Polymer-Attached Thiazolium Salts-Catalyzed Addition of Aldehydes to α, β-Unsaturated Carbonyl Compounds

A homogeneous catalyst, 3-benzyl-5-(2-hydroxyethyl)-4-methyl-1,3-thiazolium chloride, for addition of aldehydes to activated double bond, was attached to 20% cross-linked polystyrene-divinylbenzene copolymer. The attached catalysts could be easily removed from the reaction mixture. Polymer-attached thiazolium salts in the presence of triethylamine are active catalysts for addition of aromatic and aliphatic aldehydes to α,β-unsaturated ketones to yield γ-diketones.     

含有陡峭势阱和凹凸非线性项的Kirchhoff型问题的多重正解*

在这篇文章中, 作者研究涉及凹凸非线性项的Kirchhoff型问题-(a + b ∫R3|▽u|²dx) Δu + λV (x)u = μf(x)|u|^q?2u + |u|^p?2u, x ∈ R³,u ∈ H¹(R³),其中a,b > 0 是常数, λ, μ > 0 是参数, 1 < q < 2, 4 < p < 6 且 V 是一个非负连续位势. 在f(x) 和 V 的合适条件下,此问题正解的存在性和集中性能够通过Nehari 流形和Ekeland 变分原理得到.