id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-816	Baladi, Houssam; Aglzim, Abdellatif; Filali, Mohammed; Tsouli, Najib	Multiple solutions for p(x)-Kirchhoff type problems with extended Robin boundary conditions	2024	23	.pdf	application/pdf	9704	490	83	+G(x, u) ) dσx ) div ( |∇u|p(x)−2∇u ) = f(x, u) + λh(x), x ∈ Ω, |∇u|p(x)−2 ∂u ∂ν + β(x)|u|p(x)−2u+ g(x, u) = 0, x ∈ ∂Ω, (1.1) where Ω is a bounded domain in RN with smooth boundary ∂Ω, ∂u ∂ν is the outer normal derivative, dσx is the measure on the boundary ∂Ω, β ∈ L1(∂Ω), β− := infx∈∂Ω β(x) > 0, g : ∂Ω × R → R is a measurable function, with G(x, t) :=∫ t 0 g(x, s) ds, p ∈ C+(Ω̄), 1 < p− := inf x∈Ω̄ p(x) ≤ p+ := max x∈Ω̄ p(x) < = o ( |t|αp+−1 ) , t→ 0, uniformly a.e. x ∈ Ω; (A5) There exists a constant µ > αp+ such that µF (x, t) := µ ∫ t 0 f(x, s) ds ≤ f(x, t)t, ∀(x, t) ∈ Ω× R; (A6) inf{x∈Ω;|t|=1} F (x, t) > 0. (A7) g(x, t) = o(|t|r1(x)−1) uniformly a.e. x ∈ ∂Ω, as t→ 0, where r1 ∈ C+(∂Ω), supx∈∂Ω r1(x) = r+1 < p− ≤ p(x) for all x ∈ ∂Ω; (A8) g(x, t) = o ( |t|r2(x)−1 ) , t→ +∞, uniformly a.e. x ∈ ∂Ω, where r2 ∈ C+(∂Ω), supx∈∂Ω r2(x) = r+2 < p− ≤ p(x) for all x ∈ ∂Ω; (A9) G(x, t) := ∫ t 0 g(x, s) ds ≥ 0, ∀(x, t) ∈ ∂Ω× R, where α and µ are given in (A1) and (A5).	cache/ejde-816.pdf	txt/ejde-816.txt