Abstract
We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.
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1 Introduction
In the present paper we study well-posedness, asymptotic behavior of solutions and stability of traveling waves to the Cauchy problem of the system of the Schrödinger equations with derivative nonlinearities on the Euclidean space \(\mathbb {R}^d\):
where \(d\in \{1,2,3\}\) is the spatial dimension, \(\partial _t:=\partial /\partial _t\) is the time derivative, \(\partial _{x_j}:=\partial /\partial _{x_j}\) is the spatial derivative with respect to the spatial variable \(x_j\) with \(j\in \{1,\cdots ,d\}\), \(\nabla :=(\partial _{x_1},\cdots ,\partial _{x_d})\) is the nabla, and \(\Delta :=\sum _{j=1}^d\partial _{x_j}^2\) is the Laplace operator on \(\mathbb {R}^d\). Here \(\alpha \), \(\beta \), \(\gamma \in \mathbb {R}\backslash \{0\}\) are real constants, \(I_{\max }:=(-T_{\min },T_{\max })\) is the maximal existence time interval, \(T_{\max }\in (0,\infty ]\) (resp. \(-T_{\min }\)) is the forward (resp.backward) existence time of the function \((u_1,u_2,u_3)\), \(u_1\), \(u_2\), \(u_3:I_{\max }\times \mathbb {R}^d\rightarrow \mathbb {C}^d\) are unknown d-dimensional complex vector-valued functions and \(u_{1,0}\), \(u_{2,0}\), \(u_{3,0}:\mathbb {R}^d\rightarrow \mathbb {C}^d\) are prescribed d-dimensional complex vector-valued functions. Let \(u_j^{(k)}:I_{\max }\times \mathbb {R}^d\rightarrow \mathbb {C}\) denote the k-th component of \(u_j\) for \(j=1,2,3\) and \(k=1,\cdots ,d\). Namely, \(u_j=(u_j^{(1)},\cdots , u_j^{(d)})\).
The system (1.1) was first derived from the bi-fluid Euler-Maxwell system by Colin and Colin [4, P301] as a model of laser-plasma interaction, in which \(u_1\) and \(u_3\) denote the scattered light and the electronic-plasma wave, respectively and \(u_2\) is the sum of the incident laser field and the gauged Brillouin component.
The system (1.1) is invariant under the following scale transformation
More presicely if U is a solution to (1.1), then so is \(U_{\lambda }\) with the rescaled initial data \(\lambda ^{-1}U_0(\lambda ^{-1}x)\). Moreover, we calculate
where \(\Vert \cdot \Vert _{\dot{H}^s}\) is the s-th order and \(L^2\)-based homogeneous Sobolev norm. If s satisfies \(s=d/2-1\), then \(\Vert U_{\lambda }(0)\Vert _{\dot{H}^s}=\Vert U_0\Vert _{\dot{H}^s}\) for any \(\lambda >0\). Therefore \(d/2-1\) is called the scaling critical Sobolev index and the current cases where \(d=1,2,3\) belong to the \(H^1\)(energy)-subcritical case.
For \(s\in \mathbb {R}\), we denote the inhomogeneous \(L^2\)-based s-th order Sobolev space by \(H^{s}=H^{s}(\mathbb {R}^d)\) endowed with the norm
where \(\langle \cdot \rangle :=1+\vert \cdot \vert \). We introduce the function space \(\mathcal {H}^s:=(H^s(\mathbb {R}^d))^{3d}\) and give the initial data \((u_{1,0},u_{2,0},u_{3,0})\) belonging to \(\mathcal {H}^s\). In [4, 15], the well-posedness of (1.1) in \(\mathcal {H}^s\) was considered by using the energy method, and they obtained the local well-posedness for \(s>d/2+3\) under the condition \(\beta \gamma > 0\). The global well-posedness for sufficiently small initial data was also obtained in [15] if the condition \(\beta +\gamma \ne 0\) holds. On the other hand, the low regularity well-posedness of (1.1) in \(\mathcal {H}^s\) was studied in [11,12,13,14] by using the iteration argument in the space of the Fourier restriction norm. In particular, the well-posedness in the energy space \(\mathcal {H}^1\) was obtained as follows.
Theorem 1.1
(Local well-posedness in \(\mathcal {H}^1\), [11, 14]) Let \(d=1,2\), or 3. Assume that \(\alpha ,\beta ,\gamma \in \mathbb {R}\backslash \{0\}\) satisfy \((\alpha -\gamma )(\beta +\gamma )\ne 0\). Then, (1.1) is locally well-posed in \(\mathcal {H}^1\). The forward (resp. backward) maximal existence time \(T_{\max }\) (resp. \(T_{\min }\)) depends only on \(\Vert U_0\Vert _{\mathcal {H}^1}\). Furthermore, if \(\alpha \), \(\beta \), and \(\gamma \) have same sign, then the local solution for sufficiently small initial data in \(\mathcal {H}^1\) can be extended globally in time.
Remark 1.1
The global well-posedness in Theorem 1.1 was obtained by using the conservation law of the charge and the energy, which will be defined below.
Remark 1.2
For the case \(d=1\), which is \(L^2\)-subcritical case, we can obtain the global well-posedness in \(\mathcal {H}^1\) without smallness condition on initial data by using the Gagliardo-Nirenberg-Sobolev inequality. However for the case \(d=2\) or 3, the global well-posedness without smallness condition for initial data was not known.
Remark 1.3
For the case \(d=4\), which is \(H^1\)-critical case, the global well-posedness and the scattering of solution for small initial data in \(\mathcal {H}^1\) are obtained in [11]
As mentioned above remarks, there are only a few results for global solutions. In [4], the authors proposed open problems, which are existence and stability of solitary waves for the system (1.1). We address these problems in the case of \(d=1,2,3\) in this paper.
For vector valued functions \(f=(f^{(1)},\cdots ,f^{(d)})\) and \(g=(g^{(1)},\cdots ,g^{(d)})\in (H^1(\mathbb {R}^d))^d\), we define the derivatives, gradient, \(L^2\)-norm, \(H^1\)-norm, and \(L^2\)-inner product as
We introduce a new unknown function \(U:I_{\max }\times \mathbb {R}^d\rightarrow \mathbb {C}^d\times \mathbb {C}^d\times \mathbb {C}^d\) and a new initial data \(U_0:\mathbb {R}^d\rightarrow \mathbb {C}^d\times \mathbb {C}^d\times \mathbb {C}^d\) defined respectively by
We define the function space \(\mathcal {H}^s:=(H^s(\mathbb {R}^d))^{3d}\) with the norm \(\Vert \cdot \Vert _{\mathcal {H}^s}\) as
We define the charge \(Q:(L^2(\mathbb {R}^d))^{3d}\rightarrow \mathbb {R}_{\ge 0}\) and the energy \(E:\mathcal {H}^1\rightarrow \mathbb {R}\) for the system (1.1) as
where L and N are the kinetic energy and the potential energy respectively defined by
Furthermore, we define the momentum \(\textbf{P}:\mathcal {H}^1\rightarrow \mathbb {R}^d\) as
where
for \(f=(f_1,\cdots f_d)\in (H^1(\mathbb {R}^d))^d\). We note that Q, E, L, N are real valued and well-defined functionals on \(\mathcal {H}^1\) and \(\textbf{P}\) is vector valued and well-defined functional on \(\mathcal {H}^1\). In particular, Q(U), E(U), and \(\textbf{P}(U)\) are conserved quantities under the flow of (1.1). Namely, if U is a smooth and rapidly decaying solution to (1.1), then it holds that
for all \(t\in I_{\max }\). For the conservation law of the charge and the energy, see Section 7 in [11]. The conservation law of the momentum is implied by the following calculation.
To guarantee the definiteness of the kinetic energy L(U), we assume \(\alpha ,\beta ,\gamma >0\) or \(\alpha ,\beta ,\gamma <0\). Note that we can assume that \(\alpha >0\) without loss of generality. Indeed, if \((u_1,u_2,u_3)\) is a solution to (1.1), then \((\widetilde{u}_1,\widetilde{u}_2,\widetilde{u}_3)\) defined by \(\widetilde{u}_j(t,x):=u_j(-t,-x)\) \((j=1,2,3)\) is a solution to
If \(\alpha <0\), then it suffices to consider this Cauchy problem instead of (1.1). Therefore, we always consider the case \(\alpha ,\beta ,\gamma >0\) from here.
For \(\omega >0\) and \(\textbf{c}\in \mathbb {R}^d\), we define the functionals \(S_{\omega ,\textbf{c}}\), \(K_{\omega ,\textbf{c}}\), and \(L_{\omega ,\textbf{c}}\) on \(\mathcal {H}^1\) as
We note that
and
hold. When \(d=1\), we write \(\textbf{P}(U)=P(U)\), \(\textbf{c}=c\), and \(\textbf{c}\cdot \textbf{P}(U)=cP(U)\).
We define the derivatives \(D_j\) of the functional \(S_{\omega ,\textbf{c}}\) as
where
By the simple calculation, we can see that
for smooth \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\) and smooth \(\psi \). Therefore, solution \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\) to the elliptic system
satisfies
Definition 1.1
(Weak solution to the stationary problem) We say that a triple of functions \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\in ~\mathcal {H}^1\) is a weak solution to the stationary problem (1.7) if it satisfies
for any \(\psi \in (H^1(\mathbb {R}^d))^d\), and then write \(S_{\omega ,\textbf{c}}'(\Phi )=0\).
We note that \(\Phi \in \mathcal {H}^1\) becomes a weak solution to (1.7) if and only if \(\Phi \) satisfies (1.8). In the next section, we will show that a weak solution \(\Phi \in \mathcal {H}^1\) becomes smooth and satisfies the elliptic system (1.7) under the suitable condition for \(\omega \) and \(\textbf{c}\) (\(\omega >\frac{\sigma |\textbf{c}|^2}{4}\)). (See Corollary 2.2 and Remark 2.1 below.) We define the sets of weak solutions to (1.7) as
We call an element \(\Psi \in \mathcal {G}_{\omega ,\textbf{c}}\) a ground state to the elliptic system (1.7).
Remark 1.4
If \(\Psi \in \mathcal {E}_{\omega ,\textbf{c}}\), then it holds \(K_{\omega ,\textbf{c}}(\Psi )=0\). Indeed, if \(\Psi =(\psi _1,\psi _2,\psi _3)\in \mathcal {E}_{\omega ,\textbf{c}}\), then we have
Remark 1.5
For \(\Psi \in \mathcal {H}^1\), we set \(\Psi _{\omega }\) for \(\omega >0\) as
By a simple calculation, we can see that \(\Psi \in \mathcal {E}_{1,\frac{\textbf{c}}{\sqrt{\omega }}}\) is equivalent to \(\Psi _{\omega }\in \mathcal {E}_{\omega ,\textbf{c}}\) and it holds that
These identities imply the equality
Remark 1.6
For \(\theta \in \mathbb {R}\), we define an unitary operator \(\Lambda (\theta )\) as
If we set \(U(t,x)=U_{\omega ,\textbf{c}}(t,x):=\Lambda (\omega t)\Phi (x-\textbf{c}t)\) for \(\Phi =(\phi _1,\phi _2,\phi _3)\in \mathcal {E}_{\omega ,\textbf{c}}\), then U is a solution to the system (1.1) with \(U_0(x)=\Phi (x)\). We call this solution U a two-parameter solitary wave solution.
We study the minimization problem
and define the set of minimizers
We note that if \(S_{\omega ,\textbf{c}}(\Psi )<\mu _{\omega ,\textbf{c}}\) and \(K_{\omega ,\textbf{c}}(\Psi )=0\) hold, then \(\Psi =(\textbf{0},\textbf{0},\textbf{0})\) by the definition of \(\mu _{\omega ,\textbf{c}}\). Furthermore, for \(\eta >0\), we define a subset of \(\mathcal {M}_{\omega ,\textbf{c}}\) as
where \(G:\mathcal {H}^1\rightarrow \mathbb {R}\) is defined by
Then
For \(\alpha ,\beta ,\gamma >0\), we introduce a positive number \(\sigma \) given by
The main results in the present paper are the followings.
Theorem 1.2
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). We assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Then we have \(\mathcal {G}_{\omega ,\textbf{c}}=\mathcal {M}_{\omega ,\textbf{c}}\ne \emptyset \). Namely, there exists at least one of ground state.
According to Theorem 1.1, the local well-posedness of (1.1) in \(\mathcal {H}^1\) is obtained under the condition \((\alpha -\gamma )(\beta +\gamma )\ne 0\). We note that \(\beta +\gamma =0\) does not occur for \(\alpha ,\beta ,\gamma >0\).
Theorem 1.3
(Global well-posedness below the ground state level and uniform boundedness) Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). We assume that \(\alpha -\gamma \ne 0\) and \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). If \(U_0\in \mathcal {H}^1\) satisfies
then \(I_{\max }=\mathbb {R}\), that is, the local solution \(U(t)\in \mathcal {H}^1\) to (1.1) with \(U(0)=U_0\) can be extended globally in time. Moreover for any \(t\in \mathbb {R}\), the estimate \(S_{\omega ,\textbf{c}}(U(t))<\mu _{\omega ,\textbf{c}}\) holds.
When \(d=2\), which is \(L^2\)-critical case, the above global well-posedness result depending on the parameters \(\omega , \textbf{c}\) can be written as the following form without the parameters.
Corollary 1.4
(Global well-posedness below the ground state without parameters in 2d) Let \(d=2\) and \(\alpha ,\beta ,\gamma >0\). We assume that \(\alpha -\gamma \ne 0\). If \(U_0\in \mathcal {H}^1\) satisfies
for some \(\Phi \in \mathcal {M}_{1,\textbf{c}_0}\) and some \(\textbf{c}_0\in \mathbb {R}^2\) with \(|\textbf{c}_0|<\frac{2}{\sqrt{\sigma }}\), namely
then the local solution \(U(t)\in \mathcal {H}^1\) to (1.1) with \(U(0)=U_0\) can be extended globally in time.
Remark 1.7
When \(d=2\), the estimate \(Q(\Phi )-E(\Phi )>0\) holds for \(\Phi \in \mathcal {M}_{1,\textbf{c}_0}\). This can be seen as follows. For \(\Phi \in \mathcal {M}_{1,\textbf{c}_0}\), the estimate
holds by Proposition 2.4 below. Therefore, we have
because \(\mu _{1,\textbf{c}_0}>0\) by Proposition 3.2 below.
Remark 1.8
If \(\Phi \in \mathcal {M}_{1,\textbf{0}}\), then the condition (1.10) is equivalent to \(Q(U_0)<Q(\Phi )\) because \(\Phi \in \mathcal {M}_{1,\textbf{0}}\) with \(d=2\) satisfies \(E(\Phi )=0\) by Proposition 2.4 below.
Remark 1.9
Because the smallness condition for initial data is not necessary except (1.10) in Corollary 1.4, this result is the improvement of global result in Theorem 1.1 for \(d=2\). (See, also Remark 1.2.)
Next we state our result of orbital stability of the set \(\mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) for arbitrary \(\eta >0\), which solves one of the open problems proposed in [4, P301].
Theorem 1.5
(Stability of the set \(\mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\)) Let \(d\in \{1,2\}\) and \(\alpha ,\beta ,\gamma >0\). We assume that \(\alpha -\gamma \ne 0\) and \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). For \(\eta >0\), if \(\mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\ne \emptyset \) holds, then the set \(\mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) is orbitally stable. More precisely, for any \(\epsilon >0\), there exists \(\delta >0\) such that if \(U_0\in \mathcal {H}^1\) satisfies
then the solution U(t) to (1.1) with \(U(0)=U_0\) exists globally in time and satisfies
Next we give orbital stability of the set \(\mathcal {M}_{\omega ,c}\) for 1-dimensional setting as a Corollary of Theorem 1.5.
Corollary 1.6
Let \(d=1\). Then the ground-state set \(\mathcal {M}_{\omega ,c}\) is orbitally stable if |c| is small enough. More precisely, for any \(\omega >0\), there exists \(c_0=c_0(\omega )\in (0,2\sqrt{\frac{\omega }{\sigma }})\) such that if \(|c|\le c_0\), then \(\mathcal {M}_{\omega ,c}\) is orbitally stable.
To prove the stability of the ground-states set, we will use the variational argument for the function \(\mu (\omega ,\textbf{c}):=\mu _{\omega ,\textbf{c}}\) with respect to \((\omega , \textbf{c})\), whose argument is used in [6]. In this argument, the second derivatives of \(\mu \) play an important role. However in our problem, there is a difficulty that the ground state \(\Phi \in \mathcal {M}_{\omega ,\textbf{c}}\) cannot be written explicitly. Therefore, we cannot calculate the second derivatives of \(\mu \) specifically. To avoid this difficulty, we use the argument by using a so-called scaling curve, which was used by Hayashi in [10]. Here the scaling curve is given as a map \(\tau \mapsto \left( (\sqrt{\omega }-\tau )^2,\frac{\textbf{c}}{\sqrt{\omega }} (\sqrt{\omega } -\tau )\right) \). By considering the restriction of \(\mu \) on the scaling curve, we can treat \(\mu \) such as a polynomial function and calculate the second derivative. Thanks to these properties and the Pohozaev’s identity (Proposition 2.4) below, we can give the sufficient condition for ground states to establish the stability. Furthermore, by checking the sufficient condition when \(d=1\) (see, Proposition 4.7), we can obtain the stability of the ground-states set \(\mathcal {M}_{\omega ,\textbf{c}}\) for small \(\textbf{c}\). In particular, our results contain the stability of the set of traveling waves.
We introduce similar results such as existence of ground state and stability of solitary wave for other Schrödinger type equations. Guo-Wu [9] and Colin-Ohta [6] studied orbital stability of a two-parameter family of solitary waves \(\{u_{\omega ,c}\}_{(\omega ,c)\in \mathbb {R}\times \mathbb {R}}\) to the single derivative nonlinear Schrödinger equation in one spatial dimension
where \(u_{\omega ,c}\) for \((\omega ,c)\in \mathbb {R}\times \mathbb {R}\) is the form of
Here \(\phi _{\omega ,c}\) is the unique ground state of the elliptic equation
If \(c^2<4\omega \), then \(\phi _{\omega ,c}\) shows an exponential decay and is explicitly written as
and the charge of \(\phi _{\omega ,c}\) is given by
The orbital stability of those solitons was proved in [9] for \(c<0\) and \(c^2<4\omega \) and in [6] for any \(c^2<4\omega \). See Kwon-Wu [16] for the endpoint case \(c^2=4\omega \). Fukaya-Hayashi-Inui [7] gave a sufficient condition for global existence of solutions to the generalized version of (1.11) by a variational argument (see also [2, 19]). For the system of nonlinear Schrödinger equations without derivative nonlinearities, see [1, 5, 8, 17] and their references. In particular, in [8], similar results as Theorems 1.2 and 1.3 are obtained for the following system of Schrödinger equations:
where \(m,M\in \mathbb {R}\backslash \{0\}\) are constants. In [8], the authors also proved the global well-posedness of (1.12) with \(d=4\) (which is \(L^2\)-critical case) for arbitrary large initial data with modification of oscillations.
We use the shorthand \(A\lesssim B\) to denote the estimate \(A\le CB\) with some constant \(C>0\). The notation \(A\sim B\) stands for \(A\lesssim B\) and \(B\lesssim A\). We give a notation table (Table 1 below) for reader’s convenience.
2 Properties of weak solutions to the stationary problem
In this section we study properties of weak solutions (not necessarily ground state) to stationary problem.
Proposition 2.1
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). If \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\in \mathcal {H}^1\) be a weak solution to (1.7), then \(\Phi \in (W^{2,p}(\mathbb {R}^d))^{3d}\) holds for any \(p\in [1,2]\), where \(W^{2,p}(\mathbb {R}^d)\) denotes the second order \(L^p\)-based Sobolev space, that is, \(W^{2,p}(\mathbb {R}^d):=\{f\in L^p(\mathbb {R}^d)\ |\ \sum _{|\alpha |\le 2}\Vert \partial _x^{\alpha }f\Vert _{L^p}<\infty \}\). In particular, \(\varphi _j \in (H^2(\mathbb {R}^d))^d\) \((j=1,2,3)\) holds.
Proof
We set \(p_0:=\max \{1,\frac{d}{2}\}\) and \(q_0:=2p_0/(2-p_0)\). Then, we have
We note that \(1\le p_0<2\), \(q_0\ge 2\) and
hold because \(d\le 3\). By the Hölder inequality and the Sobolev inequality, we have
By the first equation of (1.7), we obtain
because
holds when \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Therefore, we have \(\varphi _1\in (W^{2,p_0}(\mathbb {R}^d))^d\). By the same argument, we also obtain \(\varphi _2, \varphi _3\in (W^{2,p_0}(\mathbb {R}^d))^d\). Since \(p_0\ge \frac{d}{2}\), the relation \(W^{2,p_0}(\mathbb {R}^d)\subset L^q(\mathbb {R}^d)\) holds for any \(q\in [2,\infty ]\) by the Sobolev embedding theorem. This implies \(\Phi \in (L^q(\mathbb {R}^d))^{3d}\) for any \(q\in [2,\infty ]\).
For any \(p\in [1,2]\), we put
We note that \(q\ge 2\). Therefore, we have \(\varphi _2 \in L^q(\mathbb {R}^d)\) and \( \Vert \varphi _1\Vert _{W^{2,p}} \lesssim \Vert (\nabla \cdot \varphi _3)\varphi _2\Vert _{L^{p}} <\infty \) by the same argument as above. Similarly, we also have \( \Vert \varphi _2\Vert _{W^{2,p}}<\infty ,\ \ \Vert \varphi _3\Vert _{W^{2,p}}<\infty . \)
Corollary 2.2
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). If \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\in \mathcal {H}^1\) be a weak solution to (1.7), then it holds
Proof
We first prove that if \(\varphi _j\in (H^m(\mathbb {R}^d))^d\) \((j=1,2,3)\) for some \(m\in \mathbb {N}\) with \(m\ge 2\), then \(\varphi _j \in (H^{m+1}(\mathbb {R}^d))^d\) \((j=1,2,3)\) holds. We note that \(H^m(\mathbb {R}^d)\) is Banach algebra since \(d<4\) and \(m\ge 2\). Therefore, by the third equation of (1.7) and the Hölder inequality, we have
for \(k=1,\cdots , d\) because \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). This means that \(\varphi _3\in (H^{m+1}(\mathbb {R}^d))^d\). Next, the first equation of (1.7) and the Hölder inequality, we have
for \(k=1,\cdots ,d\) because \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). This means that \(\varphi _1\in (H^{m+1}(\mathbb {R}^d))^d\). Similarly, we also have \(\varphi _2\in (H^{m+1}(\mathbb {R}^d))^d\).
By Proposition 2.1, we have \(\varphi _1\), \(\varphi _2, \varphi _3\in (H^2(\mathbb {R}^d))^d\). Therefore, we obtain \(\varphi _1\), \(\varphi _2, \varphi _3\in (H^m(\mathbb {R}^d))^d\) for any \(m\in \mathbb {N}\) by the induction. \(\square \)
Remark 2.1
By Corollary 2.2, the weak solution \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\in \mathcal {H}^1\) is also a classical solution and derivatives of \(\Phi \) vanish at infinity because the embedding
holds for \(m>\frac{d}{2}+r\). In particular, \(\Phi \in (L^{\infty }(\mathbb {R}^d))^{3d}\) holds and \(\varphi _3\) satisfies
We can also obtain the exponential decay of the solution to (1.7).
Proposition 2.3
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). If \((\varphi _1,\varphi _2,\varphi _3)\in \mathcal {H}^1\) be a weak solution to (1.7), then we have
for
where
Proof
For \(\epsilon >0\), we define \(\theta _{\epsilon }:\mathbb {R}^d\rightarrow \mathbb {R}_{> 0}\) as
We note that \(\theta _{\epsilon }\) and \(\nabla \theta _{\epsilon }\) are bounded on \(\mathbb {R}^d\). Indeed, by a simple calculation, we can see that \(0\le \theta _{\epsilon }(x)\le e^{\frac{p}{\epsilon }}\) and
for any \(x\in \mathbb {R}^d\). We define the real-valued inner product \((\cdot ,\cdot )\) on \((L^2(\mathbb {R}^d))^d\) as \((f,g):=\textrm{Re}(f,g)_{L^2(\mathbb {R}^d)}\) for \(f,g\in (L^2(\mathbb {R}^d))^d\). Because \((\varphi _1,\varphi _2,\varphi _3)\) be a weak solution to (1.7), we have \(\varphi _1\), \(\varphi _2\), \(\varphi _3\in \left( \bigcap _{m=1}^{\infty }H^m(\mathbb {R}^d)\right) ^d\) and
by Corollary 2.2. By using the integration by parts for the first terms of right hand side, these equations can be written as
where
for \(j=1,2,3\) and
Then, we obtain
By (2.4) and the Young inequality, we have
and
where \(\delta _j>0\), \(\eta _j>0\) \((j=1,2,3)\) will be chosen later. Therefore, we get
We choose \(\delta _1,\delta _2,\delta _3>0\) as
Then, we can see that
by (2.3). Therefore, we can choose \(\eta _1,\eta _2,\eta _3>0\) such that \(A_1,A_2,A_3>0\) and \(B_1,B_2,B_3>0\).
On the other hand, by (2.1), there exists \(R>0\) such that for any \(x\in \mathbb {R}^d\) with \(|x|>R\),
holds, where \(M:=\min _{1\le j\le 3}\{A_j,B_j\}>0\) and \(\delta >0\) is small constant. For \(j=1,2\), we divide the integral in the definition of \(J_j\) as
For the first term, by (2.5) and the Young inequality, we obtain
For the second term, by the Hölder inequality and the Sobolev inequality, there exists \(C>0\) such that
For \(j=3\), we note that by the integration by parts,
For the second term, we have
by (2.4) and the same argument as for \(j=1,2\) because \(\varphi _3\) satisfies
As a result, we obtain
By choosing \(\delta =\frac{1}{3+p}\), we have
Let \(\epsilon \rightarrow +0\) and by using the monotone convergence theorem, the desired estimate (2.2) is obtained. \(\square \)
Remark 2.2
Because the solution \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\) to (1.7) satisfies
We also obtain
by using (2.2).
The following proposition (Pohozaev’s identity) plays an important role to prove the stability in Sect. 4.
Proposition 2.4
(Pohozaev’s identity) Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). If \(\Phi =(\varphi _1,\varphi _2,\varphi _3)\in \mathcal {H}^1\) is a weak solution to (1.7), then \(\Phi \) satisfies
Proof
For \(\lambda >0\), we set
Then, we have
Because \(\Phi \) be a weak solution to (1.7), \(\nabla \varphi _j\) and \(\Delta \varphi _j\) \((j=1,2,3)\) are decaying exponentially as \(|x|\rightarrow \infty \) by Proposition 2.3 (and Remark 2.2). Namely, \(\frac{d}{2}\Phi +(x\cdot \nabla ) \Phi \in \mathcal {H}^1\) holds. Therefore, we obtain
On the other hand, we can easily see that
It implies that
and we get the desired identity. \(\square \)
3 Existence of a ground state to the stationary problem
In this section we give a proof of existence of a ground state to (1.7), which addresses a open problem proposed by Colin-Colin [4].
We prove coercivity of the functional \(L_{\omega , \textbf{c}}\) given by (1.4), which plays an important role to prove Proposition 3.2 and Lemma 4.6 below.
Proposition 3.1
(Coecivity of the functional \(L_{\omega ,\textbf{c}}\)) Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Then, there exists a constant \(C=C_{\omega }>0\) independent of \(\textbf{c}\) such that the estimate
holds for any \(U\in \mathcal {H}^1\), where \(L_{\omega , \textbf{c}}\) is defined by (1.4).
Proof
Let \(U=(u_1,u_2,u_3)\in \mathcal {H}^1\) and \(\textbf{c}=(c_1,\cdots ,c_d)\in \mathbb {R}^d\). For parameters \(A_1,A_2,A_3>0\), we set
By a direct computation, for \(j\in \{1,2,3\}\), the following identities hold:
This implies that by (1.4), the following holds:
Because \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\), there exist \(A_1,A_2,A_3>0\) such that
Indeed, if we choose \(A_1,A_2,A_3>0\) as
then (3.1) holds and we can get
where the implicit constant does not depend on \(\textbf{c}\).
Next we prove positivity of \(\mu _{\omega ,\textbf{c}}\) and a property of the Nehari functional \(K_{\omega ,\textbf{c}}\) under \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\).
Proposition 3.2
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Then, the following properties hold:
-
(i)
\(\mu _{\omega ,\textbf{c}}>0\).
-
(ii)
If \(U\in \mathcal {H}^1\backslash \{(\textbf{0},\textbf{0},\textbf{0})\}\) satisfies \(K_{\omega ,\textbf{c}}(U)<0\) (resp \(\le 0)\), then \(L_{\omega ,\textbf{c}}(U)>6\mu _{\omega ,\textbf{c}}\) (resp \(\ge 6\mu _{\omega ,\textbf{c}})\).
Proof
We first prove (i). By (1.5) and the definition of \(\mu _{\omega ,\textbf{c}}\), it holds
Therefore, it suffices to show that there exists \(C_1>0\) such that
holds for any \(\Psi =(\psi _1,\psi _2,\psi _3)\in \mathcal {H}^1\backslash \{(\textbf{0},\textbf{0},\textbf{0})\}\) with \(K_{\omega ,\textbf{c}}(\Psi )=0\). If \(K_{\omega ,\textbf{c}}(\Psi )=0\), then by the definition of \(L_{\omega ,\textbf{c}}\), the Hölder inequality, and the Sobolev inequality, we have
Furthermore by Proposition 3.1, we obtain
This implies that there exists \(\widetilde{C_1}>0\) such that \(\Vert \Psi \Vert _{\mathcal {H}^1}\ge \widetilde{C_1}\) for any \(\Psi \in \mathcal {H}^1\backslash \{(\textbf{0},\textbf{0},\textbf{0})\}\) with \(K_{\omega ,\textbf{c}}(\Psi )=0\). By setting \(C_1:=\widetilde{C_1}^2C(4\omega -\sigma |\textbf{c}|^2)\), where \(C=C_{\omega }>0\) appears in Proposition 3.1, we get \(L_{\omega ,\textbf{c}}(\Psi )\ge C_1\).
Next, we prove (ii). We assume \(U\in \mathcal {H}^1\backslash \{(\textbf{0},\textbf{0},\textbf{0})\}\) and \(K_{\omega ,\textbf{c}}(U)<0\). For \(\lambda \in \mathbb {R}\), by a simple calculation, we have
Therefore, if we set \(\lambda \) as
then \(K_{\omega ,\textbf{c}}(\lambda U)=0\) holds. We can see that \(\lambda \in (0,1)\) from the fact that \(L_{\omega ,\textbf{c}}(U)+3N(U)=K_{\omega ,\textbf{c}}(U)<0\) and \(L_{\omega ,\textbf{c}}(U)>0\). As a result, by using (3.2), we get
If we assume \(K_{\omega ,\textbf{c}}(U)\le 0\) instead of \(K_{\omega ,\textbf{c}}(U)<0\), then there is a possibility that \(\lambda =1\) holds. Therefore, we can only get \(\mu _{\omega ,\textbf{c}}\le \frac{1}{6}L_{\omega ,\textbf{c}}(U)\).
Proposition 3.3
(Existence of a minimizer of \(\mathcal {M}_{\omega ,\textbf{c}}\)) Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Let \(\{U_n\}\subset \mathcal {H}^1\) be a sequence satisfying
Then, there exists \(\{y_n\}\subset \mathbb {R}^d\) such that the sequence \(\{U_n(\cdot -y_n)\}\) has a subsequence which converges to some \(V\in \mathcal {H}^1\backslash \{(\textbf{0},\textbf{0},\textbf{0})\}\) strongly in \(\mathcal {H}^1\). Furthermore it holds that \(V\in \mathcal {M}_{\omega ,\textbf{c}}\). Namely \(\mathcal {M}_{\omega ,\textbf{c}}\) is not empty.
In addition, if \(\{U_n\}\) satisfies
for some \(\eta >0\), where G is defined by (1.9), then it holds that \(V\in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\).
We remark that the latter statement in this proposition will be applied to prove Theorem 1.5 in Sect. 4.2.
To prove Proposition 3.3, we use the following Lieb’s compactness theorem.
Lemma 3.4
(Lieb’s compactness theorem [18]) Let \(\{F_n\}\) be a bounded sequence in \(\mathcal {H}^1\). Assume that there exists \(p\in (0,2^*)\) such that \(\limsup _{n\rightarrow \infty }\Vert F_n\Vert _{L^p}>0\). Then there exist \(\{y_n\}\subset \mathbb {R}^d\) and \(F\in \mathcal {H}^1\backslash \{(\textbf{0},\textbf{0},\textbf{0})\}\) such that \(\{F_n(\cdot -y_n)\}\) has a subsequence that converges to F weakly in \(\mathcal {H}^1\), where \(2^*=\infty \) if \(d=1\) or 2, \(2^*=\frac{2d}{d-2}\) if \(d\ge 3\).
Lemma 3.5
(Special case of Brezis–Lieb’s lemma [3]) Assume that \(\{f_n\}\) be a bounded sequence in \(L^2(\mathbb {R}^d)\) and \(f_n\rightharpoonup f\) weakly in \(L^2(\mathbb {R}^d)\). Then it holds that
Remark 3.1
Lemma 3.5 also holds if we replace \(L^2(\mathbb {R}^d)\) by \((L^2(\mathbb {R}^d))^d\).
Proof of Proposition 3.3
We divide the proof into 6 steps as follows.
Step 1. We prove the boundedness of \(\{U_n\}\) in \(\mathcal {H}^1\). By (1.5), we have
Thus \(\{L_{\omega ,\textbf{c}}(U_n)\}\) is a bounded sequence. This and Proposition 3.1 imply that \(\{U_n\}\) is bounded in \(\mathcal {H}^1\).
Step 2. We prove \(\limsup _{n\rightarrow \infty }\Vert U_n\Vert _{L^4}>0\). We write \(U_n=(u_{1n},u_{2n},u_{3n})\) (\(u_{jn}\in (H^1(\mathbb {R}^d))^d\)). By Step 1, there exists \(C>0\) such that \(\Vert U_n\Vert _{\mathcal {H}^1}\le C\) holds for any \(n\in \mathbb {N}\). Therefore, by the Hölder inequality, we have
Assume \(\lim _{n\rightarrow \infty }\Vert U_n\Vert _{L^4}=0\). Then we obtain \(\lim _{n\rightarrow \infty }N(U_n)=0\). This implies that
This contradicts to (3.4) because \(\mu _{\omega ,\textbf{c}}>0\) by Proposition 3.2 (i). Therefore we get \(\lim _{n\rightarrow \infty }\Vert U_n\Vert _{L^4}\ne 0\). Namely,
Step 3. In this step, we apply Lemma 3.4. By Step 1 and Step 2, \(\{U_n\}\) satisfies the assumptions in Lemma 3.4 for \(p=4\). Therefore, there exist \(\{y_n\}\subset \mathbb {R}^d\) and \(V\in \mathcal {H}^1\backslash \{(\textbf{0},\textbf{0},\textbf{0})\}\) such that \(\{U_n(\cdot -y_n)\}\) has a subsequence \(\{V_{n}\}\) that converges to V weakly in \(\mathcal {H}^1\). We set
We can assume \(V_n\ne V\) for any \(n\in \mathbb {N}\) without loss of generality.
Step 4. We prove
and
First, we show (3.5). As in the proof of Proposition 3.1, we can write
for some positive constants \(C_{1,j}\), \(C_{2,j}\), \(C_{3,j,k}\) and constants \(c_{j,k}\in \mathbb {C}\). Because \(V_n\rightharpoonup V\) weakly in \(\mathcal {H}^1\), the sequences \(\{v_{jn}\}\) and \(\{\partial _kv_{jn}\}\) are bounded in \((L^2(\mathbb {R}^d))^d\), and it holds that
by taking subsequences. Therefore, by applying Lemma 3.5, we get (3.5).
Next, we show (3.6). We can write
Because \(V_n\rightarrow V\) weakly in \(\mathcal {H}^1\), the sequences \(\{v_{1n}\}\), \(\{v_{2n}\}\), and \(\{v_{3n}\}\) are bounded in \((H^1(\mathbb {R}^d))^d\). This implies that the sequences \(\{\nabla \cdot v_{3n}\}\) and \(\{v_{1n}\cdot \overline{v_{2n}}\}\) are bounded in \(L^2(\mathbb {R}^d)\) because
and
hold by the Hölder inequality and the Sobolev inequality. Therefore, by taking subsequences, we obtain
Furthermore, by (3.9) and applying Lemma 3.5, we have
Because \(K_{\omega ,\textbf{c}}(U)=L_{\omega ,\textbf{c}}(U)+3N(U)\), we get (3.6) by (3.5) and (3.10).
Step 5. In this step, we prove \(K_{\omega ,\textbf{c}}(V)\le 0\) by contradiction. We assume \(K_{\omega ,\textbf{c}}(V)>0\). (Then, \(V\ne 0\) by the definition of \(K_{\omega ,\textbf{c}}\).) Because
we have
by (3.6). Therefore, if \(n\in \mathbb {N}\) is large enough, then it holds \(K_{\omega ,\textbf{c}}(V_n-V)<0\). Since \(V_n-V\ne 0\), this and Proposition 3.2 (ii) imply that \(L_{\omega ,\textbf{c}}(V_n-V)>6\mu _{\omega ,\textbf{c}}\) for sufficiently large \(n\in \mathbb {N}\). Because by (3.4),
holds and we obtain
by (3.5). But this leads to a contradiction because by Proposition 3.1 and \(V\ne 0\),
holds.
Step 6. Finally. we prove \(V_n\rightarrow V\) strongly in \(\mathcal {H}^1\) and \(V\in \mathcal {M}_{\omega ,\textbf{c}}\). By Step 5 and Proposition 3.2 (ii), we have \(L_{\omega , \textbf{c}}(V)\ge 6\mu _{\omega ,\textbf{c}}\). On the other hand, by (3.7) and (3.8), we obtain
Therefore, we get \(L_{\omega ,\textbf{c}}(V)\le 6\mu _{\omega ,\textbf{c}}\) by (3.11). As a result, we have \(L_{\omega ,\textbf{c}}(V)= 6\mu _{\omega ,\textbf{c}}\). Furthermore, by Proposition 3.1, (3.5), and (3.11), we obtain
Namely \(V_n\rightarrow V\) strongly in \(\mathcal {H}^1\). Because \(L_{\omega ,\textbf{c}}(V)= 6\mu _{\omega ,\textbf{c}}\) (In particular \(L_{\omega ,\textbf{c}}(V)\le 6\mu _{\omega ,\textbf{c}}\)), we have \(K_{\omega ,\textbf{c}}(V)\ge 0\) by Proposition 3.2 (ii). This and Step 5 imply \(K_{\omega ,\textbf{c}}(V)=0\). Thus, we obtain
by (1.5) and it implies \(V\in \mathcal {M}_{\omega ,\textbf{c}}\).
Now, we assume that the additional condition (3.3) holds. Namely, we assume
By the definitions of Q and \(\textbf{P}\),
holds. We also have
These imply \(G(V)=\limsup _{n\rightarrow \infty }G(U_n)\ge \eta \) and we get \(V\in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\).
Proposition 3.6
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Then, we have \(\mathcal {M}_{\omega ,\textbf{c}}=\mathcal {G}_{\omega ,\textbf{c}}\).
Proof
We first prove \(\mathcal {M}_{\omega ,\textbf{c}}\subset \mathcal {G}_{\omega ,\textbf{c}}\). Let \(\Psi =(\psi _1,\psi _2,\psi _3)\in \mathcal {M}_{\omega ,\textbf{c}}\). Since \(\Psi \) is a minimizer of \(S_{\omega ,\textbf{c}}(\Phi )\) under the condition \(K_{\omega ,\textbf{c}}(\Phi )=0\), there exists a Lagrange multiplier \(\eta \in \mathbb {R}\) such that
Therefore, we have
We also note that the following identities hold:
Because \(\Psi \ne 0\), we obtain \(\partial _{\lambda }K_{\omega ,\textbf{c}}(\lambda \Psi )\big |_{\lambda =1}<0\) by Proposition 3.1. This implies \(\eta =0\) and we get
This means \(\Psi \in \mathcal {E}_{\omega ,\textbf{c}}\). Furthermore, for any \(\Theta \in \mathcal {E}_{\omega ,\textbf{c}}\), we have \(K_{\omega ,\textbf{c}}(\Theta )=0\) by Remark 1.4. Thus, by the definition of \(\mu _{\omega ,\textbf{c}}\), we obtain
Therefore, we get \(\Psi \in \mathcal {G}_{\omega ,\textbf{c}}\).
Next, we prove \(\mathcal {G}_{\omega ,\textbf{c}}\subset \mathcal {M}_{\omega ,\textbf{c}}\). Let \(\Psi \in \mathcal {G}_{\omega ,\textbf{c}}\). By Proposotion 3.3, there exists \(\Phi \in \mathcal {M}_{\omega ,\textbf{c}}\). Because \(\mathcal {M}_{\omega ,\textbf{c}}\subset \mathcal {G}_{\omega ,\textbf{c}}\subset \mathcal {E}_{\omega ,\textbf{c}}\), we have \(\Phi \in \mathcal {E}_{\omega ,\textbf{c}}\). This implies that
On the other hand, by Remark 1.4, we have \(K_{\omega ,\textbf{c}}(\Psi )=0\) since \(\Psi \in \mathcal {G}_{\omega ,\textbf{c}}\subset \mathcal {E}_{\omega ,\textbf{c}}\). Therefore, by the definition of \(\mu _{\omega ,\textbf{c}}\), we get
As a result, we obtain \(S_{\omega ,\textbf{c}}(\Psi )=\mu _{\omega ,\textbf{c}}\) and this implies \(\Psi \in \mathcal {M}_{\omega ,\textbf{c}}\).
Proof of Theorem 1.2
Theorem 1.2 can be proved by combining Proposition 3.3 and Proposition 3.6.
Next we prepare fundamental identities about the ground-state energy level \(\mu _{\omega ,\textbf{c}}\), which will be applied to prove the stability results in Sect. 4.
Proposition 3.7
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Then, we have
Proof
By Propositions 3.3 and 3.6, there exists \(\Psi \in \mathcal {M}_{1,\frac{\textbf{c}}{\sqrt{\omega }}}=\mathcal {G}_{1,\frac{\textbf{c}}{\sqrt{\omega }}}\), which implies that \(\Psi \in \mathcal {E}_{1,\frac{\textbf{c}}{\sqrt{\omega }}}\). Furthermore, we set
Then, we have \(\Psi _{\omega }\in \mathcal {E}_{\omega ,\textbf{c}}\) and by Remark 1.5,
Let \(\Theta \in \mathcal {E}_{\omega ,\textbf{c}}\). We put
Then, we can see that \(\Theta ^{\omega }\in \mathcal {E}_{1,\frac{\textbf{c}}{\sqrt{\omega }}}\) and
On the other hand, because \(\Psi \) is a minimizer of \(\mathcal {M}_{1,\frac{\textbf{c}}{\sqrt{\omega }}}\), we obtain
This implies that by (3.12) and (3.13), \(S_{\omega ,\textbf{c}}(\Psi _{\omega })\le S_{\omega ,\textbf{c}}(\Theta )\). This implies \(\Psi _{\omega }\in \mathcal {G}_{\omega ,\textbf{c}}=\mathcal {M}_{\omega ,\textbf{c}}\). As a result, we get
Proposition 3.8
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). If \(\Phi \in \mathcal {M}_{\omega ,\textbf{c}}\), then it holds that
Proof
Because \(\Phi \) is a minimizer of \(\mu _{\omega ,\textbf{c}}\), it holds
We also note that \(\Phi \) is a weak solution to (1.7) by Proposition 3.6. Therefore, by Proposition 2.4, the identity
holds. The identities (3.15) and (3.16) imply
Hence we obtain
On the other hand, by Proposition 3.7, it holds that \(\mu _{\omega ,\textbf{c}}=\omega ^{2-\frac{d}{2}}\mu _{1,\frac{\textbf{c}}{\sqrt{\omega }}}\). Therefore, we get (3.14).
4 Proof of global well-posednessand stability results
In this section, we give the proofs of Theorems 1.3, 1.5 and Corollaries 1.4, 1.6.
4.1 Proof of global well-posedness
We define the subsets in the energy space \(\mathcal {H}^1\)
We prove that \(\mathcal {A}^{\pm }_{\omega ,\textbf{c}}\) is invariant under the flow of (1.1).
Proposition 4.1
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Then the sets \(\mathcal {A}^{\pm }_{\omega ,\textbf{c}}\) are invariant under the flow of (1.1). More precisely, if \(U_0\) belongs to \(\mathcal {A}^{+}_{\omega ,\textbf{c}}\) (resp \(\mathcal {A}^{-}_{\omega ,\textbf{c}})\), then the solution U(t) of (1.1) with \(U(0)=U_0\) also belongs to \(\mathcal {A}^{+}_{\omega ,\textbf{c}}\) (resp \(\mathcal {A}^{-}_{\omega ,\textbf{c}})\) for all \(t\in I_{\max }\).
Proof
Let \(U_0\in \mathcal {A}^{+}_{\omega ,\textbf{c}}\) and U be the solution to (1.1) with \(U(0)=U_0\) on \(I_{\max }\). Because \(S_{\omega ,\textbf{c}}(U_0)<\mu _{\omega ,\textbf{c}}\) holds and \(S_{\omega ,\textbf{c}}\) is a conserved quantity, we have \(S_{\omega ,\textbf{c}}(U(t))<\mu _{\omega ,\textbf{c}}\) for all \(t\in I_{\max }\). Furthermore, because \(U_0\ne 0\) and Q is a conserved quantity, we obtain \(Q(U(t))=Q(U_0)\ne 0\). It implies that \(U(t)\ne 0\) for all \(t\in I_{\max }\). Therefore, it suffices to show that \(K_{\omega ,\textbf{c}}(U(t))>0\) for all \(t\in I_{\max }\). Assume \(K_{\omega ,\textbf{c}}(U(t_*))<0\) for some \(t_*\in I_{\max }\). Then, by the continuity of \(t\mapsto K_{\omega ,\textbf{c}}(U(t))\), there exists \(t_0\in (0,t_*)\) such that \(K_{\omega ,\textbf{c}}(U(t_0))=0\). This implies \(U(t_0)=0\) because \(S_{\omega ,\textbf{c}}(U(t_0))<\mu _{\omega ,\textbf{c}}\). It is a contradiction. The invariance of \(\mathcal {A}^{-}_{\omega ,\textbf{c}}\) can be proved by the same manner.
Now we prove our global well-posedness results (Theorem 1.3 and Corollary 1.4).
Proof of Theorem 1.3
Let U be a solution on the maximal existence time interval \(I_{\max }\) to (1.1) with \(U(0)=U_0\in \mathcal {H}^1\backslash \{0\}\) satisfying
That is \(U_0\in \mathcal {A}^{+}_{\omega ,\textbf{c}}\). By Proposition 4.1, \(U(t)\in \mathcal {A}^{+}_{\omega ,\textbf{c}}\) holds for any \(t\in I_{\max }\). This implies \(K_{\omega ,\textbf{c}}(U(t))>0\) and by (1.5) we have
Thus by Proposition 3.1, we obtain
for any \(t\in I_{\max }\) and some \(C>0\) independent of t, since \(S_{\omega ,\textbf{c}}\) is a conserved quantity, which implies that \(|I_{\max }|=\infty \) by the blow-up alternative.
Proof of Corollary 1.4
We assume \(\Phi \in \mathcal {M}_{1,\textbf{c}_0}\) with \(|\textbf{c}_0|<\frac{2}{\sqrt{\sigma }}\) and set \(\textbf{c}:=\sqrt{\omega }\ \textbf{c}_0\) for \(\omega >0\). Furthermore, we can assume \(Q(U_0)>0\) because \(U_0=0\) when \(Q(U_0)=0\). Thanks to Theorem 1.3, it is enough to show that
hold for some \(\omega >0\). Because
and \(Q(U_0)>0\), the second inequality in (4.2) holds for sufficiently large \(\omega >0\).
Next, to prove the first inequality in (4.2), we define the function \(f:(0,\infty )\rightarrow \mathbb {R}\) as
Because \(\Phi \in \mathcal {M}_{1,\textbf{c}_0}\), we have
by Proposition 2.4 with \(d=2\). This identity and Proposition 3.7 with \(d=2\) imply that
Therefore, we obtain
This says that if \(Q(U_0)<Q(\Phi )-E(\Phi )\), then \(f(\omega )>0\) holds for sufficiently large \(\omega >0\).
We define the subsets in the energy space \(\mathcal {H}^1\)
The following Proposition will be used to prove the stability of ground states set in next subsection.
Proposition 4.2
Let \(d\in \{1,2,3\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). Then, \(\mathcal {A}^{+}_{\omega ,\textbf{c}}=\mathcal {B}^{+}_{\omega ,\textbf{c}}\) and \(\mathcal {A}^{-}_{\omega ,\textbf{c}}=\mathcal {B}^{-}_{\omega ,\textbf{c}}\) hold, where \(\mathcal {A}^{+}_{\omega ,\textbf{c}}\) and \(\mathcal {A}^{-}_{\omega ,\textbf{c}}\) are defined in (4.1).
Proof
We first prove \(\mathcal {A}^{+}_{\omega ,\textbf{c}}= \mathcal {B}^{+}_{\omega ,\textbf{c}}\). Let \(U\in \mathcal {A}^{+}_{\omega ,\textbf{c}}\). Because \(S_{\omega ,\textbf{c}}(U)<\mu _{\omega ,\textbf{c}}\) and \(K_{\omega ,\textbf{c}}(U)>0\) hold we obtain \( N(U)>-2\mu _{\omega ,\textbf{c}}\) by (1.6). This says \(U\in \mathcal {B}^{+}_{\omega ,\textbf{c}}\). Conversely, let \(V\in \mathcal {B}^{+}_{\omega ,\textbf{c}}\). We assume \(K_{\omega ,\textbf{c}}(V)\le 0\). Then, by Proposition 3.2 (ii), we have \(L_{\omega ,\textbf{c}}(V)\ge 6\mu _{\omega ,\textbf{c}}\). Since \(N(V)>-2\mu _{\omega ,\textbf{c}}\), this and (1.5) imply
which contradicts to \(V\in \mathcal {B}^{+}_{\omega ,\textbf{c}}\). Therefore, we get \(K_{\omega ,\textbf{c}}(V)>0\), which implies \(V\in \mathcal {A}^{+}_{\omega ,\textbf{c}}\).
Next, we prove \(\mathcal {A}^{-}_{\omega ,\textbf{c}}= \mathcal {B}^{-}_{\omega ,\textbf{c}}\). Let \(U\in \mathcal {A}^{-}_{\omega ,\textbf{c}}\). Then, \(K_{\omega ,\textbf{c}}(U)<0\) holds and we have \(L_{\omega ,\textbf{c}}(U)>6\mu _{\omega ,\textbf{c}}\) by Proposition 3.2 (ii). Therefore, we obtain
This says \(U\in \mathcal {B}^{-}_{\omega ,\textbf{c}}\). Conversely, let \(V\in \mathcal {B}^{-}_{\omega ,\textbf{c}}\). Then, by (1.6) and \(S_{\omega ,\textbf{c}}(V)<\mu _{\omega ,\textbf{c}}\), we have
This says \(V\in \mathcal {A}^{-}_{\omega ,\textbf{c}}\).
Remark 4.1
Let \(\alpha \), \(\beta \), \(\gamma >0\), \(\omega >\frac{\sigma |c|^2}{4}\), and \(U_0\in \mathcal {B}^{+}_{\omega ,\textbf{c}}\). Then, we can obtain the global solution to (1.1) with \(U|_{t=0}=U_0\) by Theorem 1.3 and Proposition 4.2. We note that \(U_0\in \mathcal {B}^{+}_{\omega ,\textbf{c}}\) satisfies either
While the case (i) corresponds to the focusing case with small nonlinear effect, the case (ii) corresponds to the defocusing case.
4.2 Proof of stability of ground states set
In this subsection, we write \(\mu _{\omega ,\textbf{c}}=\mu (\omega ,\textbf{c})\). For fixed \((\omega , \textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) with \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\), we define the function \(h=h_{\omega ,\textbf{c}}\) on \(\bigl (-\infty ,\sqrt{\omega }\bigl )\) as
where to obtain the second equality, we have used Proposition 3.7. Namely, the function h is a restriction of \(\mu \) on a so-called scaling curve
We note that
and h is strictly decreasing on \(\bigl (-\infty ,\sqrt{\omega }\bigl )\). By using these identities and (3.14), the following lemma is obtained.
Lemma 4.3
Let \(d\in \{1,2\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). If \(\Phi \in \mathcal {M}_{\omega ,\textbf{c}}\), then it holds that
For \(\eta >0\), we define the function \(F=F_{\omega ,\textbf{c},\eta }\) on \(\bigl (-\infty ,\sqrt{\omega }\bigl )\) as
Lemma 4.4
Let \(d\in \{1,2\}\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\). There exists \(\tau _*=\tau _*(\omega ,\textbf{c},\eta )>0\) such that \(F(\tau )\ge \frac{\eta }{4\omega }\) holds for any \(\tau \in (-\tau _*,\tau _*)\).
Proof
Because the definition of the function h is independent of \(\Phi \in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\), we have
In particular, F is continuous because \(h''(\tau )\) is polynomial. On the other hand, we have
for any \(\Phi \in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) by Lemma 4.3. This implies \(F(0)\ge \frac{\eta }{2\omega }\). Therefore, we obtain the conclusion by the continuity of F on \(\tau =0\).
For \(\tau _0\in \Bigl (0,\sqrt{\omega }\Bigl )\), we set
where the double-signs correspond. The following proposition plays an important role to prove the stability result.
Proposition 4.5
Let \(d\in \{1,2\}\), \(\alpha ,\beta ,\gamma >0\), and \(\eta >0\). Assume that \((\omega ,\textbf{c})\in \mathbb {R}\times \mathbb {R}^d\) satisfies \(\omega >\frac{\sigma |\textbf{c}|^2}{4}\) and \(\mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\ne \emptyset \). Then for any \(\tau _0\in (0,\tau _*)\), there exists \(\delta =\delta (\tau _0,\tau _*,\omega ,\textbf{c},\eta )>0\) such that if \(U_0\in \mathcal {H}^1\) satisfies
then it holds \(U_0\in \mathcal {B}^{+}_{\omega _+,\textbf{c}_+}\cap \mathcal {B}^{-}_{\omega _-,\textbf{c}_-}\), where \(\tau _*>0\) is given in Lemma 4.4 and \(\mathcal {B}^{\pm }_{\omega _{\pm },\textbf{c}_{\pm }}\) are defined in (4.3). Furthermore, such \(\delta \) tends to 0 as \(\tau _0\rightarrow 0\).
Proof
We assume that \(U_0\in \mathcal {H}^1\) satisfies (4.4) with sufficiently small \(\delta \in (0,1)\), which will be chosen later. By the definition of the infimum, there exists \(\Phi _{\omega ,\textbf{c}}\in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) such that
Therefore, we have
and
Here we note that the above implicit constants do not depend on \(U_0\) and \(\Phi _{\omega ,\textbf{c}}\). Indeed, by Proposition 3.1 and (1.5), there exists \(C=C(\omega ,\textbf{c})\) such that
This implies that
We divide the proof into the following two steps:
Step 1. We first prove that for any \(\tau _0 \in (0,\tau _*)\),
By (4.5), (4.6), and Lemma 4.3, we obtain
We can assume that the coefficient which appears in \(O(\delta )\) does not depend on \(\tau _0\) (but depends on \(\tau _*\)) by using \(0<\tau _o<\tau _*\).
On the other hand, by the Taylor expansion, there exists \(\theta \in (-\tau _0, \tau _0)\) such that
Therefore, we obtain
Since \(|\theta |<\tau _0<\tau _*\), by Lemma 4.4, we have \(F(\theta )\ge \frac{\eta }{4\omega }\). Here we choose \(\delta =\delta (\tau _0,\tau _*,\omega ,\textbf{c},\eta )>0\) as
Then we have
Therefore we get (4.7) by the definition of h. We also note that \(\delta \) tends to 0 as \(\tau _0\rightarrow 0\) because the coefficient which appears in \(O(\delta )\) does not depend on \(\tau _0\).
Step 2. Next, we prove that for any \(\tau _0\in (0,\tau _*)\),
Because h is a decreasing function on \(\Bigl (-\infty ,\sqrt{\omega }\Bigl )\), it holds that
for any \(0<\tau <\tau _0\). Since \(K_{\omega ,\textbf{c}}(\Phi _{\omega ,\textbf{c}})=0\), by using (1.6), we have
Therefore, by the third inequality of (4.5), there exists \(C=C(\omega ,\textbf{c})>0\) such that
By choosing \(\delta =\delta (\tau _0,\tau _*,\omega ,\textbf{c})>0\) further smaller as
we have
which is equivalent to (4.8). This completes the proof of the proposition.
For \(n\in \mathbb {N}\), we set
Let \(\tau _*\) be given in Lemma 4.4. Then by Proposition 4.5, for \(n>\frac{1}{\tau _*}\), there exists \(\delta _n=\delta (1/n,\tau _{*},\eta ,\omega ,\textbf{c})>0\) satisfying \(\delta _n\rightarrow 0\) as \(n\rightarrow \infty \), such that if \(U_0\in \mathcal {H}^1\) satisfies
then \(U_0\in \mathcal {B}^{+}_{\omega _{n+},\textbf{c}_{n+}}\cap \mathcal {B}^{-}_{\omega _{n-},\textbf{c}_{n-}}\).
Now we give a proof of the stability of the ground-states set \(\mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\).
Proof of Theorem 1.5
We prove the stability by contradiction. We assume that the ground-states set \(\mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) is unstable. Then, there exists \(\epsilon >0\) such that the following holds:
For each \(n\in \mathbb {N}\) sufficiently large with \(n>\frac{1}{\tau _*}\), there exists \(U_{n,0}\in \mathcal {H}^1\) such that
and one of the following (i) or (ii) are satisfied, where \(\delta _n>0\) is given as above.
-
(i)
The time global solution to (1.1) with initial data \(U_{n,0}\) does not exists.
-
(ii)
There exist a time global solution \(U_n\) to (1.1) with initial data \(U_{n,0}\) and \(t_n>0\) such that
$$\begin{aligned} \inf _{\Phi \in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )}\Vert U_n(t_n)-\Phi \Vert _{\mathcal {H}^1}\ge \epsilon . \end{aligned}$$(4.11)
Because \(U_{n,0}\in \mathcal {B}^{+}_{\omega _{n+},\textbf{c}_{n+}}\cap \mathcal {B}^{-}_{\omega _{n-},\textbf{c}_{n-}}\) (in particular \(U_{n,0}\in \mathcal {B}^{+}_{\omega _{n+},\textbf{c}_{n+}}\)) by (4.10) and Proposition 4.5, there exists a solution to (1.1) with initial data \(U_{n,0}\) globally in time by Theorem 1.3 and Proposition 4.2. Therefore, the case (i) does not occur.
Now we assume the case (ii). By Propositions 4.1 and 4.2, we have \(U_n(t_n)\in \mathcal {B}^{+}_{\omega _{n+},\textbf{c}_{n+}}\cap \mathcal {B}^{-}_{\omega _{n-},\textbf{c}_{n-}}\) because \(U_{n,0}\in \mathcal {B}^{+}_{\omega _{n+},\textbf{c}_{n+}}\cap \mathcal {B}^{-}_{\omega _{n-},\textbf{c}_{n-}}\). This implies
Thus as \(n\rightarrow \infty \) by the continuity of h, we obtain
Furthermore, by (4.10) and the definition of the infimum, there exists \(\Phi _{\omega ,\textbf{c},n}\in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) such that
Since \(S_{\omega ,\textbf{c}}\) is a conserved quantity and \(\Phi _{\omega ,\textbf{c},n}\) is a minimizer of \(\mu _{\omega ,\textbf{c}}\), by (4.13), we have
as \(n\rightarrow \infty \). By using (1.6), (4.12), and (4.14), we get
as \(n\rightarrow \infty \). In addition, because Q and \(\textbf{P}\) are conserved quantities, by (4.13), we obtain
These identities and the relation \(\Phi _{\omega ,\textbf{c},n}\in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) imply
Therefore, by (4.14), (4.15), (4.16), and Proposition 3.3, there exist \(\{y_n\}\subset \mathbb {R}^d\) and \(V\in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\) such that \(\{U_n(t_n,\cdot -y_n)\}\) has a subsequence which converges strongly to V in \(\mathcal {H}^1\). Therefore, by setting \(\Phi _n:=V(\cdot +y_n)\) and taking subsequence, we obtain
This contradicts to (4.11) since \(\Phi _n\in \mathcal {M}_{\omega ,\textbf{c}}^*(\eta )\).
4.3 Stability of ground state with small speed for \(d=1\)
To prove Corollary 1.6, we introduce \(\widetilde{\mathcal {M}}_{\omega ,c_*}\) for \(c_*>0\) defined by
and prepare the following lemma.
Lemma 4.6
Let \(d=1\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,c)\in \mathbb {R}\times \mathbb {R}\) satisfies \(\omega >\frac{\sigma c^2}{4}\). Then the following holds:
-
(i)
There exists \(c_1=c_1(\omega )>0\) such that \(\widetilde{\mathcal {M}}_{\omega ,c_1(\omega )}\) is bounded in \(\mathcal {H}^1\). Namely, it holds that
$$\begin{aligned} A_{\omega }:=\sup _{\Phi \in \widetilde{\mathcal {M}}_{\omega ,c_1(\omega )}} \Vert \Phi \Vert _{\mathcal {H}^1}\in (0,\infty ). \end{aligned}$$ -
(ii)
There exists \(c_2=c_2(\omega )>0\) such that it holds that
$$\begin{aligned}B_{\omega }:=\inf _{\Phi \in \widetilde{\mathcal {M}}_{\omega ,c_2(\omega )}}\omega Q(\Phi )>0. \end{aligned}$$
Proof
We first prove (i). By Theorem 1.2, there exists an element \(\Phi _{\omega }\in \mathcal {M}_{\omega ,0}\). Thus we see that \(A_{\omega }>0\). Because \(K_{\omega , 0}(\Phi _{\omega })=0\) and \(N(\Phi _{\omega })=-2\mu _{\omega ,0}\) by (1.6), for \(\lambda \in \mathbb {R}\), we have
Since \(\mu _{\omega ,0}> 0\) by Proposition 3.2, we can choose \(\lambda \) as \(\lambda =\lambda _0\), where \(\lambda _0:=1+\frac{1}{3\mu _{\omega ,0}}cP(\Phi _{\omega })\). Then, it holds that \(K_{\omega ,c}(\lambda _0 \Phi _{\omega })=0\). Because
if \(|c|< \mu _{\omega ,0}\Vert \Phi _{\omega }\Vert _{\mathcal {H}^1}^{-2}\) holds, then we have \(0<\lambda _0 <2\). Therefore, by the definitions of \(\mu _{\omega ,c}\) and \(\lambda _0\), we get
On the other hand, by (1.5) and Proposition 3.1, it holds that
for any \(\Phi \in \mathcal {M}_{\omega ,c}\), where \(C_{\omega }>0\) is a constant which depends on \(\omega \) but does not depend on c.
By the above argument, we put
Then, for any \(\Phi \in \mathcal {M}_{\omega ,c}\) with \(|c|\le c_1\), we have
Next, we prove (ii). Let \(\Phi \in \mathcal {M}_{\omega ,c}\). By (1.6), the Hölder inequality, and the Sobolev inequality, we obtain
where the constant \(C>0\) does not depend on \(\omega \) and c. Therefore, by (4.17), we have
because \(\Phi \ne 0\) in \(\mathcal {H}^1\). In particular, if \(|c|\le \sqrt{\frac{2\omega }{\sigma }}\), then it holds that \(\frac{2C_{\omega }\omega }{C}\le \Vert \Phi \Vert _{\mathcal {H}^1}\). This implies that
where \(C'=\min \{1,\frac{\alpha }{2},\frac{\beta }{2},\frac{\gamma }{2}\}\). Hence, at least one of
holds. If the former estimate holds, then by setting \(c_2=c_2(\omega ):=\sqrt{\frac{2\omega }{\sigma }}>0\), the proof of (ii) is completed as \(B_{\omega }=\frac{\widetilde{B}_{\omega }}{2}\). On the other hand, if the latter estimate holds, then by Proposition 2.4 with \(d=1\) and \(K_{\omega ,c}(\Phi )=0\), the estimates \(\omega Q(\Phi )=L(\Phi )\ge \frac{\widetilde{B}_{\omega }}{2}\) hold.
Corollary 1.6 follows from Theorem 1.5 and the following proposition.
Proposition 4.7
(Stability of the set \(\mathcal {M}_{\omega ,c}\)) Let \(d=1\) and \(\alpha ,\beta ,\gamma >0\). Assume that \((\omega ,c)\in \mathbb {R}\times \mathbb {R}\) satisfies \(\omega >\frac{\sigma c^2}{4}\). Then, there exist \(\eta _0=\eta _0(\omega )>0\) and \(c_0=c_0(\omega )>0\) such that if \(|c|\le c_0\), then \(\mathcal {M}_{\omega ,c}^*(\eta _0)=\mathcal {M}_{\omega ,c}(\ne \emptyset )\) holds. Namely, the estimate
holds for any \(\Phi \in \mathcal {M}_{\omega ,c}\) with \(|c|\le c_0\).
Proof
Let \(c_1=c_1(\omega )>0\) and \(c_2=c_2(\omega )>\) are the constants given in Lemma 4.6. If \(|c|\le \min \{c_1,c_2\}\), then by Lemma 4.6, the estimates \(\omega Q(\Phi )\ge B_{\omega }\) and \(|P(\Phi )|\le 3\Vert \Phi \Vert _{\mathcal {H}^1}^2\le 3A_{\omega }^2 \) hold for any \(\Phi \in \mathcal {M}_{\omega ,c}\), which implies that \(2\omega Q(\Phi )+2cP(\Phi ) \ge 2B_{\omega }-6|c|A_{\omega }^2\). Here we set
Then the estimate \(\omega Q(\Phi )+cP(\Phi ) \ge \eta _0\) holds for any \(\Phi \in \mathcal {M}_{\omega ,c}\) with \(|c|\le c_0\).
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Acknowledgements
The authors express deep gratitude to Professor Norihisa Ikoma for fruiteful comments. The first author is supported by Grant-in-Aid for Young Scientists Research (No. 21K13825). The second author is supported by JST CREST Grant Number JPMJCR1913, Japan and Young Scientists Research (No. 19K14581), Japan Society for the Promotion of Science.
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Hirayama, H., Ikeda, M. Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities. Calc. Var. 63, 170 (2024). https://doi.org/10.1007/s00526-024-02782-w
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DOI: https://doi.org/10.1007/s00526-024-02782-w