Some weighted fourth-order Hardy-Henon equations

TL;DR

This paper investigates sharp constants and optimizers in weighted Hardy-Henon inequalities, explicitly characterizes solutions to related fourth-order equations, and explores solution invariance and perturbation effects.

Contribution

It provides explicit forms of solutions to weighted fourth-order Hardy-Henon equations and reveals new solution invariances when the weight parameter is an even integer.

Findings

Explicit form of the unique radial positive solution U_{λ,α}.

Existence of new solutions related to linearized problems for even integer α.

Analysis of the remainder term and solutions to perturbed equations.

Abstract

By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*} \int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx \geq S^{rad}(N,\alpha)\left(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^{p^*_{\alpha}} dx\right)^{\frac{2}{p^*_{\alpha}}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<\alpha<2$, $p^*_{\alpha}=\frac{2(N-\alpha)}{N-4+\alpha}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{\lambda,\alpha}$ to the weighted fourth-order Hardy (for $\alpha>0$) or H\'{e}non (for $\alpha<0$) equation: \begin{equation*} \Delta(|x|^{\alpha}\Delta u)=|x|^{-\alpha} u^{p^*_{\alpha}-1},\quad u>0 \quad \mbox{in}\quad \mathbb{R}^N. \end{equation*} %Furthermore, we characterize all the solutions to the linearized problem related to above equation at $U_{1,\alpha}$. For $\alpha\neq 0$, it is known the solutions of above equation are invariant for dilations $\lambda^{\frac{N-4+\alpha}{2}}u(\lambda x)$ but not for translations. However we show that if $\alpha$ is an even integer, there exist new solutions to the linearized problem, which related to above equation at $U_{1,\alpha}$, that "replace" the ones due to the translations invariance. This interesting phenomenon was first shown by Gladiali, Grossi and Neves [Adv. Math. 249, 2013, 1-36] for the second-order H\'{e}non problem. Finally, as applications, we investigate the reminder term of above inequality and also the existence of solutions to some related perturbed equations.