Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems with continuous potentials

In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials:{-ε~2?u +(1 + δP(x))u = μ1 u~3+ βuv~2 in ?,-ε~2?v +(1 + δQ(x))v = μ2 v~3+ βu~2 v in ?,u 0, v 0 in ?,(?u)/(?v)=(?ν)/(?ν)=0on ??,(A_ε)where ? is a smooth bounded domain in R~N for N = 2, 3, δ, ε, μ_1 and μ_2 are positive parameters, β∈ R,P(x) and Q(x) are two smooth potentials defined on ?, the closure of ?. Due to Liapunov-Schmidt reduction method, we prove that(A_ε) has at least O(1/(ε| ln ε|)~N) synchronized and O(1/(ε| ln ε|)~(2 N)) segregated vector solutions for ε and δ small enough and some β∈ R. Moreover, for each m ∈(0, N) there exist synchronized and segregated vector solutions for(A_ε) with energies in the order of ε~(N-m). Our results extend the result of Lin et al.(2007) from the Lin-Ni-Takagi problem to the nonlinear Schr¨odinger elliptic systems with continuous potentials.     

不含三角形图的正常染色路和正常染色圈

图G为边染色图,对G中的任一顶点v,定义v的色度dc（v）：G中与顶点v相关联的边中不同染色的数目.用δc（G）表示图G的最小色度,即δc（G）=min{dc（v）：v∈G}.若图G为不含三角形的边染色图,且δc（G）≥2,则G含长为4d-2的正常染色路或长至少为2d-2的正常染色圈.     

Amidyl Radical Directed Remote Allylation of Unactivated sp3 C−H Bonds by Organic Photoredox Catalysis

The development of visible‐light‐mediated allylation of unactivated sp³ C?H bonds is reported. The remote allylation was directed by the amidyl radical, which was generated by photocatalytic fragmentation of a pre‐functionalized amide precursor. Both aromatic and aliphatic amide derivatives could successfully deliver the remote C?H allylation products in good yields. A variety of electron deficient allyl sulfone systems could be used as δ‐carbon radical acceptor.     

Solutions for conformally invariant fractional Laplacian equations with multi-bumps centered in lattices

In this paper, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent:

${(?\mathrm{\Delta })}^{s}u=K(x)u^{\frac{N+2s}{N?2s}},u>0\mathrm{in}{\mathbb{R}}^N,$

where $s\in (0,1)$ and $N>2+2s$, $K>0$ is periodic in $(x_1,\dots ,x_k)$ with $1\le k<\frac{N?2s}{2}$. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in ${\mathbb{R}}^k$, including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the bumps locate in lattices in ${\mathbb{R}}^k$, the restriction on $1\le k<\frac{N?2s}{2}$ is in some sense optimal, since we can show that for $k\ge \frac{N?2s}{2}$, no such solutions exist.     

Oxime as a general photocage for the design of visible light photo-activatable fluorophores

Photoactivatable fluorophores have been widely used for tracking molecular and cellular dynamics with subdiffraction resolution. In this work, we have prepared a series of photoactivatable probes using the oxime moiety as a new class of photolabile caging group in which the photoactivation process is mediated by a highly efficient photodeoximation reaction. Incorporation of the oxime caging group into fluorophores results in loss of fluorescence. Upon light irradiation in the presence of air, the oxime-caged fluorophores are oxidized to their carbonyl derivatives, restoring strong fluorophore fluorescence. To demonstrate the utility of these oxime-caged fluorophores, we have created probes that target different organelles for live-cell confocal imaging. We also carried out photoactivated localization microscopy (PALM) imaging under physiological conditions using low-power light activation in the absence of cytotoxic additives. Our studies show that oximes represent a new class of visible-light photocages that can be widely used for cellular imaging, sensing, and photo-controlled molecular release.

Photoactivatable fluorophores have been widely used for tracking molecular and cellular dynamics with subdiffraction resolution.