分数阶Choquard方程变号解的存在性  

Existence of Sign-Changing Solutions for a Fractional Choquard Equation

作　　者：高金华 

机构地区：[1]云南师范大学数学学院,云南 昆明

出　　处：《应用数学进展》2022年第7期4089-4109,共21页Advances in Applied Mathematics

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。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.

关 键 词：隐函数定理 基态能量 物理背景 卷积算子 变分原理 分数阶 非线性分析 

分 类 号：O175[理学—数学]