id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-291	Carriao, Paulo Cesar; Costa, Augusto Cesar dos Reis; Miyagaki, Olimpio Hiroshi; Vicente, Andre	Kirchhoff-type problems with critical Sobolev exponent in a hyperbolic space	2022	12	.pdf	application/pdf	5182	305	81	We consider the functional J : H1 0,r(Ω)→ R associated with problem (2.1), J(v) = a 2 ‖v‖2 + b 4 ‖v‖4 − λ q ∫ Ω pα|v|q − 1 6 ∫ Ω |v|6, (2.2) whose Gateaux derivative is J ′(v)w = (a+ b‖v‖2) ∫ Ω ( ∇v ·∇w+ 3 4 p2vw ) −λ ∫ Ω pα|v|q−2vw− ∫ Ω |v|4vw. From (2.13), we have d dt J(tvε)|t=tε = 0, thus, atε‖vε‖2 + bt3ε‖vε‖4 − λtq−1 ε ∫ Ω pα|vε|q − t5ε ∫ Ω |vε|6 = 0, EJDE-2021/53 KIRCHHOFF-TYPE PROBLEMS IN A HYPERBOLIC SPACE 7 which implies a‖vε‖2 + bt2ε‖vε‖4 − λtq−2 ε ∫ Ω pα|vε|q − t4ε ∫ Ω |vε|6 = 0.	cache/ejde-291.pdf	txt/ejde-291.txt