Abstract
In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator:
where \(a,b\geq0\). When Ω is the unit ball or \(\mathbb{R}^{n}\), we prove that the solutions \((u,v)\) are radially symmetric and decreasing. When Ω is the parabolic domain on \(\mathbb{R}^{n}\), we prove that the solutions \((u,v)\) are increasing. Furthermore, if Ω is the \(\mathbb{R}^{n}_{+}\), then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.
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1 Introduction
This paper is mainly devoted to investigating the properties of the solutions of the following system involving the fractional Laplacian operators:
with
and
where P.V. stands for the Cauchy principle value, \(C_{n,\alpha },C_{n,\beta}>0\) and \(0<\alpha, \beta<2\). To make sense for the integrals, we require \(u\in C_{\mathrm {loc}}^{1,1} \cap L_{\alpha}\), \(v\in C_{\mathrm{loc}}^{1,1} \cap L_{\beta}\), where
and
For more background on the fractional Laplacian operator \((-\Delta )^{\frac{\alpha}{2}}\), we refer to [1–4]. We mention that there are also several applications involving the fractional Laplacian in mathematical physics [5–8], finance [9], image processing [10], and so on.
Since the fractional Laplacian is nonlocal, that is, it does not act by pointwise differentiation but as a global integral with respect to a singular kernel, this is the main difficulty in studying problems involving it. To circumvent this difficulty, Caffarelli and Silvestre [11] introduced the extension method (CS extension) to overcome the difficulty of nonlocality. Their idea is to localize the fractional Laplacian by constructing a Dirichlet to Neumann operator of a degenerate elliptic equation. We can also use the integral equation method, the method of moving planes in integral forms, and regularity lifting to investigate equations involving the fractional Laplacian. Recently, Chen, Li, and Li [12] developed a new method that can handle directly these nonlocal operators. They used this property to develop some techniques needed in the direct method of moving planes in the whole space \(\mathbb{R}^{n}\) and the upper half-space \(\mathbb{R}^{n}_{+}\), such as the narrow region principle and decay at infinity. The direct method of moving planes is very useful, and a series of fruitful results have been obtained. For more articles concerning the method of moving planes for nonlocal equations and systems, mainly for integral equations, we refer to [13–21].
In this paper, following the ideas of [12], among others, we consider the properties of the solutions to system (1.1) for different domains Ω. More precisely, we get the following four theorems. Firstly, we consider the case where Ω is the unit ball. For simplicity, we denote \(B=B_{1}(0)\). We have
Theorem 1.1
Let \(u\in C(\overline{B})\cap\mathcal {C}_{\mathrm{loc}}^{1,1}(B)\) and \(v\in C(\overline{B})\cap\mathcal {C}_{\mathrm{loc}}^{1,1}(B)\) be positive solutions of the system
with \(M>f'(\cdot)\), \(g'(\cdot)>0\), where M is a positive constant. Then u is radially symmetric and decreasing about the origin.
Remark 1.1
Li [22] considered the similar problem with \(f(v)=v^{p}\) and \(g(u)=u^{p}\). So Theorem 1.1 can be regarded as an extension of the result in [22].
We denote by
the parabolic domain on \(\mathbb{R}^{n}\).
Theorem 1.2
Let \(u\in L_{\alpha}(\mathbb{R}^{n})\cap\mathcal {C}_{\mathrm {loc}}^{1,1}(\Omega)\) and \(v\in L_{\beta}(\mathbb{R}^{n})\cap\mathcal {C}_{\mathrm{loc}}^{1,1}(\Omega)\) be positive solutions of the system
with \(M>f'(\cdot)\), \(g'(\cdot)>0\), where M is a positive constant. Then u, v are increasing in \(x_{n}\).
Now we consider the whole space case.
Theorem 1.3
Let \(u\in L_{\alpha}(\mathbb{R}^{n})\cap\mathcal {C}_{\mathrm {loc}}^{1,1}(\mathbb{R}^{n}), v \in L_{\beta}(\mathbb{R}^{n})\cap \mathcal {C}_{\mathrm{loc}}^{1,1}(\mathbb{R}^{n})\) be positive solutions of the system
Suppose that, for \(\gamma,\nu>0\),
and
Then \(u(x)\) and \(v(x)\) are radially symmetric and decreasing about some point \(x_{0}\) in \(\mathbb{R}^{n}\).
Now we consider the nonexistence of positive solutions to system (1.1) in the half-space.
Theorem 1.4
Let \(u\in L_{\alpha}(\mathbb{R}_{+}^{n})\cap\mathcal {C}_{\mathrm{loc}}^{1,1}(\mathbb {R}_{+}^{n})\) and \(v\in L_{\beta}(\mathbb{R}_{+}^{n})\cap\mathcal {C}_{\mathrm{loc}}^{1,1}(\mathbb{R}_{+}^{n})\) be nonnegative solutions of the system
Suppose
and \(M>f'(\cdot)\), \(g'(\cdot)>0\), where M is a positive constant with \(f(0)=0\), \(g(0)=0\). Then \(u(x)\equiv0\) and \(v(x)\equiv0\) in \(\mathbb{R}^{n}\).
Remark 1.2
In Sect. 2, we introduce two maximum principles, namely, the narrow region principle and decay at infinity. This two maximum principles play a key role in the proof of Theorems 1.1–1.4. We give detailed proofs of our main theorems in Sect. 3.
2 Two maximum principles
Let \(T_{\lambda}\) be a hyperplane in \(\mathbb{R}^{n}\). Without loss of generality, we assume that
where \(x'=(x_{2},x_{3},\ldots,x_{n})\). Let
be the reflection of x about the plane \(T_{\lambda}\). Set
For simplicity of notations, we denote \(U_{\lambda}(x)\) by \(U(x)\) and \(V_{\lambda}(x)\) by \(V(x)\).
Lemma 2.1
(Narrow region principle)
Let Ω be a bounded narrow region in \(\Sigma_{\lambda}\) that is contained in
for small l. Let \(U, V\in L_{\alpha}(\mathbb{R}^{n})\cap\mathcal {C}_{\mathrm {loc}}^{1,1}(\Omega)\) and suppose that U, V are lower semicontinuous on Ω̅. Assume that \(C_{1}(x)\) and \(C_{4}(x)\) are bounded from below in Ω, whereas \(C_{2}(x), C_{3}(x)<0\) are bounded from below in Ω. If U, V satisfy the system
then, for sufficiently small l, we have
These conclusions hold for an unbounded domain Ω if we further assume that
Furthermore, if there exists \(x_{0}\in\Omega\) such that
then
Remark 2.1
After finish this paper, we have found that this lemma was given by Niu and Wang [23]. For completeness, we give a full proof of the lemma with some changes.
Proof of Lemma 2.1
Suppose on the contrary, that (2.2) is false. Without loss of generality, we assume that there exists a point such that \(U(x)<0\). Since \(U(x)\) is lower semicontinuous on Ω̅, there exists \(x_{0}\in \Omega\), such that
Now let \(\Sigma_{\lambda}^{c}=\mathbb{R}^{n}\setminus \Sigma_{\lambda}\). Then by the definition of \((-\Delta)^{\frac{\alpha}{2}}\) we have
To estimate \(I_{1}\), we notice that
Since \(U(y)\geq0\) and \(y\in\Sigma_{\lambda} \setminus \Omega\), by (2.4) we have
We can see that
which implies
Choose \(x_{0}^{*}=(3l+(x_{0})_{1},x')\) in \(\Sigma_{\lambda}^{c}\), where \(x_{0}=((x_{0})_{1},x')\). It is easy to see that \(B_{l}(x_{0}^{*})\subset\Sigma _{\lambda}^{c}\). Moreover, there exists \(C>0\) such that
Combining the previous estimate with (2.7), we have
Combining (2.8) with (2.1), we have
which is equivalent to
Since we can choose l small enough and \(C_{1}\) is bounded from below, we have \(\frac{C}{l^{\alpha}}+C_{1}(x_{0})>0\). Since
by the condition \(C_{2}(x)<0\) and (2.4) we get that
On the other hand, V is lower semicontinuous on Ω̅; hence there exists \(\overline{x}_{0} \) such that
Similarly to (2.8), we derive that
Combining (2.8) and (2.12) with (2.1), we have
We notice that \(C_{4}(x)\) is bounded from below in Ω and \(C_{2}(x), C_{3}(x)<0\) are bounded from below in Ω. Choosing l small enough, we can derive that \(V(\overline{x}_{0})\geq0\). This yields a contradiction with (2.11). So (2.2) holds.
Furthermore, if Ω is an unbounded domain, then by the decay condition of U, V it is easy to see that the negative minimum of U, V cannot be taken at infinity.
Now we prove (2.3). Without loss of generality, we assume that there exists \(x_{0}\in\Omega\) such that \(U(x_{0})=0\). Then, due to (2.2) and the fact that \(C_{2}(x_{0})V(x_{0})\leq0\), combining (2.5) with the first equation of (2.1), we have
If \(U(y)\not\equiv0\), \(y\in\Sigma_{\lambda}\), then noticing that \(U(y)\geq 0\), \(y\in\Sigma_{\lambda}\), we have
This yields a contradiction. So
By (2.10) we immediately get \(V(x)\equiv0, x\in\Sigma_{\lambda}\). So since U and V are antisymmetric functions, we have (2.3). □
Lemma 2.2
(Decay at infinity)
Let Ω be an unbounded domain in \(\Sigma_{\lambda}\). Let \(U, V\in L_{\alpha}(\mathbb{R}^{n}) \cap C_{\mathrm{loc}}^{1,1}(\Omega)\) be lower semicontinuous on Ω̅. Assume that
where \(C_{1}(x)\), \(C_{4}(x)\) are nonnegative on Ω, whereas \(C_{2}(x),C_{3}(x)<0\) on Ω, and furthermore,
Then there exists a constant \(R_{0}>0\) (depending on \(C_{i}(x)\) but independent of U, V) such that if
then
Remark 2.2
This lemma is quite the same as that of Niu and Wang [23] but with some difference. In [23], \(C_{1}\), \(C_{4}\) should satisfying
So we conclude that the condition of our lemma is different with [23], even they both have the same result.
Proof of Lemma 2.2
Without loss of generality, we assume that there exist a point \(x_{0}\in \Omega\) such that
Then as in the proof as (2.7), we have
Choose a point in \(\Sigma_{\lambda}^{c}:\widetilde{x}^{\ast }=(3|\widetilde {x}|+x_{1},x')\), where \(\widetilde{x}=(x_{1},x')\). Then \(B_{|\widetilde {x}|}(\widetilde{x}^{\ast})\subset\Sigma_{\lambda}^{c}\), and there exists \(C>0\) such that
So combining (2.17) with (2.16), we have
By (2.13) we have that
Combining this with (2.18), we get that
Now by the conditions \(C_{1}(\widetilde{x})\geq0\) and \(C_{2}(\widetilde {x})<0\) we easily calculate that
Noticing that \(U(\widetilde{x})<0\) and \(C_{2}(\widetilde{x})<0\), we get \(V(\widetilde{x})<0\). Then since V are lower semicontinuous on Ω̅. there exists \(\overline{x_{0}}\) such that
Similarly to (2.18), we derive that
Combining (2.19) and (2.21) with (2.13), we have
Choosing \(|\widetilde{x}|\), \(|\overline{x}|\) large enough, by (2.14) we can derive \(V(\overline{x})\geq0\). This yields a contradiction with (2.20). So the lemma is proved. □
3 Proof of Theorems 1.1-1.4
Proof of Theorem 1.1
Let \(T_{\lambda}\), \(x_{\lambda}\), \(u_{\lambda}\), \(\Sigma_{\lambda}\), and \(U_{\lambda}\), \(V_{\lambda}\) be defined as in the previous section.
Step 1: We will show that for \(\lambda>-1\) and sufficiently close to −1, we have
By the first equation in system (1.2) and the mean value theorem it is easy to see that
where \(\xi(x)\) is between \(v(x)\) and \(v_{\lambda}(x)\). Similarly, we also have
where \(\eta(x)\) is between \(u(x)\) and \(u_{\lambda}(x)\). Choosing \(C_{1}=a\), \(C_{2}=-f'(\xi(x))\), \(C_{3}=b\), and \(C_{4}=-g'(\eta(x))\), by the narrow region principle (Lemma 2.1) we get (3.1).
Step 2: Define
Then we claim that
Suppose the claim is not true. If \(\lambda_{0} < 0\), then we will show that the plane can be moved to the right a little more so that inequality (3.1) will still valid. More precisely, there exists small \(\epsilon> 0\) such that, for all \(\lambda\in[\lambda_{0},\lambda_{0}+\epsilon)\), inequality (3.1) holds, which contradicts the definition of \(\lambda_{0}\).
First, since \(U_{\lambda_{0}}\) and \(V_{\lambda_{0}}\) are not identically zero, from the proof of the narrow region principle (Lemma 2.1) we have
Thus \(U_{\lambda_{0}}\) and \(V_{\lambda_{0}}\) can take the minimum values if \(x\in\Sigma_{\lambda_{0}-\delta}\cap B\). More precisely, for any \(\delta>0\),
By the continuity of \(U_{\lambda}\) and \(V_{\lambda}\) with respect to λ there exists \(\epsilon> 0\) such that
We can see that \((\Sigma_{\lambda} \setminus \Sigma_{\lambda_{0}-\delta })\cap B\) is a narrow region if ϵ and δ are small enough. Then by the narrow region principle (Lemma 2.1) we have
This contradicts the definition of \(\lambda_{0}\). Therefore we prove the claim (3.3). It follows that
or, more apparently,
Since the \(x_{1}\)-direction can be chosen arbitrarily, it follows that \(u,v\) are radially symmetric about the origin. The monotonicity is a consequence of the fact that
for all \(-1<\lambda\leq0\). This completes the proof of Theorem 1.1. □
Proof of Theorem 1.2
Let \(T_{\lambda}\), \(x_{\lambda}\), \(u_{\lambda}\), and \(U_{\lambda}\), \(V_{\lambda}\) be defined as in the previous section. Let
Step 1: Similarly to (3.1), we can get that, for \(\lambda>0\) sufficiently close to 0, we have
Step 2: Define
Then we must have
Otherwise, suppose that \(\lambda_{0} <+\infty\), Then we claim that
If (3.5) were not true, then by the narrow region principle (Lemma 2.1) we would have
We will show that the plane \(T_{\lambda}\) can be moved further to the right. More precisely, there exists small \(\epsilon>0\) such that, for all \(\lambda\in[\lambda_{0},\lambda_{0}+\epsilon)\),
This is a contraction with the definition of \(\lambda_{0}\).
If (3.6) holds, then for any \(\delta>0\),
By the continuity of \(U_{\lambda}\), \(V_{\lambda}\) with respect to λ there exists \(\epsilon> 0\) such that
We can see that \(\Sigma_{\lambda} \setminus \Sigma_{\lambda_{0}-\delta }\) is a narrow region since ϵ and δ are small enough. Then by the narrow region principle (Lemma 2.1) we have
This contradicts the definition of \(\lambda_{0}\). Then the claim (3.5) holds, which implies
which contradicts the fact that \(u,v>0\) on Ω. We have shown that \(\lambda_{0}=\infty\) and \(U\geq0\), \(V\geq0\). This shows that \(u(x)\) and \(v(x)\) are increasing in \(x_{n}\), which completes the proof of Theorem 1.2. □
Proof of Theorem 1.3
Start moving the plane \(T_{\lambda}\) from −∞ to that right along the \(x_{1}\)-direction.
Step 1: We will show that for λ sufficiently negative,
By the first equation in system (1.4) and the mean value theorem it is easy to see that
where \(\xi(x)\) is between \(v(x)\) and \(v_{\lambda}(x)\). Similarly, we also have
where \(\eta(x)\) is between \(u(x)\) and \(u_{\lambda}(x)\). In fact, \(0< v_{\lambda}(x)\leq\xi(x)\leq v(x)\) and \(0< u_{\lambda}(x)\leq\eta (x)\leq u(x)\).
At those points, for \(|x|\) sufficiently large, the decay assumptions (1.5) and (1.6) immediately yield
and going through a similar proof, we have
Then by the decay at infinity (Lemma 2.2) there exists a constant \(R_{0}>0\) such that for \(\lambda<-R_{0}\) in Lemma 2.2, one of \(U_{\lambda}(x)\) and \(V_{\lambda}(x)\) must be positive in \(\Sigma _{\lambda}\). Without loss of generality, we assume that
Now we can also prove \(U_{\lambda}(x)\geq0\), \(x\in\Sigma_{\lambda}\). If not, then by the decay condition of \(u(x)\) there must exist a point \(x_{0}\in \Sigma_{\lambda}\) such that
From previous arguments (2.18) and (3.9) we have
and then
since \(f'(\cdot)>0\), \(V_{\lambda}(x_{0})\geq0\). We can derive that \(U_{\lambda}(x_{0})\geq0\), which contradicts with (3.10). So (3.8) holds.
Step 2: Keep moving the planes to the right to the limiting positive \(T_{\lambda_{0}}\) as long as (3.8) holds.
Let
We have that
Otherwise, if \(\lambda_{0}=\infty\), then the solution \(u(x)\) is increasing with respect to \(x_{1}\). This contradicts condition (1.5), so that \(\lambda_{0}<\infty\).
Then we claim that
If (3.11) were not true, then by the proof of the narrow region principle (Lemma 2.1) we would have
We will show that the plane \(T_{\lambda}\) can be moved further to the right. More precisely, there exists small \(\epsilon>0\) such that, for all \(\lambda\in[\lambda_{0},\lambda_{0}+\epsilon)\),
This is a contraction with the definition of \(\lambda_{0}\).
If (3.12) is true, then let \(R_{0}\) be determined in the decay at infinity (Lemma 2.2). It follows that, for any \(\delta>0\),
Since we have the continuity of \(U_{\lambda}(x)\) and \(V_{\lambda}(x)\) with respect to λ, there exists \(\epsilon>0\) such that, for any \(\lambda\in[\lambda_{0},\lambda+\epsilon)\),
Suppose (3.13) is not true. If \(x_{0}\) and \(\overline{x_{0}}\) are the negative minima of \(U_{\lambda}(x)\) and \(V_{\lambda}(x)\) in \(\Sigma _{\lambda}\), then by decay at infinity (Lemma 2.2) and (3.14) we can get that they are all in the bounded narrow region \((\Sigma_{\lambda_{0}+\epsilon} \setminus \Sigma_{\lambda_{0}-\delta })\cap B_{R_{0}}(0)\) for δ and ϵ small enough, which contradicts the narrow region principle (Lemma 2.1). So (3.13) has to be true, which is a contradiction with the definition of \(\lambda_{0}\).
Now we have proved the claim (3.11). Since the \(x_{1}\)-direction can be chosen arbitrarily, we get that \(u(x)\) and \(v(x)\) are radially symmetric and decreasing about some point \(x_{0}\). This completes the proof of Theorem 1.3. □
Proof of Theorem 1.4
First, we claim that
To prove (3.15), we assume that \(u(x)\not\equiv0\). If there exists \(x_{0}\in\mathbb{R}^{n}_{+} \) such that \(u(x_{0})=0\), then we have that
On the other hand, by the function (1.7) and the condition on \(f(x)\) and \(g(x)\) we have that \((-\Delta)^{\frac{\alpha }{2}}u=f(v)-mv\geq0\). This yields a contradiction, so we have that
If \(u\equiv0\) in \(\mathbb{R}^{n}_{+}\), by (1.7) we have \(f(v(x))\equiv mv\) in \(\mathbb{R}^{n}_{+}\). Together with the condition \(f'(s)> m\) and \(f(0)=0\), we can obtain that \(v\equiv0\) in \(\mathbb {R}^{n}_{+}\). Hence if \(u(x)\) attains 0 somewhere in \(\mathbb{R}^{n}_{+}\), then \(u(x)=v(x)\equiv0\). Similarly, we can also derive that if \(v(x)\) attains 0 somewhere in \(\mathbb{R}^{n}_{+}\), then \(u(x)=v(x)\equiv0\), so (3.15) holds. Now we assume that
Denote \(T_{\lambda}=\{x\in\mathbb{R}^{n}_{+}\mid x_{n}=\lambda,\lambda>0\} ,\Sigma _{\lambda}=\{x\in\mathbb{R}^{n}_{+}\mid 0< x_{n}<\lambda\}\). Let \(x^{\lambda}=(x_{1},\ldots,x_{n-1},2\lambda-x_{n})\) be the reflection of x about the plane \(T_{\lambda}\), and let \(U_{\lambda}(x)=u_{\lambda}(x)-u(x)\) and \(V_{\lambda}(x)=v_{\lambda}(x)-v(x)\).
Step 1. For \(\lambda>0\) sufficiently close to 0, set \(\Sigma=\Sigma _{\lambda}\cup\mathbb{R}^{n}_{-}\), where \(\mathbb{R}^{n}_{-}=\{x\in\mathbb {R}^{n}\mid x_{n}\leq0\}\).
First, we know that \(U(x)=u_{\lambda}(x)-u(x)\geq-u(x)\) and \(V(x)=v_{\lambda}(x)-v(x)\geq-v(x)\), so by condition (1.8) we have
Then by the narrow region principle (Lemma 2.1) it is easy to see that
since \(\Sigma_{\lambda}\) is a narrow region.
Step 2. Next, we move the plane \(T_{\lambda}\) along the \(x_{n}\)-axis to the right as long as (3.17) holds and set
We claim that
Otherwise, if \(\lambda_{0}<\infty\), then by the proof of the narrow region principle (Lemma 2.1) we have
Then going through similar arguments as in (3.11), we can see that
If we choose the point \(\bar{x}=(x_{1},x_{2},\ldots,x_{n-1},0) \) in the hyperplane \(\{x_{n}=0\}\), then \(\bar{x}^{\lambda_{0}}\in\mathbb {R}^{n}_{+}\), which implies
and
This contradicts with (3.16).
Therefore we have proved the claim \(\lambda_{0}=\infty\), and consequently the solutions \(u(x)\) and \(v(x)\) are increasing with respect to \(x_{n}\). We recall that condition (1.8) tells us that \(\varliminf_{|x|\rightarrow\infty}u(x)=0\) and \(\varliminf_{|x|\rightarrow\infty}v(x)=0\). So the claim (3.16) is not true, and thus \(u(x)\equiv0\) and \(v(x)\equiv0\), \(x\in\mathbb{R}^{n}\). □
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Acknowledgements
Both authors want to take this chance to thank Dr. Yan Li for sending us the doctoral thesis.
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The first author is partially supported by the NSF of China (Grant No. 11471033), the Innovation Foundation of Anhui Normal University (No. 2018XJJ111), and the Research Cultivation Foundation of Anhui Normal University (No. 2018XJJ93).
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Qu, M., Yang, L. Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian. J Inequal Appl 2018, 297 (2018). https://doi.org/10.1186/s13660-018-1874-9
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DOI: https://doi.org/10.1186/s13660-018-1874-9