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arXiv 0812.0928math.AP
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各向异性Sobolev不等式的极值函数

Extremal functions for the anisotropic Sobolev inequalities

Abdallah El Hamidi, J. M. Rakotoson

发表信息：
Annales de l'Institut Henri Poincaré Analyse non linéaire 24, 5 (2006) 741-756
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摘要：

多个非负解的存在性各向异性临界问题- \sum_{i=1}^{N} \frac{\partial}{\partial x_i}（|\frac{\partial u}{\partial x_i}|在适当的各向异性索伯列夫空间中证明了（^{p_i-2} \frac{\partial u}{\partial x_i}）= |u| ^{p^*-2} u {in} \mathbb{R}^N.解对应于某个最佳Sobolev常数的极值函数。在我们的研究中的主要工具是著名的P. -的集中紧性引理的改编。L.狮子到各向异性算子。此外，我们证明了一组非平凡的解决方案$\calS$包括在$L^\infty(\R^N)$和位于一个球的半径$τ>0$在$L^{p^*}(\R^N)$。

英文摘要：

The existence of multiple nonnegative solutions to the anisotropic critical problem - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} (| \frac{\partial u}{\partial x_i} |^{p_i-2} \frac{\partial u}{\partial x_i}) = |u|^{p^*-2} u {in} \mathbb{R}^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions $\calS$ is included in $L^\infty(\R^N)$ and is located outside of a ball of radius $τ>0$ in $L^{p^*}(\R^N)$.

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