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Normalized solutions for a coupled fractional Schrödinger system in low dimensions
Boundary Value Problems volume 2020, Article number: 166 (2020)
Abstract
We consider the following coupled fractional Schrödinger system:
with \(0< s<1\), \(2s< N\le 4s\) and \(1+\frac{2s}{N}< p<\frac{N}{N-2s}\), under the following constraint:
Assuming that the parameters \(\mu _{1}\), \(\mu _{2}\), \(a_{1}\), \(a_{2}\) are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter \(\beta >0\).
1 Introduction
In this paper, we consider the following fractional Schrödinger system with \(1+\frac{2s}{N}< p<\frac{N}{N-2s} \text{and} 2s<N\le 4s\):
under the constraint
The parameters \(\mu _{1}>0\), \(\mu _{2}>0\) and \(\beta >0\).
More precisely, we analyze the existence of solutions \((\lambda _{1},\lambda _{2},u,v)\in \mathbb{R}^{2}\times H^{s}( \mathbb{R}^{N})\times H^{s}(\mathbb{R}^{N})\) to the system (1) satisfying the additional condition (2) for different ranges of the coupling parameter \(\beta >0\).
Condition (2) is called the normalization condition, which imposes a normalization on the \(L^{2}\)-masses of u and v. The solutions to the system (1) under the constraint (2) are usually referred as normalized solutions. In order to obtain the solution to the system (1) satisfying the normalization condition (2), one need to consider the critical point with the \(H_{a}\) (see (4)). Then \(\lambda _{1}\) and \(\lambda _{2}\) appear as Lagrange multipliers with respect to the mass constraint, which cannot be determined a priori, but are part of the unknown.
The normalized solutions of nonlinear Schrödinger equations and systems have gradually attracted the attention of a large number of researchers in recent years, both for the pure mathematical research and in view of its very important applications in many physical problems; see for more details [1–4].
On the one hand, for the nonlinear Schrödinger equation with \(s=1\), in [5], the author studied existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities p, q which satisfy \(2< q\leq 2+\frac{4}{N}\leq p\), \(p\neq q\). In [6], the author studied existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities q, 2∗.
On the other hand, for the nonlinear Schrödinger system with \(s=1\), Thomas et al. [3] recently proved the existence of positive solutions for the system with any arbitrary number of components in three-dimensional space. In [7], the authors considered the existence of multiple positive solutions to the nonlinear Schrödinger systems set on \(H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})\). In [8], the authors proved the existence of solutions \((\lambda _{1},\lambda _{2},u,v)\in \mathbb{R}^{2}\times H^{1}( \mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})\) to systems of coupled Schrödinger equations.
The fractional Schrödinger equation is introduced by Laskin [9, 10] through expanding the Feynman path integral from Brownian-like to Lévy-like mechanical paths. The path integral over the Lévy-like quantum-mechanical paths allows one to develop the generalization of the quantum mechanics. The existence of normalized solution for fractional Schrödinger system is an interesting problem.
The fractional Laplacian \((-\Delta )^{s}\) with \(s\in (0,1)\) of a function \(f:\mathbb{R}^{N}\rightarrow \mathbb{R}\) is expressed by the formula
where \(\mathrm{P.V.}\) stands for the Cauchy principal value, and \(C_{N,s}\) is a normalization constant.
It can also be defined as a pseudo-differential operator
where \(\mathscr{F}\) is the Fourier transform. For more details about the fractional Laplacian we refer to [11–15] and the references therein. The nature function space associated with \((-\Delta )^{s}\) in N dimension is
equipped with the norm
where, by the Fourier transform,
The energy functional associated with (1) is
on the constraint \(H_{a_{1}}\times {H_{a_{2}}}\). For \(a\in \mathbb{R}^{+}\), we define
We prove the existence of normalized solution for different ranges of the coupling parameter \(\beta >0\). Our first main theorem, which is the generalization of the corresponding result (\(s=1\), \(N=3\), \(p=2\)) given in [3], it stated as follows.
Theorem 1.1
Assume \(0< s<1\), \(2s< N\le 4s\) and \(1+\frac{2s}{N}< p<\frac{N}{N-2s}\). Let \(a_{1}\), \(a_{2}\), \(\mu _{1}\) and \(\mu _{2}>0\) be fixed, and let \(\beta _{1}>0\) be defined by
If \(0<\beta <\beta _{1}\), then (1) has a solution \((\tilde{\lambda }_{1},\tilde{\lambda }_{2},\tilde{u},\tilde{v})\) with \((\tilde{u},\tilde{v})\) on the constraint \(H_{a_{1}}\times {H_{a_{2}}}\), such that \(\tilde{\lambda }_{1},\tilde{\lambda }_{2}>0\) and ũ and ṽ are both positive and radial.
For the next result, we introduce a Pohozaev-type constraint as follows:
where
We define a Rayleigh-type quotient as
where
Theorem 1.2
Assume \(0< s<1\), \(2s< N\le 4s\) and \(1+\frac{2s}{N}< p<\frac{N}{N-2s}\). Let \(a_{1}\), \(a_{2}\), \(\mu _{1}\) and \(\mu _{2}>0\) be fixed, and let \(\beta _{2}>0\) be defined by
If \(\beta >\beta _{2}\), then (1) has a solution \((\bar{\lambda }_{1},\bar{\lambda }_{2},\bar{u},\bar{v})\) with \((\bar{u},\bar{v})\) on the constraint \(H_{a_{1}}\times {H_{a_{2}}}\), such that \(\bar{\lambda }_{1},\bar{\lambda }_{2}>0\) and ū and v̄ are both positive and radial. Moreover, \((\bar{\lambda }_{1},\bar{\lambda }_{2},\bar{u},\bar{v})\) is a solution in the sense that
holds.
Remark 1.1
In the system (1) with prescribed \(L^{2}\) constraint, the problem appears to be more complicated as the Lagrange multipliers \(\lambda _{i}\) are also need to be determined simultaneously. The exponent \(2p\in (2+\frac{4s}{N},\frac{2N}{N-2s})\) brings another difficulty as it is \(L^{2}\)-supercritical and \(E(u,v)\) is unbounded from below on the \(L^{2}\) constraint. To overcome these difficulties, the idea introduced by Jeanjean in [3, 16] can be adopted to our system: A minimax argument can be applied to E, allowing one to construct a Palais–Smale sequence on the constraint satisfying the Pohozaev identity in limit sense. This leads to the boundedness of the Palais–Smale sequence. Some a priori estimates on \(\lambda _{i}\) and a Liouville-type result for the fractional Laplacian (Lemma 2.7) ensure \(H^{s}\)-convergence of the Palais–Smale sequence.
We do not know if the results are still true in high dimensions. Since \(u\in H^{s}(\mathbb{R}^{N})\), when the Liouville-type result is applied, we require that \(2\le \frac{N}{N-2s}\) to get our results. It should be interesting to consider the problem in high dimension, even in the Laplacian case.
Remark 1.2
The quantities \(\beta _{1}\) given in (5) and \(\beta _{2}\) given in (10) are complicated, however, when \(N=3\), \(s=1\), \(p=2\), the condition (5) becomes
and (10) becomes
which are the conditions given in [3].
The paper is organized as follows. In Sect. 2, we introduce some important lemmas. In Sect. 3, we prove Theorem 1.1 and in Sect. 4, the proof of Theorem 1.2 is given.
2 Preliminaries
In this section, we will show some facts about the fractional NLS equation, which are used later. First, we need the fractional Gagliardo–Nirenberg–Sobolev inequality, which can be found in [12, 13]. For the reader’s convenience, we give the proof here.
Lemma 2.1
(The fractional Gagliardo–Nirenberg–Sobolev inequality)
Here \(N>2s\), \(0<\alpha <\frac{4s}{N-2s}\) and \(C_{\mathrm{opt}}>0\) denotes the sharp constant (depending on α, N and s).
Proof
We consider the “Weinstein functional” given by
defined for \(u\in H^{s}(\mathbb{R}^{N})\) with \(u\not \equiv 0\). Set \(u^{\lambda ,\mu }=\mu u(\lambda x)\), then we can obtain
Since \(J(u)> 0\), there exists a minimizing sequence \(\{u_{n}\}^{\infty }_{n=1}\subset H^{s}(\mathbb{R}^{N})\cap L^{ \alpha +2}(\mathbb{R}^{N})\). Therefore, it follows that \(0\le \eta =\inf_{u\in H^{s}(\mathbb{R}^{N})\setminus \{0\}} J(u)= \lim_{n\rightarrow \infty }J(u_{n})<\infty \). Since \(\int _{\mathbb{R}^{N}}|(-\Delta )^{\frac{s}{2}} |u||^{2}\,dx\le \int _{ \mathbb{R}^{N}}|(-\Delta )^{\frac{s}{2}}u|^{2}\,dx\), we may assume that \(u_{n}\ge 0\). By Schwarz symmetrization, we may also assume that \(u_{n}=u_{n}(|x|)\). If we take \(\lambda _{n}= ( \frac{\|u_{n}\|^{2}_{2}}{\|(-\Delta )^{\frac{s}{2}}u_{n}\|_{2}^{2}} )^{\frac{1}{2s}}\), \(\mu _{n}= \frac{(\|u\|_{2})^{\frac{2s}{N}-1}}{(\|(-\Delta )^{\frac{s}{2}}u_{n}\|_{2})^{\frac{2s}{N}}}\), then we can obtain a sequence \(v_{n}=u^{\lambda _{n},\mu _{n}}\) satisfying
Since \(\{v_{n}\}\) is bounded in \(H^{s}(\mathbb{R}^{N})\), there exists a v such that \(v_{n}\rightharpoonup v\) weakly in \(H^{s}(\mathbb{R}^{N})\). Because of the radial symmetry of \(v_{n}\), by Sobolev embedding, we can obtain \(v_{n}\rightarrow v\) strongly in \(L^{\alpha +2}(\mathbb{R}^{N})\). By weak convergence, we have \(\|v\|^{2}_{2}\leq 1\) and \(\|(-\Delta )^{\frac{s}{2}}v\|^{2}_{2}\leq 1\). Hence
Therefore \(\|v\|^{2}_{2}=\|(-\Delta )^{\frac{s}{2}}v\|^{2}_{2}=1\), hence \(v_{n}\rightarrow v\) strongly in \(H^{s}(\mathbb{R}^{N})\), this also proves that \(\eta >0\).
Since v minimizes the functional J, it follows that v satisfies the Euler–Lagrange equation
By applying \(\|v\|^{2}_{2}=\|(-\Delta )^{\frac{s}{2}}v\|^{2}_{2}=1\), then we can derive that
for all \(\varphi \in C^{\infty }_{0}(\mathbb{R}^{N})\). Therefore v satisfies
Taking \(\hat{v}=[\eta (\frac{\alpha }{2}+1)]^{-\frac{1}{\alpha }}v\), it satisfies
Let \(C^{-1}_{\mathrm{opt}}=\inf_{u\neq 0}J(u)\). Then the inequality is established. □
It is well known that, when \(N>2s\),
Consider the general fractional Laplacian equation
with \(f\in C^{2}(\mathbb{R})\). Assume that \(u\in {H}^{s}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) is a solution to (13), the Pohozaev identity for (13) is proved in [17].
Theorem 2.2
([17])
Let \(u\in {H}^{s}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})\) be a solution to (13) and \(F(u)\in L^{1}(\mathbb{R}^{N})\). Then
where \(F(u)=\int ^{u}_{0}f(t)\,dt\).
Let us consider the scalar problem
It is shown in [13] that there is a unique positive radial solution \(w_{0}\in H^{s}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N}) \) to (15) for \(1< p<\frac{N}{N-2s}\) and \(N>2s\); see Proposition 3.1 in [13].
We set
By the Pohozaev identity for (15), we can get
Remark 2.1
For the constant \(C_{\mathrm{opt}}\) in Gagliardo–Nirenberg–Sobolev inequality (11) with \(\alpha =2p-2\), it can be evaluated by \(w_{0}\)
which implies that
For \(a,\mu >0\) fixed, we search for \((\lambda ,w)\in \mathbb{R}\times H^{s}(\mathbb{R}^{N})\), with \(\lambda >0\) in \(\mathbb{R}\), solving
Solutions to (19) can be found as the critical points of \(I_{\mu }:H^{s}(\mathbb{R}^{N})\rightarrow \mathbb{R}\), defined by
constrained on the \(L^{2}\)-sphere \(H_{a}:=\{u\in H^{s}(\mathbb{R}^{N}):\int _{\mathbb{R}^{N}}|u|^{2}=a^{2} \}\), and λ appears as the Lagrange multiplier. It is well known that it can be obtained from \(w_{0}\) by scaling.
Lemma 2.3
Equation (19) has a unique positive solution \((\lambda _{a,\mu },w_{a,\mu })\) defined by
Furthermore, \(w_{a,\mu }\) satisfies
Proof
We can directly check that \(w_{a,\mu }\) satisfies the equation (19) with \(\lambda =\lambda _{a,\mu }\) and \(w_{a,\mu }\) is the unique positive radial solution of (19) by [13]. By direct calculation,
we get (22). We have
and combined with (17), we get (21). Combining (21) and (22), we obtain (23). □
Let us introduce the set
When \(1+\frac{2s}{N}< p<\frac{N}{N-2s}\), we have the following lemma.
Lemma 2.4
Assume that \(1+\frac{2s}{N}< p<\frac{N}{N-2s}\), if w is a solution of (19), then \(w\in \mathcal{P}(a,\mu )\). In addition the positive solution of (19) minimizes \(I_{\mu }\) on \(\mathcal{P}(a,\mu )\).
Proof
Let \((w,\lambda )\in H_{a}\times \mathbb{R}\) be a solution of (19). By the Pohozaev identity (14),
and combined with
we get
thus, \(w\in \mathcal{P}(a,\mu )\).
In the following, we prove that the positive solution \(w_{a,\mu }\) of (19) minimizes \(I_{\mu }\) on \(\mathcal{P}(a,\mu )\). For any \(u\in \mathcal{P}(a,\mu )\), by the Gagliardo–Nirenberg–Sobolev inequality (11) and the fact that \(\|u\|_{L^{2}}=a\), we have
Together with (24), we obtain
Therefore, for any \(u\in \mathcal{P}(a,\mu )\),
It is clear that equality in (27) is obtained by \(w_{a,\mu }\) due to the Pohozaev identity (14) and the fact that \(C_{\mathrm{opt}}\) is achieved by \(w_{a,\mu }\) (see [13]). Therefore
□
Lemma 2.5
For \(1+\frac{2s}{N}< p<\frac{N}{N-2s}\), let \(u\in H_{a}\) be arbitrary but fixed. Define \((l\star u)(x):=e^{\frac{Nsl}{2}}u(e^{sl}x)\), then we have
-
(i)
\(\|(-\Delta )^{\frac{s}{2}} (l\star u)\|_{L^{2}}\rightarrow 0\) and \(I_{\mu }(l\star u)\rightarrow 0\) as \(l\rightarrow -\infty \),
-
(ii)
\(\|(-\Delta )^{\frac{s}{2}} (l\star u)\|_{L^{2}}\rightarrow +\infty \) and \(I_{\mu }(l\star u)\rightarrow -\infty \) as \(l\rightarrow +\infty \),
-
(iii)
\(f_{u}(l)=I_{\mu }(l\star u)\) reaches its unique maximum value at \(l(u)\in \mathbb{R}\) with \(l(u)\star u\in \mathcal{P}(a,\mu )\).
Proof
By direct calculation, we have
thus, \(\|(-\Delta )^{\frac{s}{2}} (l\star u)\|_{L^{2}}\rightarrow 0\) as \(l\rightarrow -\infty \), and \(\|(-\Delta )^{\frac{s}{2}} (l\star u)\|_{L^{2}}\rightarrow +\infty \) as \(l\rightarrow +\infty \).
Now we compute \(f_{u}(l)\),
thus, \(I_{\mu }(l\star u)\rightarrow 0\) as \(l\rightarrow -\infty \). Due to \(p>1+\frac{2s}{N}\), we have \(I_{\mu }(l\star u)\rightarrow -\infty \) as \(l\rightarrow +\infty \). (i), (ii) are proved. To show the third claim, by (28), we have
Therefore \(f'_{u}(l)=0\) is equivalent to
So there exists a unique \(l_{0}\in \mathbb{R}\) such that \(f'_{u}(l)|_{l=l_{0}}=0\) and \(l_{0}\star u\in \mathcal{P}(a,\mu )\). Furthermore, we have
Note that
This implies that \(f_{u}(l)\) gets its unique maximum value at \(l_{0}(u)\). If \(u\in \mathcal{P}(a,\mu )\), then, by (30), \(l_{0}=0\). □
When \(\mu _{0}=({C_{0}}/{a^{2}})^{p-1}\) in (19), by Lemma 2.3, \(\lambda _{a,\mu _{0}}=1\), i.e., \(w_{a,\mu _{0}}\) is the unique positive solution of the following equation:
and hence is a minimizer of \(I_{\mu _{0}}\) on \(\mathcal{P}(a,\mu _{0})\). Our next result shows that this level can also be characterized as the infimum of a Rayleigh-type quotient.
Lemma 2.6
where
and \(R_{0}\) is defined in (9).
Proof
If \(u\in \mathcal{P}(a,\mu _{0})\), then
Therefore,
which proves that
On the other hand, for all \(l\in \mathbb{R}\) and \(u\in H_{a}\), direct calculation shows that
By Lemma 2.5, we know that, for \(u\in H_{a}\) arbitrary but fixed, there exists a unique \(l_{0}(u)\in \mathbb{R}\) such that \(l_{0}(u)\star u\in \mathcal{P}(a,\mu _{0})\), and \(I_{\mu _{0}}(l\star u)\) reaches its unique maximum at \(l_{0}(u)\star u\). Hence, for every \(u\in H_{a}\), we have
which proves that
□
Next, we give a Liouville-type result for fractional Laplacian. A similar Liouville-type result for Laplacian can be found in [18].
Lemma 2.7
Let \(u\in {H^{s}(\mathbb{R}^{N})}\) with \(N>2s\),
-
(i)
If u satisfies
$$ \textstyle\begin{cases} (-\Delta )^{s}{u}\geq 0\quad \textit{in } {\mathbb{R}^{N}}, \\ u\in {L^{q}\bigl({\mathbb{R}^{N}}\bigr)},\quad q\in {(0, \frac{N}{N-2s}]}, \\ u\geq 0, \end{cases} $$then \(u\equiv 0\).
-
(ii)
If u satisfies
$$ \textstyle\begin{cases} (-\Delta )^{s}{u}\geq {u^{q}}\quad \textit{in } {\mathbb{R}^{N}}, \\ u\geq 0, \quad \textit{and}\quad q\in (1,\frac{N}{N-2s}], \end{cases} $$then \(u\equiv 0\).
Proof
We prove (i) by contradiction. If \(u\not \equiv 0\), by the maximum principle, we have \(u>0\) in \({\mathbb{R}^{N}}\). Let \(v(x)=\frac{1}{|x|^{N-2s}}u(\frac{x}{|x|^{2}})\), then \(v(x)>0\) in \(\mathbb{R}^{N}\setminus \{0\}\), and \(v(x)\) satisfies
so \((-\Delta )^{s}{v}\geq 0\) in the distribution sense. Since \(u\in H^{s}(\mathbb{R}^{N})\subset {L_{2s}}(\mathbb{R}^{N})\), where
we can see that \(v\in L_{2s}(\mathbb{R}^{N})\). By Theorem 1 in [19], there exists a constant \(C>0\) such that
Therefore, we obtain
For \(q\in (0,\frac{N}{N-2s}]\), we can compute
which is a contradiction to \(u\in L^{q}({\mathbb{R}^{N}})\). So \(u\equiv 0\).
To prove (ii), let φ be the first eigenfunction of
where \(B_{1}(0)\) is the unit ball in \({\mathbb{R}^{N}}\), \(\varphi >0\) in \(B_{1}(0)\) and \(\lambda _{1}>0\) is the first eigenvalue of \((-\Delta )^{s}\) in \(B_{1}(0)\). For any \(R>0\) but fixed, let \(\varphi _{R}(x)=\varphi (\frac{x}{R})\), then
We can compute
in the above, we have use the fact that \((-\Delta )^{s}\varphi _{R}<0\) in \(B_{R}^{c}(0)\). Therefore
When \(q\in (1,\frac{N}{N-2s})\), we have
So we have \(u\equiv 0\).
When \(q=\frac{N}{N-2s}\), we have
with C independent of R by (34), so \(u\in L^{q}(\mathbb{R}^{N})\). By (i), we obtain \(u\equiv 0\). □
3 Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1. We work in a radial setting. That is, we find the critical point of the functional E constrained on \(H^{\mathrm{rad}}_{a_{1}}\times {H^{\mathrm{rad}}_{a_{2}}}\), where for any \(a>0\), we define
and \(H^{s}_{r}(\mathbb{R}^{N})\) is the subset of \(H^{s}(\mathbb{R}^{N})\) containing all the functions which are radial with respect to the origin. We know that \(H^{s}_{r}(\mathbb{R}^{N})\hookrightarrow L^{p}(\mathbb{R}^{N})\) is compact when \(2< p<\frac{2N}{N-2s}\). Due to the Palais principle of symmetric criticality, the critical points of E constrained on \(H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}\) are true critical points of E constrained in the full product \(H_{a_{1}}\times H_{a_{2}}\).
For \(a_{1}\), \(a_{2}\), \(\mu _{1}\) and \(\mu _{2}>0\), let \(\beta _{1}>0\) be defined by (5).
Lemma 3.1
For \(0<\beta <\beta _{1}\),
where \(I_{\mu _{i}}(w_{a_{i},\mu _{1}})\), \(i=1,2\) is defined by (23).
Proof
For \((u_{1},u_{2})\in \mathcal{P}(a_{1},\mu _{1}+\beta )\times \mathcal{P}(a_{2},\mu _{2}+\beta )\}\), we have
by Lemma 2.4. From (23) and (5), it is easy to get, when \(0<\beta <\beta _{1}\),
Therefore,
□
Now we fix \(0<\beta <\beta _{1}\) and choose \(\varepsilon >0\) such that
Denote
and, for \(i=1,2\),
Lemma 3.2
For \(i=1,2\), there exist \(\rho _{i}<0\) and \(R_{i}>0\), depending on ε and β, such that
-
(i)
\(0<\varphi _{i}(\rho _{i})<\varepsilon \) and \(\varphi _{i}(R_{i})\leqslant 0\);
-
(ii)
\(\tilde{\varphi }_{i}(l)>0\) for any \(l<0\), \(\tilde{\varphi }_{i}(0)=0\) and \(\tilde{\varphi }_{i}(l)<0\) for any \(l>0\). In particular, \(\tilde{\varphi }_{i}(\rho _{i})>0\) and \(\tilde{\varphi }_{i}(R_{i})<0\).
Proof
By Lemma 2.4 and Lemma 2.5, we have
thus, \(\varphi _{i}(l)\rightarrow 0^{+}\) as \(l\rightarrow -\infty \), and \(\varphi _{i}(l)\rightarrow -\infty \) as \(l\rightarrow +\infty \). Therefore, there exist \(\rho _{i}<0\) and \(R_{i}>0\), such that \(0<\varphi _{i}(\rho _{i})<\varepsilon \) and \(\varphi _{i}(R_{i})\leqslant 0\).
then
which implies that (ii) holds. □
Let \(Q:=[\rho _{1},R_{1}]\times [\rho _{2},R_{2}]\), and let
We introduce the minimax class
Lemma 3.3
We have
Proof
For every \((u_{1},u_{2})\in H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}\), we have
Then, from Lemma 3.2,
By Lemma 2.3, we have
Then, due to \(l_{1}\star (l_{2}\star w)=(l_{1}+l_{2})\star w\) for every \(l_{1},l_{2}\in \mathbb{R}\) and \(w\in H^{s}(\mathbb{R})\), we have
As a consequence of Lemma 2.5,
Therefore, we have
Similarly, we have
and
Hence, the conclusion of Lemma 3.3 holds. □
Lemma 3.4
For every \(\gamma \in \Gamma \), there exists \((t_{1,\gamma },t_{2,\gamma })\in Q\) such that \(\gamma (t_{1,\gamma },t_{2,\gamma })\in \mathcal{P}(a_{1},\mu _{1}+ \beta )\times \mathcal{P}(a_{2},\mu _{2}+\beta )\).
Proof
For \(\gamma \in \Gamma \), we use the notation \(\gamma (t_{1},t_{2})=(\gamma _{1}(t_{1},t_{2}),\gamma _{2}(t_{1},t_{2})) \in H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}\). Considering the map \(F_{\gamma }:Q\rightarrow \mathbb{R}^{2}\) defined by
From
we deduce that
Now, we will show that \(F_{\gamma }(t_{1},t_{2})=(0,0)\) has a solution in Q for every \(\gamma \in \Gamma \). Since
By Lemma 3.2, we get \((0,0)\notin F_{\gamma _{0}}(\partial Q)\), and \((0,0)\) is the only solution to \(F_{\gamma _{0}}(t_{1},t_{2})=(0,0)\) in Q. It is easy to compute
Now, for any \(\gamma \in \Gamma \), since \(F_{\gamma }(\partial ^{+}Q)=F_{\gamma _{0}}(\partial ^{+}Q)\), therefore, \((0,0)\notin F_{\gamma }(\partial Q)\), we get
Hence, there exists a \((t_{1,\gamma },t_{2,\gamma })\in Q\) such that \(F_{\gamma }(t_{1,\gamma },t_{2,\gamma })=(0,0)\). □
Lemma 3.5
There exists a bounded Palais–Smale sequence \((u_{n},v_{n})\) for E on \({H}^{\mathrm{rad}}_{a_{1}}\times {H}^{\mathrm{rad}}_{a_{2}}\) at the level
satisfying the additional condition
where \(o(1)\rightarrow 0\) as \(n\rightarrow \infty \). Furthermore, there exists \(\bar{C}>0\) such that
and \(u_{n}^{-},v_{n}^{-}\rightarrow 0\) a.e. in \(\mathbb{R}^{N}\) as \(n\rightarrow \infty \).
Proof
The idea comes from [3]. Equation (39) is simply from Lemma 3.4. We consider the augmented functional \(\tilde{E}:\mathbb{R}\times H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}} \rightarrow \mathbb{R}\) defined by \(\tilde{E}(l,u_{1},u_{2}):=E(l\star u_{1},l\star u_{2})\). Let
and
Since, for any \(\gamma (t_{1},t_{2})=(\gamma _{1}(t_{1},t_{2}),\gamma _{2}(t_{1},t_{2})) \in \Gamma \), \((0,\gamma _{1}(t_{1},t_{2}),\gamma _{2}(t_{1},t_{2}))\in \tilde{\Gamma }\), we have \(\tilde{c}\leqslant c\). On the other hand, for any \(\tilde{\gamma }\in \tilde{\Gamma }\) and \((t_{1},t_{2})\in Q\), we have
and \((l(\cdot )\star \gamma _{1}(\cdot ),l(\cdot )\star \gamma _{2}( \cdot ))\in \Gamma \) due to \(\tilde{\gamma }=\tilde{\gamma }_{0}\) on ∂Q, so \(c\leqslant \tilde{c}\). Hence, \(c=\tilde{c}\).
Now take a sequence of \(\{\tilde{\gamma }_{n}\}\subset \tilde{\Gamma }\) such that
We may also assume that \(\tilde{\gamma }_{n}=(l_{n},\gamma _{1,n},\gamma _{2,n})\) satisfies the following two additional properties: for all \((t_{1},t_{2})\in Q\):
-
\(l_{n}(t_{1},t_{2})\equiv 0\),
-
\(\gamma _{1,n}(t_{1},t_{2})\ge 0\), \(\gamma _{2,n}(t_{1},t_{2})\ge 0\), a.e. in \(\mathbb{R}^{N}\).
The first property comes from the fact that
and the second one is the consequence of \(\tilde{E}(l,|u|,|v|)\le \tilde{E}(l,u,v)\) and the definition of c̃.
Applying Theorem 3.2 in [20], there exists a Palais–Smale sequence \((l_{n},u_{n},v_{n})\) for Ẽ on \(\mathbb{R}\times H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}\) at level c̃, such that
-
\(\lim_{n\to +\infty }\tilde{E}(l_{n},u_{n},v_{n})=\tilde{c}=c\),
-
\(\lim_{n\to +\infty }|l_{n}|+\operatorname{dist}((u_{n},v_{n}), \tilde{\gamma }_{n}(Q))=0\),
-
For all \(u,v\in H_{r}^{s}(\mathbb{R}^{N})\) with \(\int _{\mathbb{R}^{N}}u_{n}u\,dx=0\), \(\int _{\mathbb{R}^{N}}v_{n}v\,dx=0\) and \(\forall l\in \mathbb{R}\),
$$ \bigl\langle \tilde{E}'(l_{n},u_{n},v_{n}),(l,u,v) \bigr\rangle =o(1) \bigl( \vert l \vert + \Vert u \Vert _{H^{s}}+ \Vert v \Vert _{H^{s}}\bigr). $$
Take \((l,u,v)=(1,0,0)\), direct calculations gives
From the above, we can get
Since \(l_{n}\to 0\) and \(p>1+\frac{2s}{N}\), we see that there exist \(\bar{C}>0\) and \(C>0\), such that
therefore \((u_{n},v_{n})\) is bounded in \(H^{s}_{r}(\mathbb{R}^{N})\times H^{s}_{r}(\mathbb{R}^{N})\). Using \(l_{n}\to 0\) and (41) again, we conclude that \((u_{n},v_{n})\) satisfies (40). Now take \((l,u,v)=(0,u,v)\) for any \((u,v)\in H^{s}_{r}(\mathbb{R}^{N})\times H^{s}_{r}(\mathbb{R}^{N})\) with \(\int _{\mathbb{R}^{N}}u_{n}u\,dx=0\), \(\int _{\mathbb{R}^{N}}v_{n}v\,dx=0\), due to the boundedness of \((u_{n},v_{n})\) and \(l_{n}\to 0\), it is easy to see that
Therefore, \((u_{n},v_{n})\) is a bounded Palais–Smale sequence for E on \(H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}\) at level c with additional condition (40). Finally, \(u_{n}^{-},v_{n}^{-}\rightarrow 0\) a.e. in \(\mathbb{R}^{N}\) as \(n\rightarrow \infty \) is a simple consequence of \(\gamma _{1,n}(t_{1},t_{2})\ge 0\), \(\gamma _{2,n}(t_{1},t_{2})\ge 0\) and \(\lim_{n\to +\infty } \operatorname{dist}((u_{n},v_{n}),\tilde{\gamma }_{n}(Q))=0\). □
From Lemma 3.5, there exist nonnegative functions ũ, ṽ in \(H_{r}^{s}(\mathbb{R}^{N})\), such that, up to a subsequence,
As a consequence \(E'|_{H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}}(u_{n},v_{n})\rightarrow 0\), there exist two sequences of real number \(\{\lambda _{1,n}\}\) and \(\{\lambda _{2,n}\}\) such that
for every \(g, h\in H^{s}({\mathbb{R}^{N}})\) with \(o(1)\rightarrow 0\), \(\text{as} n\rightarrow \infty \).
Lemma 3.6
Both \(\{\lambda _{1,n}\}\) and \(\{\lambda _{2,n}\}\) are bounded sequences and at least one of them is converging, up to a sequence, to a positive value.
Proof
By using \((u_{n},0)\) and \((0,v_{n})\) as test functions in (43), we get
with \(o(1)\rightarrow 0\), \(\text{as} n\rightarrow \infty \). Hence the boundedness of \(\{\lambda _{i,n}\}\) follows from the boundedness of \(u_{n}\), \(v_{n}\) in \(H^{s}({\mathbb{R}^{N}})\) and in \(L^{2p}({\mathbb{R}^{N}})\). Furthermore, since \((u_{n},v_{n})\) satisfies (40),
therefore by (42),
for \(1+\frac{2s}{N}< p<\frac{N}{N-2s}\) and every n sufficiently large. Therefore, at least one sequence of \(\{\lambda _{i,n}\}\) is positive and bounded away from 0. This shows that at least one sequence of \(\{\lambda _{i,n}\}\) is converging, up to a sequence, to a positive value. □
Next, we consider converging subsequence \(\lambda _{1,n}\rightarrow \tilde{\lambda }_{1}\in {\mathbb{R}}\) and \(\lambda _{2,n}\rightarrow \tilde{\lambda }_{2}\in {\mathbb{R}}\), as \(n\rightarrow \infty \). The sign of \(\tilde{\lambda }_{i}\) plays an important role for the strong convergence of \(u_{n}\), \(v_{n}\) in \(H^{s}(\mathbb{R}^{N})\).
Lemma 3.7
If \(\tilde{\lambda }_{1}>0\) (resp. \(\tilde{\lambda }_{2}>0\)), then \(u_{n}\rightarrow \tilde{u}\) (resp. \(v_{n}\rightarrow \tilde{v}\)) strongly in \(H^{s}(\mathbb{R}^{N})\).
Proof
Let us suppose that \(\tilde{\lambda }_{1}>0\). By the weak convergence of \(u_{n}\) in \(H^{s}(\mathbb{R}^{N})\) and the strong convergence in \(L^{2p}(\mathbb{R}^{N})\), it is easy to get from (43)
with \(o(1)\rightarrow 0\) and \(n\rightarrow \infty \). Since \(\tilde{\lambda }_{1}>0\), this is equivalent to the strong convergence of \(u_{n}\) in \(H^{s}(\mathbb{R}^{N})\). The proof in the case \(\tilde{\lambda }_{2}>0\) is similar. □
Having arrived at the end of this section, we give the proof of Theorem 1.1.
Proof of Theorem 1.1
By the convergence of \(\{\lambda _{1,n}\}\) and \(\{\lambda _{2,n}\}\), and the weak convergence \((u_{n},v_{n})\rightharpoonup (\tilde{u},\tilde{v})\), we see that \((\tilde{\lambda }_{1},\tilde{\lambda }_{2},\tilde{u},\tilde{v})\) is a solution of (1) with at least one \(\tilde{\lambda }_{i}\) positive. We will show that both \(\tilde{\lambda }_{1}\), \(\tilde{\lambda }_{2}\) are positive, hence by Lemma 3.7, \(\tilde{u}\in H_{a_{1}}\), \(\tilde{v}\in H_{a_{2}}\) and the proof is complete.
We prove by contradiction. Without loss of generality, by Lemma 3.7, we may assume that \(\tilde{\lambda }_{1}>0\) and \(\tilde{\lambda }_{2}\le 0\). Since \((\tilde{\lambda }_{1},\tilde{\lambda }_{2},\tilde{u},\tilde{v})\) is a solution of (1) and \(\tilde{u},\tilde{v}\geq 0\), we have
and since \(2s< N\le 4s\), i.e., \(2\le \frac{N}{N-2s}\), from Lemma 2.7(i), we can deduce that \(\tilde{v}\equiv 0\). In particular, this implies that ũ solves
so that \(\tilde{u}=w_{a_{1},\mu _{1}}\in \mathcal{P}(a_{1},\mu _{1})\). However, due to strong convergence of \(u_{n}\), \(v_{n}\) in \(L^{2p}(\mathbb{R}^{N})\), we obtain due to (40),
This is a contradiction with Lemma 3.5. Therefore, both \(\tilde{\lambda }_{1}\), \(\tilde{\lambda }_{2}\) are positive. □
4 Proof of Theorem 1.2
In this section, we prove Theorem 1.2. The proof is divided into two parts. Firstly, we show the existence of a positive solution \((\bar{u},\bar{v})\), and secondly we characterize it as a ground state. The proof of the theorem is based on a mountain pass argument. For \((u,v)\in H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}\), we consider the function
where \(l\star (u,v)=(l\star {u},l\star {v})\). If \((u,v)\in {H^{\mathrm{rad}}_{a_{1}}}\times {H^{\mathrm{rad}}_{a_{2}}}\), then \(l\star (u,v)\in {H^{\mathrm{rad}}_{a_{1}}}\times {H^{\mathrm{rad}}_{a_{2}}}\) for any \(l\in \mathbb{R}\). Similar to Lemma 2.5, we have the following lemma.
Lemma 4.1
Let \((u,v)\in {H^{\mathrm{rad}}_{a_{1}}\times {H^{\mathrm{rad}}_{a_{2}}}}\). Then
The next lemma enlightens the mountain pass structure of the problem.
Lemma 4.2
There exists \(K>0\) sufficiently small such that
where
Proof
By the Gagliardo–Nirenberg–Sobolev inequality (11),
for every \((u,v)\in {H^{\mathrm{rad}}_{a_{1}}}\times {H^{\mathrm{rad}}_{a_{2}}}\), where \(C>0\) depends on \(\mu _{1},\mu _{2},\beta ,a_{1},a_{2}>0\), but not on the choice of \((u,v)\). Now if \((u_{1},v_{1})\in B\) and \((u_{2},v_{2})\in A\) (with K to be determined), we have
provided \(K>0\) is sufficiently small. Furthermore if necessary, we can make K smaller, then
for every \((u_{2},v_{2})\in A\). □
For the next part, we shall introduce a suitable minimax class. Define
Recall from Lemma 2.3 that \(w_{a,\mu }\) is the unique positive radial solution of (19). By Lemma 4.1, there exist \(l_{1}<0\) and \(l_{2}>0\) such that
At last, we define
Similarly to the proof of Lemma 3.5, we derive
Lemma 4.3
There exists a bounded Palais–Smale sequence \((u_{n},v_{n})\) for E on \(H^{\mathrm{rad}}_{a_{1}}\times H^{\mathrm{rad}}_{a_{2}}\) at the level
satisfying the additional condition
with \(o(1)\rightarrow 0\) as \(n\rightarrow \infty \). Furthermore, \(u^{-}_{n},v^{-}_{n}\rightarrow 0\) a.e. in \(\mathbb{R}^{N}\) as \(n\rightarrow \infty \).
Lemma 4.4
Let \(\beta _{2}\) be defined in (10), if \(\beta >\beta _{2}\), then
Proof
By Lemma 2.3, direct computation gives
Therefore, it is easy to get
Due to (10) and (23), if \(\beta >\beta _{2}\), (52) is satisfied. □
Existence of a positive solution at level d
We will prove the existence of positive solution at level d by contradiction. By Lemma 4.3, up to a subsequence, we may assume that
Then it can be easily derived that \((\bar{u},\bar{v})\) is a solution of (1) for some constants \(\bar{\lambda }_{1},\bar{\lambda }_{2}\in \mathbb{R}\). Moreover, Lemma 3.6 and Lemma 3.7 are applicable. We may assume that \(\bar{\lambda }_{1}>0\) and \(u_{n}\to \bar{u}\) strongly in \(H^{s}(\mathbb{R}^{N})\). If \(\bar{\lambda }_{2}\le 0\), we can derive that \(\bar{v}\equiv 0\) and \(\bar{u}=w_{a_{1},\mu _{1}}\) as in the proof of Theorem 1.1. By \(G(u_{n},v_{n})\to 0\) and strong convergence in \(L^{2p}(\mathbb{R}^{N})\), \(d=I_{\mu _{1}}(w_{a_{1},\mu _{1}})\). We can consider the path
Obviously, \(\bar{\gamma }\in \bar{\Gamma }\). Then, by Lemma 4.4,
which is a contradiction. Therefore, \(\bar{\lambda }_{2}>0\) and \(v_{n}\to \bar{v}\) strongly in \(H^{s}(\mathbb{R}^{N})\). This shows that \((\bar{\lambda }_{1},\bar{\lambda }_{2},\bar{u},\bar{v})\) is a solution of (1) with \(\bar{\lambda }_{1},\bar{\lambda }_{2}>0\) and \((\bar{u},\bar{v})\in H_{a_{1}}\times H_{a_{2}}\).
Obviously, we can see that \(G(\bar{u},\bar{v})=0\), i.e., \((\bar{u},\bar{v})\in F\).
Variational characterization of \((\bar{u},\bar{v})\)
In the following, we will prove that
where F and \(\mathcal{R}\) are defined in (6) and (8). Recall the definition of A in (49) and D in (51), let us define
For any \((u_{1},v_{1})\in A^{+}\) and \((u_{2},v_{2})\in D^{+}\), let
Lemma 4.5
The sets \(A^{+}\) and \(D^{+}\) are connected by arcs, so that
for every \((u_{1},v_{1})\in A^{+}\) and \((u_{2},v_{2})\in D^{+}\).
Proof
Equality (53) follows easily once we show that \(A^{+}\) and \(D^{+}\) are connected by arcs ( as \(l*(\bar{u},\bar{v})\) is a path from \(A^{+}\) to \(D^{+}\)). Let \((u_{1},v_{1}),(u_{2},v_{2})\in {H^{\mathrm{rad}}_{a_{1}}}\times {H^{\mathrm{rad}}_{a_{2}}}\) be nonnegative functions such that
for some \(\alpha >0\). For \(l\in \mathbb{R}\) and \(\theta \in [0,\frac{\pi }{2}]\),
Set \(h=(h_{1},h_{2})\), we have \(h_{1}(l,\theta ), h_{2}(l,\theta )\geq 0\) a.e. in \(\mathbb{R}^{N}\). It is not difficult to check that
for all \((l,\theta )\in \mathbb{R}\times [0,\frac{\pi }{2}]\). We can deduce that
Therefore
and it is continuous in \(\theta \in [0,\frac{\pi }{2}]\), so there is a constant \(C>0\) independent of l, θ, such that
Thus we can define the function
for \((l,\theta )\in \mathbb{R}\times [0,\frac{\pi }{2}]\).
Notice that \(\hat{h}(l,\theta )\in {H^{\mathrm{rad}}_{a_{1}}}\times {H^{\mathrm{rad}}_{a_{2}}}\) for every \((l,\theta )\), we see that
with \(c_{0}= \frac{\min \{a^{2}_{1},a^{2}_{2}\}}{\max \{a^{2}_{1},a^{2}_{2}\}}\) and \(c_{1}= \frac{\max \{a^{2}_{1},a^{2}_{2}\}}{\min \{a^{2}_{1},a^{2}_{2}\}}\).
For \(u,v\ge 0\) and \(t\in [0,\frac{\pi }{2}]\),
Therefore, we have, for some constant \(C>0\) independent of l, θ,
for all \((l,\theta )\in \mathbb{R}\times [0,\frac{\pi }{2}]\). Let \((u_{1},v_{1}),(u_{2},v_{2})\in A^{+}\), and let ĥ be as previously. From (55), we can deduce there exists \(l_{0}>0\) such that
for all \(\theta \in [0,\frac{\pi }{2}]\), where K is defined in Lemma 4.2. For the choice of \(l_{0}\), let
It is not difficult to check that \(\sigma _{1}\) is a continuous path connecting \((u_{1},v_{1})\) and \((u_{2},v_{2})\) and lying in \(A^{+}\). For the case that condition (54) is not satisfied, suppose for instance
Then, by Lemma 4.1, there exists \(l_{1}<0\) such that
Therefore, to connect \((u_{1},v_{1})\) and \((u_{2},v_{2})\) by a path in \(A^{+}\), we can at first connect \((u_{1},v_{1})\) with \(l_{1}\star (u_{1},v_{1})\) along arc \(l*(u_{1},v_{1})\), then connect \(l_{1}\star (u_{1},v_{1})\) with \((u_{2},v_{2})\). This shows that \(A^{+}\) is path connected. In a similar way, we can prove that \(D^{+}\) is also path connected. □
From the previous notation,
we define its radial subset and positive radial subset
where
For \((u,v)\in H_{a_{1}}\times H_{a_{2}}\), let us set
where \(l\star {(u,v)}=(l\star u,l\star v)\) for short. Similar to the proof of Lemma 2.5, we have the following lemma.
Lemma 4.6
For every \((u,v)\in H_{a_{1}}\times H_{a_{2}}\), there exists a unique \(l_{(u,v)}\in {\mathbb{R}}\) such that \(l_{(u,v)}\star {(u,v)}\in F\). Moreover, \(l_{(u,v)}\) is the unique critical point of \(\Psi _{(u,v)}\), which is a strict maximum.
Lemma 4.7
We have \(\inf_{F}E=\inf_{F^{+}}E=\inf_{F_{\mathrm{rad}}^{+}}E\).
Proof
We prove the lemma by contradiction. Suppose there exists \((u,v)\in F\) such that
For any \(u\in {H^{s}(\mathbb{R}^{N})}\), since \(\|(-\Delta )^{\frac{s}{2}}|u|\|_{L^{2}}\le \|(-\Delta )^{\frac{s}{2}} u\|_{L^{2}}\), we get \(E(|u|,|v|)\le E(u,v)\) and \(G(|u|,|v|)\le G(u,v)=0\). Thus, there exists \(l_{0}\leq 0\) such that \(G(l_{0}\star (|u|,|v|))=0\). We obtain
Therefore
which contradicts \(l_{0}\leq 0\). Thus \(\inf_{F}E=\inf_{F^{+}}E\).
Next, if there exists \((u,v)\in F^{+}\) such that
For \(u\in {H^{s}(\mathbb{R}^{N})}\), let \(u^{\ast }\) denotes its Schwarz spherical rearrangement. According to the property of Schwarz symmetrization, we have \(E(u^{\ast },v^{\ast })\leq {E(u,v)}\) and \(G(u^{\ast },v^{\ast })\leq {G(u,v)}=0\). Thus there exists \(l_{0}\leq 0\) such that \(G(l_{0}\star (u^{\ast },v^{\ast }))=0\). Similarly, we get
which contradicts \(l_{0}\leq 0\). Thus \(\inf_{F^{+}}E=\inf_{F_{\mathrm{rad}}^{+}}E\). □
Proof of Theorem 1.2
We have showed that \((\bar{\lambda }_{1},\bar{\lambda }_{2},\bar{u},\bar{v})\) is a solution of (1). Since \((\bar{u},\bar{v})\in F^{+}_{\mathrm{rad}}\), we just need to show that
Then \(E(\bar{u},\bar{v})=\inf_{F}E\) follows from Lemma 4.7. Choose any \((u,v)\in F^{+}_{\mathrm{rad}}\). Let us consider the function \(\Psi _{(u,v)}(l)=E(l*(u,v))\). By Lemma 4.1 there exists \(l_{0}\gg 1\) such that \((-l_{0})\star (u,v)\in A^{+}\) and \(l_{0}\star (u,v)\in D^{+}\). Therefore, the continuous path
connects \(A^{+}\) with \(D^{+}\), and by Lemma 4.5 and Lemma 4.6, we can deduce that
Since this holds for all the elementary quantities in \(F^{+}_{\mathrm{rad}}\), we have
Finally, it remains to show that
The proof of (58) is similar to the case for the single equation; see Lemma 2.6. □
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Li, M., He, J., Xu, H. et al. Normalized solutions for a coupled fractional Schrödinger system in low dimensions. Bound Value Probl 2020, 166 (2020). https://doi.org/10.1186/s13661-020-01463-9
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DOI: https://doi.org/10.1186/s13661-020-01463-9