Abstract

In this paper, we study the following nonlocal system exhibiting singular nonlinearity and weighted singular terms: { ( − Δ ) p s u + v u p − 1 = h ( x ) u − α , u > 0 in Ω ; u = 0 , in R N ∖ Ω , ( − Δ ) p s v = u p , v > 0 in Ω ; v = 0 , in R N ∖ Ω , where 0 < s < 1 , p > 1 , α > 0 , and Ω ⊂ R N , with N > s p , is an open bounded domain with C 1 , 1 boundary ∂ Ω . The function h : Ω → R + exhibits growth of negative powers of the distance function d ( x ) := dist ( x , ∂ Ω ) near the boundary, that is, h ( x ) ∼ d − β ( x ) for some β ≥ 0 , when x is close to the boundary ∂ Ω . For β < s p , we discuss the existence of a positive weak solution ( u , v ) ∈ W loc s , p ( Ω ) × W loc s , p ( Ω ) using the classical method of regularization and the fixed point theorem together. Indeed, we found some essential uniform a priori estimates for the approximating sequence before proceeding to the limits. Moreover, we address the uniqueness of finite energy solutions, that is, ( u , v ) ∈ W 0 s , p ( Ω ) × W 0 s , p ( Ω ) , and demonstrate that this solution pair is a saddle point of a suitable functional when α < 1 . We also provide the boundary behavior of the weak solutions in terms of the distance function. Finally, we establish the nonexistence of a weak solution for the case where β ≥ s p .

Information

Item Type:	Journal
Divisions:
ePrint ID:	5440
Date Deposited:	2025-06-21
Further Information:	Google Scholar
URI:	https://www.univ-soukahras.dz/en/publication/article/5440

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