On the Hang-Yang conjecture for GJMS equations on Sn\mathbb S^n

This work concerns a Liouville type result for positive, smooth solution $v$ to the following higher-order equation $${\mathbf P}^{2m}_n (v) = \frac{n-2m}2 Q_n^{2m} (\varepsilon v+v^{-\alpha} )$$ on $\mathbb S^n$ with $m \geq 2$, $3 \leq n < 2m$, $0<\alpha \leq (2m+n)/(2m-n)$, and $\varepsilon >0$. Here ${\mathbf P}^{2m}_n$ is the GJMS operator of order $2m$ on $\mathbb S^n$ and $Q_n^{2m} =(2/(n-2m)) {\mathbf P}^{2m}_n (1)$ is constant. We show that if $\varepsilon >0$ is small and $0<\alpha \leq (2m+n)/(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon =0$ and $0<\alpha < (2m+n)/(2m-n)$. In the special case $n=3$, $m=2$, and $\alpha=7$, such Liouville type result was recently conjectured by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020). As a by-product, we obtain the sharp (subcritical and critical) Sobolev inequalities $$\Big( \int_{\mathbb S^n} v^{1-\alpha} d\mu_{\mathbb S^n} \Big)^{\frac {2}{\alpha -1}} \int_{\mathbb S^n} v {\mathbf P}^{2m}_n (v) d\mu_{\mathbb S^n} \geq \frac{\Gamma (n/2 + m)}{\Gamma (n/2 - m )} | \mathbb S^n|^\frac{\alpha + 1}{\alpha - 1}$$ for the GJMS operator ${\mathbf P}^{2m}_n$ on $\mathbb S^n$ under the conditions $n \geq 3$, $n=2m-1$, and $\alpha \in(0,1) \cup (1, 2n+1]$. A log-Sobolev type inequality, as the limiting case $\alpha=1$, is also presented.Comment: 31 pages, 2 figure

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

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.Comment: 28 pages, 0 figur