On finite energy solutions of fractional order equations of the Choquard type
On finite energy solutions of fractional order equations of the Choquard type
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DOI：
10.3934/dcds.2019064
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Y. Lei
中科院分区：
文献类型：
--
作者：
Y. Lei
有限能量解是Choquard方程的一类重要的解。本文研究有限能量弱解的正则性。对于非局部分数阶方程，包含Riesz势和Bessel势的积分系统起着关键作用。将正则性提升引理应用于该积分系统，可以发现某些弱可积解具有较好的正则性。此外，我们还证明了这种可积解与有限能量解之间的关系。在此基础上，我们证明了弱有限能量解在一定条件下也是经典解。最后指出，虽然找不到基态解，但具有临界指数的最小能量可以用Sobolev型不等式的锐常数来表示。
Finite energy solutions are the important class of solutions of the Choquard equation. This paper is concerned with the regularity of weak finite energy solutions. For nonlocal fractional-order equations, an integral system involving the Riesz potential and the Bessel potential plays a key role. Applying the regularity lifting lemma to this integral system, we can see that some weak integrable solution has the better regularity properties. In addition, we also show the relation between such an integrable solution and the finite energy solution. Based on these results, we prove that the weak finite energy solution is also the classical solution under some conditions. Finally, we point out that the least energy with the critical exponent can be represented by the sharp constant of some inequality of Sobolev type though the ground state solution cannot be found.