Classification of nonnegative solutions to fractional Schrödinger-Hatree-Maxwell type system

1.   Introduction

In this paper, we consider the following fractional Schrödinger-Hatree-Maxwell type system

where $0 < \alpha\leq2$, $n\geq2$, $0 < \sigma_1, \sigma_2 < n$, $f_i$, $g_i$ ($i = 1, 2$) are strictly increasing on $[0, +\infty)$, and $f_i(t)\cdot t^{-p_{i}}$, $g_i(t)\cdot t^{-q_{i}}$ ($i = 1, 2$) are nonincreasing on $(0, +\infty)$.

We assume $u, v\in C^{1, 1}_{loc}\cap\mathcal{L}_{\alpha}(\mathbb{R}^{n})$ if $0 < \alpha < 2$, and $u, v\in C^{2}(\mathbb{R}^{n})$ if $\alpha = 2$, where

The nonlocal fractional Laplacians $(-\Delta)^{\frac{\alpha}{2}}$ with $0 < \alpha < 2$ are defined by ^{[10,16,20,51,54]}

for functions $u, v\in C^{1, 1}_{loc}\cap\mathcal{L}_{\alpha}(\mathbb{R}^{n})$, where $C_{\alpha, n} = \big(\int_{\mathbb{R}^{n}}\frac{1-\cos(2\pi\zeta_{1})}{|\zeta|^{n+\alpha}}d\zeta\big)^{-1}$ is the normalization constant. The fractional Laplacians $(-\Delta)^{\frac{\alpha}{2}}$ can also be defined equivalently ^[17] by Caffarelli and Silvestre's extension method ^[6] for $u, v\in C^{1, 1}_{loc}\cap\mathcal{L}_{\alpha}(\mathbb{R}^{n})$.

We should mention that the fractional Laplacian is different from the fractional differentiation. Both are defined through a singular convolution integral, but the former is guaranteed to be the positive definition via the Riesz potential as the standard Laplace operator, while the latter via the Riemann-Liouville integral is not. It is noted that the fractional Laplacian can not be interpreted by the fractional differentiation in the sense of either Riemann-Liouville or Caputo. Both the fractional Laplacian and the fractional differentiation have found applications in many complicated engineering problems. In particular, the fractional Laplacian attracts new attentions in recent years owing to its unique capability describing anomalous diffusion problems ^[34].

The fractional Laplacian can be regarded as an infinitesimal generator of a stable Lévy process, which has many applications in probability, optimization and finance ^[1,4]. It is also widely used to simulate various physical phenomena, such as anomalous diffusion and quasi-geostrophic flows, turbulence and water waves, molecular dynamics and relativistic quantum mechanics of stars (see ^[2,22,23,36] and the references therein). But, it is difficult to study the fractional Laplacians because of its non-local feature. In order to overcome this difficulty, Chen, Li and Ou ^[18] put forward the method of moving planes in integral forms. Subsequently, Caffarelli and Silvestre ^[6] introduced an extension method to conquer this difficulty, which transformed this nonlocal problem into a local one in higher dimensions. This extension method provides a powerful tool and leads to a very active study in equations containing the fractional Laplacians, and obtains a series of fruitful results (see ^[3,21] and the references therein).

Chen, Li and Li developed a direct method of moving planes for the fractional Laplacians in ^[16,25]. Rather than use the extension method of Caffarelli and Silvestre ^[6], they worked directly on the non-local operator to establish strong maximum principles for anti-symmetric functions and narrow region principles, and then for nonnegative solutions they obtained classification and Liouville type results. The direct method of moving planes introduced in ^[16] has been used to study more general nonlocal operators with general nonlinearities ^[15,25]. In the early 1950s, the method of moving planes was originally invented by Alexanderoff. Later, it was further developed by Serrin ^[51], Gidas, Ni and Nirenberg ^[33], Caffarelli, Gidas and Spruck ^[5], Chen and Li ^[11], Li and Zhu ^[40], Lin ^[43], Chen, Li and Ou ^[18], Chen, Li and Li ^[16], and so on. For more literatures on the classification of solutions and Liouville type theorems for various PDE and IE problems through the methods of moving planes or spheres, please refer to ^{[7,9,10,14,20,24,27,29,30,44,45,46,47,52]} and the references therein.

In ^[20], Chen, Li and Zhang introduced another direct method—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 ^[49], Chen and Li ^[12] and Li and Zhu ^[40]. It can be applied to obtain the explicit form of solutions directly instead of deriving radial symmetry of solutions and then classifying radial solutions.

There are many literatures on the qualitative properties of solutions to Hartree and Choquard equations of fractional or higher order, please see e.g., Cao and Dai ^[7], Chen and Li ^[13], Dai, Fang, et al., ^[24], Dai and Qin ^[29], Dai and Liu ^[26], Lei ^[38], Liu ^[44], Le ^[37], Ma and Zhao ^[48], Xu and Lei ^[53] and the references therein.

System (1.1) is closely related to the following integral system

where the Riesz potential's constants $R_{\alpha, n}: = \frac{\Gamma\big(\frac{n-\alpha}{2}\big)}{\pi^{\frac{n}{2}}2^{\alpha}\Gamma(\frac{\alpha}{2})}$ ^[50].

In the special case $f_i(t) = t^{p_{i}}$, $g_i(t) = t^{q_{i}}$ ($i = 1, 2$), system (1.4) turns into the following Schrödinger-Hatree-Maxwell type system

When $p_{1} = q_{1} = 2$ and $p_{2} = q_{2} = 1$, PDEs of type (1.1) arise in the Hartree-Fock theory of the nonlinear Schrödinger equations^[41]. When $f_{1}(v(x)) = v^{2}(x)$, $f_{2}(v(x)) = v(x)$, $g_{1}(u(x)) = u^{2}(x)$, $g_{2}(u(x)) = u(x)$, and $\sigma_{1} = \sigma_{2} = 2\alpha$, the solution $(u, v)$ to problem (1.1) is also a ground state or a stationary solution to the following $\dot{H}^{\frac{\alpha}{2}}$-critical focusing dynamic Schrödinger-Hartree system

In the special case $\alpha = 2$, the above Schrödinger-Hartree equations have many interesting applications in the quantum theory of large systems of non-relativistic bosonic atoms and molecules ^[32].

When $\sigma_1, \sigma_2 = 2\alpha$, $\alpha\in(0, \frac{n}{2})$, $p_{1} = q_{1} = 2$, $p_{2} = q_{2} = 1$, Dai, Fang, et al., ^[24] classified all the positive $H^{\frac{\alpha}{2}}(\mathbb{R}^{n})$ weak solutions to (1.5) by using the method of moving planes in integral forms for the equivalent integral equation system (1.4) due to Chen, Li and Ou ^[18,19]. They also classified all the $L^{\frac{2n}{n-\alpha}}(\mathbb{R}^{n})$ integrable solutions to the equivalent integral equations. For $0 < \alpha < \min\{2, \frac{n}{2}\}$, Dai, Fang and Qin ^[25] classified all the $C^{1, 1}_{loc}\cap\mathcal{L}_{\alpha}$ solutions to (1.5) with $\sigma_1, \sigma_2 = 2\alpha$, $p_{1} = q_{1} = 2$, $p_{2} = q_{2} = 1$ by applying a variant (for nonlocal nonlinearity) of the direct method of moving planes for fractional Laplacians. The qualitative properties of solutions to general fractional order or higher order elliptic equations have also been extensively studied, for instance, see Chen, Fang and Yang ^[10], Chen, Li and Li ^[16], Chen, Li and Ou ^[18], Caffarelli and Silvestre ^[6], Chang and Yang ^[9], Dai and Qin ^[29], Cao, Dai and Qin ^[8], Dai, Liu and Qin ^[28], Fang and Chen ^[30], Lin ^[43], Wei and Xu ^[52] and the references therein.

Our main theorem is the following classification theorem for PDEs system (1.1).

Theorem 1.1. Let $(u, v)\in C^0(\mathbb{R}^n)\times C^0(\mathbb{R}^n)$ be a pair of nonnegative solution to the problem (1.4). Assume that $f_1, f_2, g_1, g_2:\mathbb [0, +\infty)\rightarrow \mathbb R_{+}$ satisfy the following conditions:

(i) $f_1(t)$, $f_2(t)$, $g_1(t)$ and $g_2(t)$ are strictly increasing on $[0, +\infty)$;

(ii) $F_1(t) = f_1(t)\cdot t^{-p_1}$, $F_2(t) = f_2(t)\cdot t^{-p_2}$, $G_1(t) = g_1(t)\cdot t^{-q_1}$ and $G_2(t) = g_2(t)\cdot t^{-q_2}$ ($p_1 = \frac{{2n}-\sigma_1}{n-\alpha}$, $p_2 = \frac{n+\alpha-\sigma_1}{n-\alpha}$, $q_1 = \frac{{2n}-\sigma_2}{n-\alpha}$, $q_2 = \frac{n+\alpha-\sigma_2}{n-\alpha}$) are nonincreasing on $(0, +\infty)$.

Then for some $z_0\in\mathbb{R}^n$, for any $y\in\mathbb{R}^{n}$, $u$, $v$ must take the following form as

for some $c_1 > 0$, $c_2 > 0$ and $d > 0$.

Furthermore, $f_1(t)$, $f_2(t)$, $g_1(t)$ and $g_2(t)$ must be the form of

where $C_1$, $C_2$, $C_3$ and $C_4$ are some positive constants.

Remark 1.2. In the conformal invariant case ($p_1 = \frac{{2n}-\sigma_1}{n-\alpha}$, $p_2 = \frac{n+\alpha-\sigma_1}{n-\alpha}$, $q_1 = \frac{{2n}-\sigma_2}{n-\alpha}$, $q_2 = \frac{n+\alpha-\sigma_2}{n-\alpha}$), and $u = v$, $\sigma_1 = \sigma_2$, the classification of nonnegative solutions to system (1.1) would provide the best constants and extremal functions for the corresponding Hardy-Littlewood-Sobolev inequality ^[28,31,42].

In the following, we introduce some notation, we define the Kelvin transforms. Take arbitrary $x\in\mathbb{R}^n$ and $\lambda > 0$,

where

For any $\lambda > 0$, we denote

and $U_{x, \lambda}(y) = u_{x, \lambda}(y)-u(y)$, $V_{x, \lambda}(y) = v_{x, \lambda}(y)-v(y)$ for any $y\in B_{\lambda}(x)\setminus\{x\}$.

The rest of our paper is organized as follows. In Section 2, we carry out our proof of Theorem 1.1. In order to prove Theorem 1.1, we divide the proof into two parts. In the first part, we will apply a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians developed by Chen, Li and Zhang ^[20] and Dai and Liu ^[26] to derive the forms for nonnegative solution $(u, v)$. In the second part, we classify nonlinearities $f_{1}$, $f_{2}$, $g_{1}$ and $g_{2}$ due to Hu and Liu ^[35].

In the following, we will use $C$ to denote general positive constants that may depend on $n$, $\alpha$, $p_{1}$, $p_{2}$, $q_{1}$, $q_{2}$, $\sigma_1$, $\sigma_2$, $u$ and $v$, and whose values may differ from line to line.

2.   Preparation of Theorem 1.1

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

2.1.   The direct method of moving spheres for nonlocal nonlinearity

Assume that, $n\geq2$, $0 < \sigma_1, \sigma_2 < n$, $0 < \alpha\leq2$, $f_{1}$, $f_{2}$, $g_{1}$ and $g_{2}$ are strictly increasing on $[0, +\infty)$, furthermore, $F_1(t) = f_1(t)\cdot t^{-p_1}$, $F_2(t) = f_2(t)\cdot t^{-p_2}$, $G_1(t) = g_1(t)\cdot t^{-q_1}$ and $G_2(t) = g_2(t)\cdot t^{-q_2}$ are nonincreasing on $(0, +\infty)$£ with $p_{1} = \frac{2n-\sigma_1}{n-\alpha}$, $p_{2} = \frac{n+\alpha-\sigma_1}{n-\alpha}$, $q_{1} = \frac{2n-\sigma_2}{n-\alpha}$ and $q_{2} = \frac{n+\alpha-\sigma_2}{n-\alpha}$. Suppose $(u, v)$ is a pair of nonnegative classical solution of (1.1) which is not identically zero. Then we can derive that $u, v > 0$ in $\mathbb{R}^{n}$ and $\int_{\mathbb{R}^{n}}\frac{f_{1}(v(x))}{|x|^{\sigma_1}}dx < +\infty$, $\int_{\mathbb{R}^{n}}\frac{g_{1}(u(x))}{|x|^{\sigma_2}}dx < +\infty$. Thus we assume $(u, v)$ is actually a positive solution from now on. For convenience, we define the following conformal transforms. Take arbitrary $x\in\mathbb{R}^n$ and $\lambda > 0$,

where

Then, because $(u, v)$ is a pair of positive classical solution of (1.1), we are able to verify that $u_{x, \lambda}, v_{x, \lambda}\in\mathcal{L}_{\alpha}(\mathbb{R}^{n})\cap C^{1, 1}_{loc}(\mathbb{R}^{n}\setminus\{x\})$ if $0 < \alpha < 2$ ($u_{x, \lambda}, v_{x, \lambda}\in C^{2}(\mathbb{R}^{n}\setminus\{x\})$ if $\alpha = 2$) and satisfies the integral property

and a similar equation as $u, v$ for any $x\in\mathbb{R}^n$ and $\lambda > 0$. Actually, without loss of generality, we may assume $x = 0$ for simplicity and get, for $0 < \alpha < 2$ ($\alpha = 2$ is similar),

this means, the conformal transforms $u_{x, \lambda}, v_{x, \lambda}\in\mathcal{L}_{\alpha}(\mathbb{R}^{n})\cap C^{1, 1}_{loc}(\mathbb{R}^{n}\setminus\{x\})$ if $0 < \alpha < 2$ ($u_{x, \lambda}, v_{x, \lambda}\in C^{2}(\mathbb{R}^{n}\setminus\{x\})$ if $\alpha = 2$) satisfies

for every $y\in\mathbb{R}^{n}\setminus\{x\}$, where $p_{1} = \frac{{2n}-\sigma_1}{n-\alpha}$, $p_2 = \frac{n+\alpha-\sigma_1}{n-\alpha}$, $q_1 = \frac{{2n}-\sigma_2}{n-\alpha}$ and $q_2 = \frac{n+\alpha-\sigma_2}{n-\alpha}$. For any $\lambda > 0$, we denote

and define

Let $U_{x, \lambda}(y) = u_{x, \lambda}(y)-u(y)$, $V_{x, \lambda}(y) = v_{x, \lambda}(y)-v(y)$ for any $y\in B_{\lambda}(x)\setminus\{x\}$. By the definitions of $u_{x, \lambda}$, $v_{x, \lambda}$ and $U_{x, \lambda}$, $V_{x, \lambda}$, we have

for every $y\in B_{\lambda}(x)\setminus\{x\}$.

We will first show that there exists a $\epsilon_{0} > 0$ (depending on $x$) sufficiently small, such that, for any $0 < \lambda\leq\epsilon_{0}$, it holds that $U_{x, \lambda}(y)\geq 0$, $V_{x, \lambda}(y)\geq0$, for every $y\in B_\lambda(x)\setminus\{x\}$.

We first need to show that the nonnegative solution $(u, v)$ to PDEs system (1.1) also satisfies the equivalent integral system (1.4).

Lemma 2.1. Suppose $(u, v)$ is a nonnegative solution to (1.1), then $(u, v)$ also satisfies the equivalent integral system (1.4), and vice versa.

The proof of Lemma 2.1 is similar to ^[25,26], so we omit the details here.

Based on Lemma 2.1, we can prove that $U_{x, \lambda}$, $V_{x, \lambda}$ has a strictly positive lower bound in a small neighborhood of $x$.

Lemma 2.2. For each fixed $x\in\mathbb{R}^{n}$, there exists a $\eta_{0} > 0$ (depending on $x$) sufficiently small such that, if $0 < \lambda\leq\eta_{0}$, then

Proof. According to the idea of ^[20], we will prove Lemma 2.2 as follows. Define

For any $|y|\geq1$, since $(u, v)$ satisfy the integral system (1.4), $f_i$, $g_i$ ($i = 1, 2$) are positive on $[0, +\infty)$, so $f(v(z))$ and $g(u(z))$ have lower bounds on compact sets, we can derive that

where $\widetilde{b}_{1}: = b_{1}\cdot \int_{B_{\frac{1}{2}}(0)}f(v(z))dz$ and $\widetilde{b}_{2}: = b_{2}\cdot \int_{B_{\frac{1}{2}}(0)}g(u(z))dz$.

It follows immediately that

for all $y\in\overline{B_{\lambda^{2}}(x)}\setminus\{x\}$. Therefore, we have if $0 < \lambda\leq\eta_{0}$ for some $\eta_{0}(x) > 0$ small enough, then

for any $y\in \overline{B_{\lambda^{2}}(x)}\setminus\{x\}$, this finishes the proof of Lemma 2.2.

For every fixed $x\in \mathbb{R}^n$, define

Now we need the main theorem, which is a variant (for nonlocal nonlinearity) of the Narrow region principle (Theorem 2.2 in ^[20]).

Theorem 2.3. (Narrow region principle) Assume $x\in\mathbb{R}^{n}$ is arbitrarily fixed. Let $\Omega$ be a narrow region in $B_{\lambda}(x)\setminus\{x\}$ with small thickness $0 < l < \lambda$ such that $\Omega\subseteq A_{\lambda, l}(x): = \{y\in\mathbb{R}^n|\, \lambda-l < |y-x| < \lambda\}$. Suppose $U_{x, \lambda}$, $V_{x, \lambda}\in\mathcal{L}_{\alpha}(\mathbb{R}^{n})\cap C^{1, 1}_{loc}(\Omega)$ if $0 < \alpha < 2$ ($U_{x, \lambda}$, $V_{x, \lambda}\in C^{2}(\Omega)$ if $\alpha = 2$) and satisfies

where $\mathcal{L}_{1}(y): = p_{2}v^{p_{2}-1}_{x, \lambda}(y)P(y)F_{2}(v(y))$, $\mathcal{L}_{2}(y): = q_{2}u^{q_{2}-1}_{x, \lambda}(y)Q(y)G_{2}(u(y))$. Then, we have

(i) there exists a sufficiently small constant $\gamma_{0}(x) > 0$, such that, for all $0 < \lambda\leq\gamma_{0}$,

(ii) there exists a sufficiently small $l_{0}(x, \lambda) > 0$ depending on $\lambda$ continuously, such that, for all $0 < l\leq l_{0}$,

Proof. Without loss of generality, we may assume $x = 0$ here for convenience. Suppose (2.7) and (2.8) do not hold, we will obtain a contradiction for any $0 < \lambda\leq\gamma_{0}$ with constant $\gamma_{0}$ small enough and any $0 < l\leq l_{0}(\lambda)$ with $l_{0}(\lambda)$ sufficiently small respectively. We divide the proof into two parts.

Part 1: For $0 < \alpha < 2$ and $\alpha = 2$, if (2.7) fails, we show that there exists some $\hat{y}\in\Omega\cap B^{-}_{\lambda}$ such that

Recall (2.6) and our hypothesis, there exists $\tilde{y}\in(\Omega\cap B^{-}_{\lambda})\subseteq A_{\lambda, l}(0): = \{y\in\mathbb{R}^{n}|\, \lambda-l < |y| < \lambda\}$ such that

We first consider the cases $0 < \alpha < 2$. Let $\tilde{U}_{0, \lambda}(y) = U_{0, \lambda}(y)-U_{0, \lambda}(\tilde{y})$, then $\tilde{U}_{0, \lambda}(\tilde{y}) = 0$ and

Using the anti-symmetry property $U_{x, \lambda}(y) = -(U_{x, \lambda})_{x, \lambda}(y)$, it holds

Consequently, it follows that

Notice that, for any $z\in B_\lambda(0)\setminus\{0\}$,

Then combining this with $U_{0, \lambda}(\tilde{y}) < 0$, we get

Now we consider $\alpha = 2$, we can also derive the same estimate as (2.11) at some point $y_{0}\in\Omega\cap B^{-}_{\lambda}$. The last, we define

then it is obvious that $\phi(y)\in[\cos1, 1]$ for any $y\in\overline{A_{\lambda, l}(0)} = \{y\in\mathbb{R}^{n}\, |\, \lambda-l\leq|y|\leq\lambda\}$ and $-\frac{\Delta\phi(y)}{\phi(y)}\geq\frac{1}{l^2}$. Define

for $y\in\overline{A_{\lambda, l}(0)}$. Then there exists a $y_{0}\in\Omega\cap B^{-}_{\lambda}$ such that

Since

it can be deduced immediately that

In summary, we have proved that for both $0 < \alpha < 2$ and $\alpha = 2$, there exists some $\hat{y}\in\Omega\cap B^{-}_{\lambda}$ such that

Similar, we have

then we know that exists a $\bar{y}$ such that

and we can derive that

Part 2: We obtain a contradiction for any $0 < \lambda\leq\gamma_{0}$ with constant $\gamma_{0}$ small enough and any $0 < l\leq l_{0}(\lambda)$ with $l_{0}(\lambda)$ sufficiently small respectively.

By (2.6), we have at the point $\hat{y}$,

where

Since $\lambda-l < |y| < \lambda$, we have

and

It is obvious that $C'_{1, \lambda}$ and $C''_{1, \lambda}$ depend on $\lambda$ continuously and monotone increasing with respect to $\lambda > 0$.

Therefore, we can derive from (2.21) and (2.22) that, for any $\lambda-l\leq|y|\leq\lambda$,

where $C_{1, \lambda}$ depends continuously on $\lambda$ and monotone increasing with respect to $\lambda > 0$.

From (2.17) and (2.19) we have

According to (2.6), we also have at the point $\bar{y}$

where

Since $\lambda-l < |y| < \lambda,$ we have

and

It is obvious that $C'_{2, \lambda}$ and $C''_{2, \lambda}$ depend on $\lambda$ continuously and monotone increasing with respect to $\lambda > 0$.

As a result, we obtain from (2.27) and (2.28) that, for any $\lambda-l\leq|y|\leq\lambda$

where $C_{2, \lambda}$ depends continuously on $\lambda$ and monotone increasing with respect to $\lambda > 0$.

From (2.18) and (2.25), we have

As a consequence, it follows from (2.17), (2.19), (2.23), (2.29) and (2.30) that

that is,

We can draw a contradiction from (2.31) directly if $0 < \lambda\leq\gamma_{0}$ for some constants $\gamma_{0}$ small enough, or if $0 < l\leq l_{0}$ for some sufficiently small $l_{0}$ depending on $\lambda$ continuously. This shows that (2.7) and (2.8) must hold. Moreover, by (2.6), we can actually deduce from $U_{x, \lambda}(y)\geq0$, $V_{x, \lambda}\geq0$ in $\Omega$ that

This completes the proof of Theorem 2.3.

The following lemma provides a start point for us to move the spheres.

Lemma 2.4. For every $x\in \mathbb{R}^n$, there exists $\epsilon_0(x) > 0$ such that, $u_{x, \lambda}(y)\geq u(y)$ and $v_{x, \lambda}(y)\geq v(y)$ for all $\lambda\in(0, \epsilon_0(x)]$ and $y\in B_\lambda(x)\setminus \{x\}$.

Proof. For every $x\in \mathbb{R}^n$, recalling that

Define $\epsilon_{0}(x): = \min\{\eta_{0}(x), \gamma_{0}(x)\}$, where $\eta_{0}(x)$ and $\gamma_{0}(x)$ are defined the same as in Lemma 2.2 and Theorem 2.3. We will show via contradiction arguments that, for any $0 < \lambda\leq\epsilon_{0}$,

Suppose (2.33) does not hold, that is, $B^{-}_{\lambda}\neq\emptyset$ and hence $U_{x, \lambda}$, $V_{x, \lambda}$ is negative somewhere in $B_{\lambda}(x)\setminus\{x\}$. For arbitrary $y\in B^{-}_{\lambda}$, we deduce from (1.1) and (2.1) that

where $\mathcal{L}_{1} = p_{2}v^{p_{2}-1}_{x, \lambda}(y)P(y)F_{2}(v(y))$.

Similarly, one can calculate

where $\mathcal{L}_{2} = q_{2}u^{q_{2}-1}_{x, \lambda}(y)Q(y)G_{2}(u(y))$.

Then (2.35) and (2.36) imply that, for all $y\in B^{-}_{\lambda}$,

Due to $\epsilon_{0}(x): = \min\{\eta_{0}(x), \gamma_{0}(x)\}$, by Lemma 2.2, we have, for any $0 < \lambda\leq\epsilon_{0}$,

Thus, by taking $l = \lambda-\lambda^{2}$ and $\Omega = A_{\lambda, l}(x)$, then it follows from (2.37)–(2.39) that all the conditions in (2.6) in Theorem 2.3 are satisfied, we can derive from (i) in Theorem 2.3 that $U_{x, \lambda}\geq0$, $V_{x, \lambda}\geq0$ in $\Omega = A_{\lambda, l}(x)$ for any $0 < \lambda\leq\epsilon_{0}(x)$. In other words, there exists a $\epsilon_0(x) > 0$ such that, for all $\lambda\in(0, \epsilon_0(x)]$,

This completes the proof of Lemma 2.4.

For each fixed $x\in \mathbb{R}^n$, we define

By Lemma 2.4, $\bar{\lambda}(x)$ is well-defined and $0 < \bar{\lambda}(x)\leq+\infty$ for any $x\in \mathbb{R}^n$.

We need the following lemma, which is crucial in our proof.

Lemma 2.5. If $\bar{\lambda}(\bar{x}) < +\infty$ for some $\bar{x}\in \mathbb{R}^n$, then

Proof. Without loss of generality, we may assume $x = 0$ for convenience. Since $(u, v)$ is a pair of positive solution to integral system (1.4), one can verify that $u_{0, \lambda}, v_{0, \lambda}$ also satisfies a similar integral system as (1.4) in $\mathbb{R}^{n}\setminus\{0\}$. In fact, by (1.4) and direct calculations, we have, for any $y\in\mathbb{R}^{n}\setminus\{0\}$,

Suppose on the contrary that $U_{0, \bar{\lambda}}\geq0$ but $U_{0, \bar{\lambda}}$ is not identically zero in $B_{\bar{\lambda}}(0)\setminus\{0\}$, then we will get a contradiction with the definition (2.40) of $\bar{\lambda}$. Now, we divide the proof into two parts.

Part 1: We prove that

Actually, if there exists a point $y^{0}\in B_{\bar{\lambda}}(0)\setminus\{0\}$ such that $U_{0, \bar{\lambda}}(y^{0}) > 0$, by continuity, there exists a small $\gamma > 0$ and a constant $c_{0} > 0$ such that

For any $y\in B_{\bar{\lambda}}(0)\setminus\{0\}$, one can deduce that

and

where

Let us define

It is easy to verify that $K_{1, \bar{\lambda}}(y, z) > 0$, $K_{2, \bar{\lambda}}(y, z) > 0$, and

and moreover,

As a result, it follows immediately that, for any $y\in B_{\bar{\lambda}}(0)\setminus\{0\}$,

therefore we arrive at (2.41).

Part 2: We draw a contradiction with the definition (2.40) of $\bar{\lambda}(0)$.

Furthermore, (2.42) also shows that there exists a $0 < \eta < \bar{\lambda}$ small enough such that, for any $y\in \overline{B_{\eta}(0)}\setminus\{0\}$,

Now we define

where $l_{0}(0, \lambda)$ is given by Theorem 2.3. For a fixed small $0 < r_{0} < \frac{1}{2}\min\{\tilde{l}_{0}, \bar{\lambda}\}$, by (2.41) and (2.43), we can define

Similarly, we can define

Since $u$ is uniformly continuous on arbitrary compact set $K\subset\mathbb{R}^{n}$ (say, $K = \overline{B_{4\bar{\lambda}}(0)}$), we can deduce from (2.45) that, there exists a $0 < \varepsilon_{0} < \frac{1}{2}\min\{\tilde{l}_{0}, \bar{\lambda}\}$ sufficiently small, such that, for any $\lambda\in[\bar{\lambda}, \bar{\lambda}+\varepsilon_{0}]$,

In order to prove (2.47), one should observe that (2.45) is equivalent to

Since $u$ is uniformly continuous on $\overline{B_{4\bar{\lambda}}(0)}$, we infer from (2.48) that there exists a $0 < \varepsilon_{0} < \frac{1}{2}\min\{\tilde{l}_{0}, \bar{\lambda}\}$ sufficiently small, such that, for any $\lambda\in[\bar{\lambda}, \bar{\lambda}+\varepsilon_{0}]$,

which is equivalent to (2.47), hence we have proved (2.47).

Similarly, we can prove

For any $\lambda\in[\bar{\lambda}, \bar{\lambda}+\varepsilon_{0}]$, let $l: = \lambda-\bar{\lambda}+r_{0}\in(0, \tilde{l}_{0})$ and $\Omega: = A_{\lambda, l}(0)$, then it follows from (2.37), (2.38) and (2.47) that all the conditions (2.6) in Theorem 2.3 are satisfied, hence we can derive from (ii) in Theorem 2.3 that

Thus, we get from (2.47) and (2.51) that, $B^{-}_{\lambda} = \emptyset$ for all $\lambda\in[\bar{\lambda}, \bar{\lambda}+\varepsilon_{0}]$, that is,

which contradicts with the definition (2.40) of $\bar{\lambda}(0)$. As a result, in the case $0 < \bar{\lambda}(0) < +\infty$, we must have $U_{0, \bar{\lambda}}\equiv0$, $V_{0, \bar{\lambda}}\equiv0$ in $B_{\bar{\lambda}}(0)\setminus\{0\}$, that is,

This finishes our proof of Lemma 2.5.

We also need the following property about the limiting radius $\bar{\lambda}(x)$.

Lemma 2.6. If $\bar{\lambda}(\bar{x}) = +\infty$ for some $\bar{x}\in \mathbb{R}^n$, then $\bar{\lambda}(x) = +\infty$ for all $x\in \mathbb{R}^n$.

Proof. Since $\bar{\lambda}(\bar{x}) = +\infty$, recalling the definition of $\bar{\lambda}$, we get

That is,

It follows immediately that

On the other hand, if we assume $\bar{\lambda}(x) < +\infty$ for some $x\in \mathbb{R}^n$, then by Lemma 2.5, one gets that

which contradicts with (2.54).

This finishes the proof of Lemma 2.6.

We are now ready to give a proof of Theorem 1.1.

3.   Proof of Theorem 1.1

We derive the forms for nonnegative solution $(u, v)$ by discussing two different possible cases.

Case (i). $\bar{\lambda}(\bar{z}) = +\infty$ for some $\bar{z}\in \mathbb{R}^n$.

By Lemma 2.6, one can derive that

Thus, for all $z\in \mathbb{R}^{n}$ and $0 < \lambda < +\infty$, we have

By a calculus lemma (Lemma 11.2 in Li ^[39]), we conclude that $u = b_1, v = b_{2}$ is a constant.

This is a contradiction, therefore Case (i) is impossible.

Case (ii). $\bar{\lambda}(z) < \infty\ \ \text{for all}\ z\in \mathbb{R}^n.$

By Lemma 2.5, we infer that

From a calculus lemma (Lemma 11.1 in Li ^[39]) and (3.1), we derive that for any $y\in\mathbb{R}^{n}$,

for some $c_1 > 0$, $d > 0$, $z_0\in\mathbb{R}^{n}$. In a similar way, for any $y\in \mathbb{R}^{n}$, we have

for some $c_2 > 0$, $d > 0$, $z_0\in\mathbb{R}^{n}$.

Now we verify that $F_{1}$, $F_{2}$, $G_{1}$ and $G_{2}$ must be constants. It follows from Lemma 2.5 and (2.42) that

where

that is,

Consequently, it follows that for any $z\in B_{\bar{\lambda}}(x)$, we have

and

Thus, for some positive constant $C_1$ and $C_2$, we have

Similarly, we deduce that for some positive constant $C_3$ and $C_4$,

This completes our proof of Theorem 1.1.

4.   Conclusions

In this paper, we obtain the forms of the nonnegative solution and classify nonlinearities by appling a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians.

Acknowledgments

The authors would like to express their sincere thanks to the referees for their valuable suggestions. This work was partially supported by the NNSF of China (No. 11801237), the Natural Foundation of Jiangxi Province (No. 20202BABL211001), and the Fundamental Research Funds for the Central Universities (No. 2020QNBJRC005), the fourth author is also supported by the Educational Committee of Jiangxi Province (No. GJJ180618).

Conflict of interest

The authors declare that they have no competing interests in this paper.