Existence and non-existence results for the higher order Hardy-H\'enon equation revisited

TL;DR

This paper investigates the existence and non-existence of solutions to higher order Hardy-Hénon equations, establishing conditions under which solutions exist or do not, and characterizing their properties.

Contribution

It provides necessary and sufficient conditions for solutions to exist, including integral and super polyharmonic properties, and constructs solutions in certain parameter regimes.

Findings

Necessary condition for solution existence: n - 2m - (2m+σ)/(p-1) > 0

No non-trivial classical solutions for certain p when p < (n+2m+2σ)/(n-2m)

Existence of positive, radially symmetric classical solutions for p ≥ (n+2m+2σ)/(n-2m)

Abstract

This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions to the higher order Hardy-H\'enon equations \[ (-\Delta)^m u = |x|^\sigma u^p \] in $\mathbf R^n$ with $p > 1$. We show that the condition \[ n - 2m - \frac{2m+\sigma}{p-1} >0 \] is necessary for the existence of distributional solutions. For $n \geq 2m$ and $\sigma > -2m$, we prove that any distributional solution satisfies an integral equation and a weak super polyharmonic property. We establish some sufficient conditions for punctured or classical solution to be a distributional solution. As application, we show that if $n \geq 2m$ and $\sigma > -2m$, there is no non-negative, non-trivial, classical solution to the equation if \[ 1 < p < \frac{n+2m+2\sigma}{n-2m}. \] At last, we prove that for for $n > 2m$, $\sigma > -2m$ and $$p \geq \frac{n+2m+2\sigma}{n-2m},$$ there exist positive, radially symmetric, classical solutions to the equation.