Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence
Abstract
Let N>p>1, s>−p, 0<μ<μ̅:=((N−p)/p)p and q>p−1. We determine the behaviour at infinity of \emph{every} positive solution of the quasilinear Hardy--Emden--Fowler equation −(rN−1|v'|p−2v')'=μ rN−1−pvp−1+rN−1+svq, r>R0. Three rates compete: the self-similar rate r−γ* with γ*=(p+s)/(q−p+1), and the two Hardy rates r−ν±, where ν± are the roots of h(ν):=νp−1(N−p−(p−1)ν)=μ. Writing p*s=(p(N+s))/(N−p) and qS=p−1+(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 rγ*v(r) is a non-constant periodic function of ln r. If qS<q<p*s−1 only two orbits survive, and if q≤ qS 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=rp+svq−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≤2 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.
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BibTeX
@misc{le2026radial,
title = {{Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09724},
eprint = {2609.09724},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09724},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence
AU - Phuong Le
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09724
UR - https://arxiv.org/abs/2609.09724
AB - Let N>p>1, s>−p, 0<μ<μ̅:=((N−p)/p)^p and q>p−1. We determine the behaviour at infinity of \emph{every} positive solution of the quasilinear Hardy--Emden--Fowler equation −(r^N−1|v'|^p−2v')'=μ r^N−1−pv^p−1+r^N−1+sv^q, r>R_0. Three rates compete: the self-similar rate r^−γ_* with γ_*=(p+s)/(q−p+1), and the two Hardy rates r^−ν_±, where ν_± are the roots of h(ν):=ν^p−1(N−p−(p−1)ν)=μ. Writing p^*_s=(p(N+s))/(N−p) and q_S=p−1+(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≤ 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+sv^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≤2 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.
ER -