分数阶Choquard方程变号解的存在性
Existence of Sign-Changing Solutions for a Fractional Choquard Equation
DOI: 10.12677/AAM.2022.117437, PDF, 下载: 56  浏览: 106 
作者: 高金华:云南师范大学数学学院,云南 昆明
关键词: 分数阶拉普拉斯算子变号解Choquard方程;Fractional Laplacian Sign-Changing Solutions Choquard Equation
摘要:

分数阶 Choquard 方程具有重要的物理背景,是近年非线性分析领域广受关注的问题之一。 在本文中,我们研究如下的分数阶 Choquard 方程

(−∆)su + V (x)u = (|x|−µ ∗ |u|p)|u|p−2u, x ∈ RN , (P)

其中 s ∈ (0, 1),N ≥ 3,µ ∈ (0, N ),,“∗” 代表卷积算子,(−∆)s 是分数阶拉普拉斯算子。 通过结合 Ekeland 变分原理和隐函数定理,我们证明了 (P ) 存在极小能量变号解 w。 此外,我们还证明了 w 的能量严格大于基态能量,但严格小于基态能量的两倍。

Abstract:

With an important physical background, the fractional Choquard equation has at- tracted great attention from the field of nonlinear analysis in recent years. In this paper, we study the following fractional Choquard equation

(−∆)su + V (x)u = (|x|−µ ∗ |u|p)|u|p−2u, x ∈ RN , (P)

where s ∈ (0, 1), N ≥ 3, µ ∈ (0, N ), , “∗” stands for the convolution and (−∆)s is the fractional Laplacian operator. By combining the Ekeland variational principle with the implicit function theorem, we prove that the problem (P ) possesses one least energy sign-changing solution w. Moreover, we show that the energy of w is strictly larger than the ground state energy and less than twice the ground state energy.

文章引用:高金华. 分数阶Choquard方程变号解的存在性[J]. 应用数学进展, 2022, 11(7): 4089-4109. https://doi.org/10.12677/AAM.2022.117437

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