Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains
This article is part of Special Issue:
Imed Bachar,
Habib Mâagli,
Corresponding Author
Imed Bachar
Mathematics Department, College of Sciences , King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia ksu.edu.sa
Search for more papers by this authorHabib Mâagli
Mathematics Department, College of Sciences and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia kau.edu.sa
Search for more papers by this authorImed Bachar,
Habib Mâagli,
Corresponding Author
Imed Bachar
Mathematics Department, College of Sciences , King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia ksu.edu.sa
Search for more papers by this authorHabib Mâagli
Mathematics Department, College of Sciences and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia kau.edu.sa
Search for more papers by this authorFirst published: 19 August 2013
Academic Editor: Daniel C. Biles
Abstract
We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem , in D limx→z∈∂D(δ(x)) 1−(α/2)u(x) = 0, where 0 < α < 2, σ1, σ2 ∈ (−1,1), D is a bounded C1,1-domain in ℝn, n ≥ 2, and δ(x) denotes the Euclidian distance from x to the boundary of D. The nonnegative weight functions a1, a2 are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution.
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