论文标题
一些加权的四阶强壮亨利方程
Some weighted fourth-order Hardy-Henon equations
论文作者
论文摘要
通过使用与Sobolev不等式相关的合适转换，我们研究了径向空间中的尖锐常数和优化器，以进行以下加权的Caffarelli-Kohn-nirenberg-type不等式：\ begin {equation*} \ int _ {\ int _ {\ mathbB { s^{rad}（n，α）\ left（\ int _ {\ mathbb {r}^n} | x | x |^{ - α} | u |^{p^*_α} dx \ right） c^\ infty_c（\ mathbb {r}^n），\ end {equation*}其中$ n \ geq 3 $，$ 4-n <α<2 $，$ p^*_α= \ frac {2（n-α）} {n-4-4+α} $。然后，我们将唯一的（缩放）径向阳性解决方案的明确形式$ u_ {λ，α} $向加权的四阶强（对于$α> 0 $）或hénon（$α<0 $）等式：\ begin {equation*} δ（| x |^αΔU）= | x |^{ - α} u^{p^*_α-1}，\ quad u> 0 \ quad \ quad \ mbox {in} \ quad \ quad \ quad \ mathbb {r}^n。 \ end {equation*}％此外，我们表征了与上述方程相关的所有解决方案，$ u_ {1，α} $。对于$α\ neq 0 $，已知上述方程的解决方案对于扩张$λ^{\ frac {n-4+α} {2}} {2}} u（λx）$不变。但是，我们表明，如果$α$是一个均匀的整数，则有针对线性问题的新解决方案，该解决方案与上述方程式在$ u_ {1，α} $上相关，它由于翻译不变性而“替换”了这些问题。这种有趣的现象首先由Gladiali，Grossi和Neves [Adv。数学。 249，2013，1-36]对于二阶Hénon问题。最后，作为应用程序，我们研究了上述不平等的提醒项，以及某些相关扰动方程的解决方案的存在。
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|^α|Δu|^2 dx \geq S^{rad}(N,α)\left(\int_{\mathbb{R}^N}|x|^{-α}|u|^{p^*_α} dx\right)^{\frac{2}{p^*_α}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<α<2$, $p^*_α=\frac{2(N-α)}{N-4+α}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{λ,α}$ to the weighted fourth-order Hardy (for $α>0$) or Hénon (for $α<0$) equation: \begin{equation*} Δ(|x|^αΔu)=|x|^{-α} u^{p^*_α-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 $U_{1,α}$. For $α\neq 0$, it is known the solutions of above equation are invariant for dilations $λ^{\frac{N-4+α}{2}}u(λx)$ but not for translations. However we show that if $α$ is an even integer, there exist new solutions to the linearized problem, which related to above equation at $U_{1,α}$, 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 Hénon problem. Finally, as applications, we investigate the reminder term of above inequality and also the existence of solutions to some related perturbed equations.
