id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-5621	Habeeb Ibrahim; Dafaalla, Mohammed; Osman Abdalla Adam Osman; Ashraf. S. ELshreif	Local and Global Weak Solutions for Elliptic Nonlinear Equations	2025	16	.pdf	application/pdf	5490	317	81	The following are recollections of estimates for um to [5, 12]:∫ Bs |∇um|pdx ≤ C ∫ Bs |∇vm|pdx, (10) and sup Bp |∇um| ≤ C (∫ Bs |∇um|pdx) ) 1 p (11) in the range p ∈ (0, s/2], with C = C(n, p, α). i | N1+ε 0 , consequently, as shown in Lemma 3 that 1/ ( ≤ {2x ∈ B1 i : s∑ m=1 |∇vm| > 2N0(λ∗ + ϵ) )1/2 ≤ C̄ε1+ε 1 λ1+ε ∫ {x∈B0 i : ∑s m=1 |∇vm|>(λ∗+ε)/4} s∑ m=1 |∇vm|1+δdx in that where C̄ = C̄(n, 1 + ϵ, (1+ϵ ϵ )1, α).	cache/ejpam-5621.pdf	txt/ejpam-5621.txt