Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence
Year: 2026
Type: preprint
Abstract: 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 $q_{S}0\}$ for every $1
Source: arXiv (Cornell University)
Author: Le, Phuong
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212178037