## Displaying similar documents to “The product of distributions on ${R}^{m}$”

### Sharp estimates for bubbling solutions of a fourth order mean field equation

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a sequence of multi-bubble solutions ${u}_{k}$ of the following fourth order equation $\phantom{\rule{2.0em}{0ex}}\phantom{\rule{2.0em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\Delta }^{2}{u}_{k}={\rho }_{k}\frac{h\left(x\right){e}^{{u}_{k}}}{{\int }_{\Omega }h{e}^{{u}_{k}}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{u}_{k}=\Delta {u}_{k}=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega ,\phantom{\rule{2.0em}{0ex}}\phantom{\rule{2.0em}{0ex}}\phantom{\rule{2.0em}{0ex}}\left(*\right)$ where $h$ is a ${C}^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^{4}$, and ${\rho }_{k}$ is a constant such that ${\rho }_{k}\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_{k}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}32{\sigma }_{3}m$ for some positive integer $m\phantom{\rule{-0.166667em}{0ex}}\ge \phantom{\rule{-0.166667em}{0ex}}1$, where ${\sigma }_{3}$ is the area of the unit sphere in ${ℝ}^{4}$. Furthermore, we obtain the following sharp estimates for ${\rho }_{k}$: $\begin{array}{cc}\hfill {\rho }_{k}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}32{\sigma }_{3}m\phantom{\rule{-0.166667em}{0ex}}& =\phantom{\rule{-0.166667em}{0ex}}{c}_{0}\sum _{j=1}^{m}\phantom{\rule{-0.166667em}{0ex}}{ϵ}_{k,j}^{2}\phantom{\rule{-0.166667em}{0ex}}\left(\sum _{l\ne j}\Delta {G}_{4}\left({p}_{j},\phantom{\rule{-0.166667em}{0ex}}{p}_{l}\right)\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}\Delta {R}_{4}\left({p}_{j},\phantom{\rule{-0.166667em}{0ex}}{p}_{j}\right)\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}\frac{1}{32{\sigma }_{3}}\Delta logh\left({p}_{j}\right)\phantom{\rule{-0.166667em}{0ex}}\right)\phantom{\rule{-2.0pt}{0ex}}\hfill \\ & \phantom{\rule{1.0em}{0ex}}+o\left(\sum _{j=1}^{m}{ϵ}_{k,j}^{2}\right)\hfill \end{array}$ where ${c}_{0}\phantom{\rule{-0.166667em}{0ex}}>\phantom{\rule{-0.166667em}{0ex}}0$, $log\frac{64}{{ϵ}_{k,j}^{4}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\underset{x\in {B}_{\delta }\left({p}_{j}\right)}{max}\phantom{\rule{-0.166667em}{0ex}}{u}_{k}\left(x\right)\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}log\left(\underset{\Omega }{\int }h{e}^{{u}_{k}}\right)$ and ${u}_{k}\phantom{\rule{-0.166667em}{0ex}}\to \phantom{\rule{-0.166667em}{0ex}}32{\sigma }_{3}\sum _{j=1}^{m}{G}_{4}\left(·,{p}_{j}\right)$ in ${C}_{\mathrm{loc}}^{4}\left(\Omega \setminus \left\{{p}_{1},...,{p}_{m}\right\}\right)$. This yields a bound of solutions as...

### Some non-multiplicative properties are $l$-invariant

Commentationes Mathematicae Universitatis Carolinae

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A cardinal function $\varphi$ (or a property $𝒫$) is called $l$-invariant if for any Tychonoff spaces $X$ and $Y$ with ${C}_{p}\left(X\right)$ and ${C}_{p}\left(Y\right)$ linearly homeomorphic we have $\varphi \left(X\right)=\varphi \left(Y\right)$ (or the space $X$ has $𝒫$ ($\equiv X⊢𝒫$) iff $Y⊢𝒫$). We prove that the hereditary Lindelöf number is $l$-invariant as well as that there are models of $ZFC$ in which hereditary separability is $l$-invariant.

### ${F}_{\sigma }$-absorbing sequences in hyperspaces of subcontinua

Commentationes Mathematicae Universitatis Carolinae

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Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟\left({ℝ}^{n}\right)=n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C\left(X\right)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence ${\left({𝒟}_{\ge n}\left(C\left(X\right)\right)\right)}_{n=2}^{\infty }$ is ${F}_{\sigma }$-absorbing in $C\left(X\right)$. As a consequence, there is a homeomorphism $h:C\left(X\right)\to {Q}^{\infty }$ such that for all $n$, $h\left[\left\{A\in C\left(X\right):𝒟\left(A\right)\ge n+1\right\}\right]={B}^{n}×Q×Q×\cdots$, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate...

### On remote points, non-normality and $\pi$-weight ${\omega }_{1}$

Commentationes Mathematicae Universitatis Carolinae

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We show, in particular, that every remote point of $X$ is a nonnormality point of $\beta X$ if $X$ is a locally compact Lindelöf separable space without isolated points and $\pi w\left(X\right)\le {\omega }_{1}$.