id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-209	Calanchi, Marta; Ruf, Bernhard	Eigenvalues and bifurcation for Neumann problems with indefinite weights	2021	14	.pdf	application/pdf	6010	349	77	φ∗, φ ∗ two associated eigenvectors, then φ∗, φ ∗ are orthogonal∫ Ω ∇φ∗∇φ∗ dx = 0, ∫ Ω a(x)φ∗φ ∗ dx = 0. (b) (First eigenvalues) λ+ 1 = inf u∈B+ ∫ Ω |∇u|2dx ≥ 0, λ−1 = − inf u∈B− ∫ Ω |∇u|2dx ≤ 0 are simple, with associated positive eigenfunctions φ+ 1 and φ−1 . If λ+ 1 := infu∈B+ ∫ Ω |∇u|2dx = 0, there is a sequence un = wn + sn, with ∫ Ω wn = 0 and sn ∈ R such that∫ Ω a(x)u2 n = 1, ∫ Ω |∇wn|2dx→ 0, as n→ +∞. Therefore wn → 0 strongly in H1(Ω) and sn is bounded: otherwise we would have (up to subsequences) 1 = ∫ Ω a(x)u2 n = ∫ Ω a(x)(s2 n + 2wnsn + w2 n)dx = s2 n (∫ Ω a(x) dx+ o(1) ) → −∞. Since sn is bounded, up to subsequences, sn → s and un → s strongly, from which we obtain 1 = ∫ Ω a(x)u2 n → s2 ∫ Ω a(x) ≤ 0, which is a contradiction.	cache/ejde-209.pdf	txt/ejde-209.txt