Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence

Abstract

Let $N>p>1$, $s>-p$, $0<\mu <\bar{\mu }:=(\frac{N-p}{p}{)}^p$ and $q>p-1$. We determine the behaviour at infinity of \emph{every} positive solution of the quasilinear Hardy--Emden--Fowler equation \[ -\big(r^{N-1}|v'|^{p-2}v'\big)'=μ\,r^{N-1-p}v^{p-1}+r^{N-1+s}v^{q}, \qquad r>R_{0}. \] Three rates compete: the self-similar rate $r^{-{\gamma }_{\ast }}$ with ${\gamma }_{\ast }=\frac{p+s}{q-p+1}$, and the two Hardy rates $r^{-{\nu }_{\pm }}$, where ${\nu }_{\pm }$ are the roots of $h(\nu ):={\nu }^{p-1}(N-p-(p-1)\nu )=\mu$. Writing $p_s^{\ast }=\frac{p(N+s)}{N-p}$ and $q_S=p-1+\frac{p+s}{{\nu }_{+}}$ for the Hardy--Sobolev and Serrin exponents, we obtain a complete picture in four regimes. If $q>p_s^{\ast }-1$, every solution realises exactly one of the three rates, with a logarithmic correction in the borderline case $h({\gamma }_{\ast })=\mu$, and we compute the leading constants. If $q=p_s^{\ast }-1$ 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 $r^{{\gamma }_{\ast }}v(r)$ is a non-constant periodic function of $\ln r$. If $q_S<q<p_s^{\ast }-1$ 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 $x=-r{v}^{\prime }/v$, $Y=r^{p+s}{v}^{q-p+1}$, 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 $p\le 2$ is needed; the analysis then rests on a universal barrier $x<{\nu }_{+}$, on an explicit Dulac function which degenerates into a first integral exactly at $q=p_s^{\ast }-1$, and on an explicit Lyapunov function.

23 pages