id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-379	Zhao, Junfang; Liu, Xiangqing	Ground state solutions for quasilinear equations of Kirchhoff type	2020	14	.pdf	application/pdf	5928	348	83	Given u, ϕ ∈ X with the property that ∫ Ω u2|∇ϕ|2 dx < +∞ and ∫ Ω |∇u|2ϕ2 dx < +∞, for example ϕ ∈ C∞0 (Ω), ϕ = u, u+ or u−, where u+ = max{u, 0}, u− = min{u, 0}, we can define the derivative of I in the direction ϕ at u, denoted by 〈DI(u), ϕ〉 as 〈DI(u), ϕ〉 = lim t→0+ 1 t (I(u+ tϕ)− I(u)) For u ∈ X, define γ+(u) = 〈DI(u), u+〉 = ∫ Ω (a|∇u+|2 + 2bu2 +|∇u+|2) dx + ∫ Ω (c|∇u|2 + du2|∇u|2) dx ∫ Ω (c|∇u+|2 + 2du2 +|∇u+|2) dx − ∫ Ω f(u+)u+ dx , γ−(u) = 〈DI(u), u−〉 = ∫ Ω (a|∇u−|2 + 2bu2 −|∇u−|2) dx + ∫ Ω (c|∇u|2 + du2|∇u|2) dx ∫ Ω (c|∇u−|2 + 2du2 −|∇u−|2) dx − ∫ Ω f(u−)u− dx (1.9) and S∗ = {u : u ∈ X, γ+(u) = 0, u+ 6= 0; γ−(u) = 0, u− 6= 0}, c∗ = inf u∈S∗ I(u) .	cache/ejde-379.pdf	txt/ejde-379.txt