Two novel fluorine-containing cyclophanes 1 and 2 incorporating a 1,4-bismethylfluorophenylene subunits have been synthesized via the template-directed clipping methodology and their structure was characterized by ~1H...Two novel fluorine-containing cyclophanes 1 and 2 incorporating a 1,4-bismethylfluorophenylene subunits have been synthesized via the template-directed clipping methodology and their structure was characterized by ~1H NMR,^(13)C NMR, MALDI-TOF-MS and elemental analysis.X-ray diffraction analysis of the solid state molecular structure showed that the complex units are linked by non-classical C-H…F hydrogen bonding.展开更多
In this article,we apply blow-up analysis to study pointwise a priori estimates for some p-Laplacian equations based on Liouville type theorems.With newly developed analysis techniques,we first extend the classical re...In this article,we apply blow-up analysis to study pointwise a priori estimates for some p-Laplacian equations based on Liouville type theorems.With newly developed analysis techniques,we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function,i.e.,the solution ofΔpu=0,x∈Ω.We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplacian equation-Δp^(u)=|u|^(λ-1)u,x∈Ω,which are then extended to the equation with general right hand term f(x,u)with certain asymptotic properties.In addition,point wise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplacian equation,in a case of p=2,are also discussed.展开更多
基金support from the National Natural Science Foundation of China(No.20872051)
文摘Two novel fluorine-containing cyclophanes 1 and 2 incorporating a 1,4-bismethylfluorophenylene subunits have been synthesized via the template-directed clipping methodology and their structure was characterized by ~1H NMR,^(13)C NMR, MALDI-TOF-MS and elemental analysis.X-ray diffraction analysis of the solid state molecular structure showed that the complex units are linked by non-classical C-H…F hydrogen bonding.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11871070 and 62273364)the Guangdong Basic and Applied Basic Research Foundation(Grant No.2020B151502120)。
文摘In this article,we apply blow-up analysis to study pointwise a priori estimates for some p-Laplacian equations based on Liouville type theorems.With newly developed analysis techniques,we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function,i.e.,the solution ofΔpu=0,x∈Ω.We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplacian equation-Δp^(u)=|u|^(λ-1)u,x∈Ω,which are then extended to the equation with general right hand term f(x,u)with certain asymptotic properties.In addition,point wise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplacian equation,in a case of p=2,are also discussed.
