The proof of the Lane–Emden conjecture in four space dimensions
The proof of the Lane–Emden conjecture in four space dimensions
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DOI：
10.1016/j.aim.2009.02.014
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发表时间：
2009-08
影响因子：
1.7
通讯作者：
P. Souplet
中科院分区：
文献类型：
--
作者：
P. Souplet
我们部分解决了一个著名的猜想的不存在正的整体解的Lane-Emden型椭圆方程组的指数对低于临界Sobolev双曲线。到目前为止，这个猜想已经被证明是径向解，或在n × 3空间维，或在临界双曲线以下的某些子区域（n × 4）。我们在这里建立了四维空间中的猜想，并得到了一个新的不存在的区域，n = 5。我们的证明是基于一个微妙的组合，包括Rellich-Pohozaev型恒等式，通过最大值原理的分量之间的比较性质，Sn−1上的Sobolev和插值不等式，以及反馈和测量参数。这样的Liouville型不存在性结果在非变分椭圆型方程组的研究中有许多应用。
We partially solve a well-known conjecture about the nonexistence of positive entire solutions to elliptic systems of Lane–Emden type when the pair of exponents lies below the critical Sobolev hyperbola. Up to now, the conjecture had been proved for radial solutions, or in n⩽3 space dimensions, or in certain subregions below the critical hyperbola for n⩾4. We here establish the conjecture in four space dimensions and we obtain a new region of nonexistence for n⩾5. Our proof is based on a delicate combination involving Rellich–Pohozaev type identities, a comparison property between components via the maximum principle, Sobolev and interpolation inequalities on Sn−1, and feedback and measure arguments. Such Liouville-type nonexistence results have many applications in the study of nonvariational elliptic systems.