分数阶Choquard方程变号解的存在性

Existence of Sign-Changing Solutions for a Fractional 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 ∈ R^N , (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 ∈ R^N , (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.