拟线性Hardy--Emden--Fowler方程的径向解：精确衰减率、Delaunay解与不存在性
Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence
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中文总结 AI 辅助
本文研究拟线性Hardy--Emden--Fowler方程径向正解在无穷远处的行为，通过斜率变量将方程化为平面系统，完整刻画了四种参数区间下解的精确衰减率、Delaunay周期解的存在性及无解条件。
中文摘要 AI 辅助
设$N>p>1$，$s>-p$，$0<\mu<\bar\mu:=\big(\frac{N-p}{p}\big)^{p}$且$q>p-1$。我们确定了拟线性Hardy--Emden--Fowler方程\\[ -\big(r^{N-1}|v'|^{p-2}v'\big)'=\mu\\,r^{N-1-p}v^{p-1}+r^{N-1+s}v^{q}, \qquad r>R_{0} \\]的\emph{每一个}正解在无穷远处的行为。三种速率相互竞争：自相似速率$r^{-\gamma_{*}}$（其中$\gamma_{*}=\frac{p+s}{q-p+1}$），以及两种Hardy速率$r^{-\nu_{\pm}}$，其中$\nu_{\pm}$是$h(\nu):=\nu^{p-1}(N-p-(p-1)\nu)=\mu$的根。记$p^{*}_{s}=\frac{p(N+s)}{N-p}$和$q_{S}=p-1+\frac{p+s}{\nu_{+}}$分别为Hardy--Sobolev指数和Serrin指数，我们在四种情形下得到了完整图景。若$q>p^{*}_{s}-1$，每个解恰好实现三种速率之一，在临界情形$h(\gamma_{*})=\mu$时有对数修正，并且我们计算了前导常数。若$q=p^{*}_{s}-1$，系统变为可积的：除了奇异自相似解和异宿“气泡”之外，还存在一族Fowler--Delaunay解，对于这些解，$r^{\gamma_{*}}v(r)$是$\ln r$的非恒定周期函数。若$q_{S}<q<p^{*}_{s}-1$，仅有两个轨道幸存；若$q\le q_{S}$，在外区域中不存在任何正解。这是Franca和Sfecci在$p=2$情形所获图景的拟线性、带权对应版本。我们使用斜率变量$x=-rv'/v$，$Y=r^{p+s}v^{q-p+1}$，在这些变量下方程变为平面系统，该系统在$1<p<N$时对每个半平面$\{x>0\}$都是实解析的，因此不需要$p\le2$之类的限制；分析依赖于一个普适障碍$x<\nu_{+}$、一个显式Dulac函数（该函数在$q=p^{*}_{s}-1$时退化为首次积分）以及一个显式Lyapunov函数。
英文摘要
Let $N>p>1$, $s>-p$, $0<μ<\barμ:=\big(\frac{N-p}{p}\big)^{p}$ and $q>p-1$. We determine the behaviour at infinity of \emph{every} positive solution of the quasilinear Hardy--Emden--Fowler equation \[ -\big(r^{N-1}|v'|^{p-2}v'\big)'=μ\,r^{N-1-p}v^{p-1}+r^{N-1+s}v^{q}, \qquad r>R_{0}. \] Three rates compete: the self-similar rate $r^{-γ_{*}}$ with $γ_{*}=\frac{p+s}{q-p+1}$, and the two Hardy rates $r^{-ν_{\pm}}$, where $ν_{\pm}$ are the roots of $h(ν):=ν^{p-1}(N-p-(p-1)ν)=μ$. Writing $p^{*}_{s}=\frac{p(N+s)}{N-p}$ and $q_{S}=p-1+\frac{p+s}{ν_{+}}$ for the Hardy--Sobolev and Serrin exponents, we obtain a complete picture in four regimes. If $q>p^{*}_{s}-1$, every solution realises exactly one of the three rates, with a logarithmic correction in the borderline case $h(γ_{*})=μ$, and we compute the leading constants. If $q=p^{*}_{s}-1$ the system becomes integrable: besides the singular self-similar solution and a heteroclinic ``bubble'' there is a one-parameter family of Fowler--Delaunay solutions, for which $r^{γ_{*}}v(r)$ is a non-constant periodic function of $\ln r$. If $q_{S}<q<p^{*}_{s}-1$ only two orbits survive, and if $q\le q_{S}$ there is no positive solution at all in an exterior domain. This is the quasilinear, weighted counterpart of the picture obtained by Franca and Sfecci for $p=2$. We work with the slope variables $x=-rv'/v$, $Y=r^{p+s}v^{q-p+1}$, in which the equation becomes a planar system that is real-analytic on the half-plane $\{x>0\}$ for every $1<p<N$, so that no restriction of the type $p\le2$ is needed; the analysis then rests on a universal barrier $x<ν_{+}$, on an explicit Dulac function which degenerates into a first integral exactly at $q=p^{*}_{s}-1$, and on an explicit Lyapunov function.
发表机构
- Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam（经济数学学院，经济与法律大学）
- Vietnam National University, Ho Chi Minh City, Vietnam（越南国立大学）
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