id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-177	Molica Bisci, Giovanni; Servadei, Raffaella; Zhang, Binlin	Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications	2022	21	.pdf	application/pdf	8698	472	76	The space Xs 0(Ω) is defined as Xs 0(Ω) := { g ∈ X : g = 0 a.e. in Rn \ Ω } , EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 5 where X denotes the linear space of Lebesgue measurable functions from Rn to R such that the restriction to Ω of any function g in X belongs to L2(Ω) and the map (x, y) 7→ (g(x)− g(y)) ∣∣ 6 qν(x) a.e. x ∈ Rn (4.43) for all j ∈ N. By (1.5), (4.42), (4.43), and the Lebesgue Dominated Convergence Theorem, we obtain that ∫ Ω f(x, uj(x))uj(x)dx→ ∫ Ω f(x, u∞(x))u∞(x) dx∫ Ω f(x, uj(x))u∞(x) dx→ ∫ Ω f(x, u∞(x))u∞(x) dx (4.44) as j → +∞, while, by (1.7) and (4.42) we obtain∫ Ω g(x)uj(x) dx→ ∫ Ω g(x)u∞(x) dx (4.45) as j → +∞.	cache/ejde-177.pdf	txt/ejde-177.txt