Existence and non-existence results for the higher order Hardy-Hénon equation revisited

Abstract

This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions to the higher order Hardy-Hénon equations \[ (-Δ)^m u = |x|^σu^p \] in ${\mathbf{R}}^n$ with $p>1$. We show that the condition \[ n - 2m - \frac{2m+σ}{p-1} >0 \] is necessary for the existence of distributional solutions. For $n\ge 2m$ 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 $n\ge 2m$ and $\sigma >-2m$, there is no non-negative, non-trivial, classical solution to the equation if \[ 1 < p < \frac{n+2m+2σ}{n-2m}. \] At last, we prove that for for $n>2m$, $\sigma >-2m$ and $$p\ge \frac{n+2m+2\sigma }{n-2m},$$ there exist positive, radially symmetric, classical solutions to the equation.

28 pages, 0 figure

References in corpus (1)

• Singular solutions for the constant $Q$-curvature problem

Cited by in corpus (4)

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• A Liouville type result for fractional GJMS equations on higher dimensional spheres
• On positivity of the Q-curvatures of conformal metrics
• Compactness of singular solutions to the sixth order GJMS equation