}^n\; (n\geq 2)$. These functions
 may be singular near the boundary and are globally comparable to a product
 of a power of the distance to the boundary by some particularly well behaved
 slowly varying function near zero. Next, we prove the existence and uniqueness
 of a positive solution for the integral equation $u=V(a u^{\sigma})$ with
 $0\leq \sigma <1$, where V belongs to a class of kernels that contains
 in particular the potential kernel of the classical Laplacian
 $V=(-\Delta)^{-1}$ or the fractional laplacian
 $V=(-\Delta)^{\alpha/2}$, $0<\alpha<2$.

Submitted September 14, 2013. Published January 07, 2014.
Math Subject Classifications: 35R11, 35B40, 35J08.
Key Words: Green function; Dirichlet Laplacian; fractional Laplacian;
           Karamata function.