Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations

We show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{\Delta u}\sts{x} -{\bm\alpha}\sts{N,\lambda} \int_{\R^N} { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}{\lambda}} }\diff{y}\, u^{p-1}\sts{x} =0,\quad x\in \R^N \end{equation*} where $u(x)$ is a real-valued function, $N\geq 3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}$ and {\bm\alpha}\sts{N,\lambda} is a constant. It generalizes the results for the whole range $0<\lambda<N$ in \cite{DY2019dcds, GWY2020na, LTX2021, MWX:Hartree} and confirms an open nondegeneracy problem in \cite{GMYZ2022cvpde}. Firstly, by the stereographic projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere $\S^N$ in \cite{FL2012}, we give an alternative proof of the existence of the extremizer of sharp Hardy-Littlewood-Sobolev inequality in $\R^N$ without use of the rearrangement inequalities in \cite{lieb2001analysis}, which is related to the existence of positive bubble solutions of (NLH). Secondly, by making use of the Green function, we obtain an integral form in $\R^N$ of the corresponding linearized equation around positive bubble solutions under suitable decay condition, and its equivalent integral form on the sphere $\S^N$ via the stereographic projection. Lastly, together with the key spherical harmonic decomposition and the Funk-Hecke formula of the spherical harmonic functions in \cite{AH2012, DX2013book, SteinW:Fourier anal}, we can obtain the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equation (NLH), which is inspired by Frank and Lieb in \cite{FL2012am,FL2012}.Comment: 26 pages. Any comment is welcom

Porous Collagen Scaffold Reinforced with Surfaced Activated PLLA Nanoparticles

Porous collagen scaffold is integrated with surface activated PLLA nanoparticles fabricated by lyophilizing and crosslinking via EDC treatment. In order to prepare surface-modified PLLA nanoparticles, PLLA was firstly grafted with poly (acrylic acid) (PAA) through surface-initiated polymerization of acrylic acid. Nanoparticles of average diameter 316 nm and zeta potential −39.88 mV were obtained from the such-treated PLLA by dialysis method. Porous collagen scaffold were fabricated by mixing PLLA nanoparticles with collagen solution, freeze drying, and crosslinking with EDC. SEM observation revealed that nanoparticles were homogeneously dispersed in collagen matrix, forming interconnected porous structure with pore size ranging from 150 to 200 μm, irrespective of the amount of nanoparticles. The porosity of the scaffolds kept almost unchanged with the increment of the nanoparticles, whereas the mechanical property was obviously improved, and the degradation was effectively retarded. In vitro L929 mouse fibroblast cells seeding and culture studies revealed that cells infiltrated into the scaffolds and were distributed homogeneously. Compared with the pure collagen sponge, the number of cells in hybrid scaffolds greatly increased with the increment of incorporated nanoparticles. These results manifested that the surface-activated PLLA nanoparticles effectively reinforced the porous collagen scaffold and promoted the cells penetrating into the scaffold, and proliferation