Electron. J. Diff. Equ., Vol. 2014 (2014), No. 08, pp. 1-10.

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

Ramzi Alsaedi, Habib Maagli, Noureddine Zeddini

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 7, 2014.
Math Subject Classifications: 35R11, 35B40, 35J08.
Key Words: Green function; Dirichlet Laplacian; fractional Laplacian; Karamata function.

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Ramzi Alsaedi
Department of Mathematics, College of Sciences and Arts
King Abdulaziz University, Rabigh Campus
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: ramzialsaedi@yahoo.co.uk
Habib Mâagli
Department of Mathematics, College of Sciences and Arts
King Abdulaziz University, Rabigh Campus
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: habib.maagli@fst.rnu.tn
Noureddine Zeddini
Department of Mathematics, College of Sciences and Arts
King Abdulaziz University, Rabigh Campus
P.O. Box 344, Rabigh 21911, Saudi Arabia
email: noureddine.zeddini@ipein.rnu.tn

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