id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-990	Allalou, Mouad; El Ouaarabi , Mohamed; Raji, Abderrahmane	Existence of weak solutions for nonlocal Dirichlet problems via Young measure theory	2024	12	.pdf	application/pdf	4418	276	79	Introduction Let D be a bounded domain in Rn with smooth boundary ∂D, and 1 < p <∞. We prove the existence of weak solutions for the nonlocal problem with Dirichlet boundary conditions, −g (∫ D E(υ) dz ) div[a(z,∇υ) + |∇υ|p−2∇υ] + |υ|p−2υ = f(z, υ) in D, υ = 0 on ∂D, (1.1) where E(υ) = ∫ D ( A(z,∇υ) + 1 p |∇υ|p ) dz. Let ψ : D × Rm → R be a Carathéodory function and wµ : D → R a sequence of measurable functions such that ∇wµ generates the Young measure σz with |σz|g(Rm) = 1 for almost every z ∈ D. Then lim inf µ→∞ ∫ D ψ(z,∇wµ) dz ≥ ∫ D ∫ Rm ψ(z, η) dσz(η) dz provided that the negative part of ψ(z,∇wµ) is equi-integrable.	cache/ejde-990.pdf	txt/ejde-990.txt