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Symmetry and nonexistence of positive solutions for an elliptic system involving the Fractional Laplacian
Quaestiones Mathematicae ( IF 0.7 ) Pub Date : 2021-01-19 , DOI: 10.2989/16073606.2020.1854363
Rong Zhang 1 , Xiaoshan Wang 1 , ZuoDong Yang 2
Affiliation  

Abstract

In this paper, we study a nonlinear elliptic system involving a fractional Laplacion:

where 0 < α < 2, p, q > 0 and max{p, q} 1, τ ≥ 0, n ≥ 2. There are two cases to be considered. The first one is where the domain is bounded, and the second one is where the domain is the whole space. First of all, we consider the above system in the star-shaped and bounded domain Ω, by using the Pohozaev identity. We prove the nonexistence of a positive solution in the critical and supercritical case p + q + 1 ≥ . In addition, we show that the positive solutions of the above system are radially symmetric and decreasing about the origin by using the method of moving planes in ℝn. Moreover, while in the subcritical case , we prove the nonexistence of a positive solution for the above system in ℝn. Then, through the doubling lemma we obtain the singularity estimates of the positive solutions on a bounded domain .



中文翻译:

涉及分数拉普拉斯算子的椭圆系统的对称性和正解的不存在性

摘要

在本文中,我们研究了一个包含分数 Laplacion 的非线性椭圆系统:

其中 0 < α < 2 , p, q > 0max{ p, q } 1 , τ ≥ 0 , n ≥ 2。有两种情况需要考虑。第一个是域有界的地方,第二个是域是整个空间的地方。首先,我们使用Pohozaev 恒等式在星形和有界域 Ω 中考虑上述系统。我们证明了在临界和超临界情况p + q + 1 ≥中不存在正解。此外,我们证明了上述系统的正解是径向对称的,并且通过使用在 ℝ n中移动平面此外,虽然在亚临界情况下,我们证明了上述系统在 ℝ n中不存在正解。然后,通过加倍引理,我们获得了有界域上正解的奇异性估计

更新日期:2021-01-19
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