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

• Main Authors: Ramzi Alsaedi, Habib Maagli, Noureddine Zeddini
• Format: Article
• Language: English
• Published: Texas State University 2014-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Green function; Dirichlet Laplacian; fractional Laplacian; Karamata function
• Online Access: http://ejde.math.txstate.edu/Volumes/2014/08/abstr.html

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 $V=(-\Delta)^{\alpha/2}$, $0<\alpha<2$.