论文标题
极端尖锐的庞加莱 - 苏布尔洛尔常数的独特性
Uniqueness of extremals for some sharp Poincaré-Sobolev constants
论文作者
论文摘要
我们研究了$ w^{1，p} _0（ω）$嵌入$ l^q（ω）$的尖锐常数，在$ 2 <p <q $的情况下。我们证明，对于平稳的连接集，当$ q> p $和$ q $足够接近$ p $时，达到尖锐常数的极端功能是独一无二的，可以达到乘法常数。这反过来赋予了循环方程的最小能量的解决方案的独特性，并具有超均匀的右侧。 通过适当调整C.-S的线性化参数来实现该结果。林。我们依靠$ p- $ laplace-type方程的解决方案的一些良好估计。
We study the sharp constant for the embedding of $W^{1,p}_0(Ω)$ into $L^q(Ω)$, in the case $2<p<q$. We prove that for smooth connected sets, when $q>p$ and $q$ is sufficiently close to $p$, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace--type equations by L. Damascelli and B. Sciunzi.
