Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian

This paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: {(−Δ)α2u(x)=(1|⋅|σ∗vp1)vp2(x),x∈Rn,(−Δ)α2v(x)=(1|⋅|σ∗uq1)uq2(x),x∈Rn,u(x)≥0,v(x)≥0,x∈Rn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$\end{document} where 0<α≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\alpha \leq 2$\end{document}, n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq 2$\end{document}, 0<σ<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\sigma <n$\end{document}, and 0<p1,q1≤2n−σn−α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$\end{document}, 0<p2,q2≤n+α−σn−α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$\end{document}. Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution (u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(u,v)$\end{document} in the critical case and nonexistence of positive solutions in the subcritical cases.

In [15], Chen, Li, and Li developed a direct method of moving planes for the fractional Laplacians (see also [22]). Instead of using the extension method of Caffarelli and Silvestre [5], they worked directly on the nonlocal operator to establish strong maximum principles for anti-symmetric functions and narrow region principles, and then obtained classification and Liouville type results for nonnegative solutions. The direct method of moving planes introduced in [15] has been applied to study more general nonlocal operators with general nonlinearities (see [14,22]). The method of moving planes was initially invented by Alexanderoff in the early 1950s. Later, it was further developed by Serrin [43], Gidas, Ni, and Nirenberg [28,30], Caffarelli, Gidas, and Spruck [4], Chen and Li [10], Li and Zhu [33], Lin [34], Chen, Li, and Ou [17], Chen, Li, and Li [15], and many others. For more literature works on the classification of solutions and Liouville type theorems for various PDE and IE problems via the methods of moving planes or spheres, please refer to [6,8,9,13,19,21,24,26,27,29,[35][36][37][38][39][40]45] and the references therein.
Chen, Li, and Zhang introduced in [19] another direct method i.e. the method of moving spheres on the fractional Laplacians, which is more convenient than the method of moving planes. The method of moving spheres was initially used by Padilla [42], Chen and Li [11], and Li and Zhu [33]. It can be applied to capture the explicit form of solutions directly rather than going through the procedure of deriving radial symmetry of solutions and then classifying radial solutions.
Our main theorem is the following complete classification theorem for PDE system (1.1).
The rest of our paper is organized as follows. In Sect. 2, we carry out our proof of Theorem 1.1. In the following, we use C to denote a general positive constant that may depend on n, α, p 1 , p 2 , q 1 , q 2 , σ , u, and v, and whose value may differ from line to line.

Proof of Theorem 1.1
In this section, we use a direct method of moving spheres for nonlocal nonlinearity with the help of the narrow region principle to classify the nonnegative solutions of PDE system (1.1).

The direct method of moving spheres for nonlocal nonlinearity
is a pair of nonnegative classical solutions of (1.1) which is not identically zero.
If there exists some point x 0 ∈ R n such that u(x 0 ) = 0, then we have On the other hand, we can deduce from system (1.1) that For any x ∈ R n and λ > 0, denote Then, since (u, v) is a pair of positive classical solutions of (1.1), one can verify that and a similar equation as u, v for any x ∈ R n and λ > 0. In fact, without loss of generality, we may assume x = 0 for simplicity and get, for 0 < α < 2 (α = 2 is similar), This means that the conformal transforms u x,λ (y), We will first show that there exists 0 > 0 (depending on x) sufficiently small such that, We first need to show that the nonnegative solution (u, v) to PDE system (1.1) also satisfies the equivalent integral system (1.4).
Conversely, assume that (u, v) is a pair of positive solutions of (1.1). For any R > 0, let where G α R is Green's function for (-) α 2 on B R (0) which is given by . Using the properties of Green's function, we can deduce for any R > 0, it follows from the maximum principle that Now, for each fixed y ∈ R n , letting R → ∞, we have On the other hand, (u 1 , v 1 ) is a pair of solutions of the following system: By the Liouville theorem, we deduce Thus, we have proved that Then we have from which we can infer immediately that C 3 = 0, C 4 = 0, therefore, we arrive at Therefore, (u, v) satisfies integral system (1.4).
Based on Lemma 2.1, we can prove that U x,λ , V x,λ have a strictly positive lower bound in a small neighborhood of x.

Lemma 2.2
For every fixed x ∈ R n , there exists η 0 > 0 (depending on x) sufficiently small such that, if 0 < λ ≤ η 0 , then Proof Using a similar argument as that in [19], one can denote For any |y| ≥ 1, since u, v > 0 also satisfy integral system (1.4), we can deduce that It follows immediately that for all y ∈ B λ 2 (x) \ {x}. Therefore, we have if 0 < λ ≤ η 0 for some η 0 (x) > 0 small enough, then This completes the proof of Lemma 2.2.
For every fixed x ∈ R n , define Now we need the following theorem, which is a variant (for nonlocal nonlinearity) of the narrow region principle (Theorem 2.2 in [19]).
The following lemma provides a starting point for us to move the spheres.