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Ground states for a coupled Schrödinger system with general nonlinearities
Boundary Value Problems volume 2020, Article number: 22 (2020)
Abstract
We study a coupled Schrödinger system with general nonlinearities. By using variational methods, we prove the existence and asymptotic behaviour of ground state solution for the system with periodic couplings. Moreover, we prove the existence and nonexistence of ground state solution for the system with non-periodic couplings via Nehari manifold method. Especially, the ground state solution with both nontrivial components is obtained, and the sign of nontrivial components is considered.
1 Introduction and main results
We study the existence, nonexistence and asymptotic behaviour of ground state solution of the coupled Schrödinger system
where \(a_{i}>0\), \(i=1,2\), \(\lambda \in (-\sqrt{a_{1}a_{2}},0)\cup (0,\sqrt{a _{1}a_{2}})\), \(0< s<1\), \(N>2s\), \(2^{\ast }=\frac{2N}{N-2s}\) and \(2<2q<p<2^{\ast }\). \((-\Delta )^{s}\) stands for fractional Laplacian, see [1, 2]. The coupled Schrödinger system arises from Hartree–Fock theory in Bose–Einstein condensates and nonlinear optics, among other physical problems [3, 4].
Solutions with both nontrivial components \((u,v)\), \(u,v\neq 0\) are called nontrivial solutions. Solutions with both positive components are called positive solutions \((u,v)\), \(u,v>0\). A nontrivial solution is called a ground state solution if its energy is minimum among all nontrivial solutions.
As is well known, there are nonlinear and linear forms of coupling terms for coupled Schrödinger systems. When \(\lambda =0\), Eqs. (1.1) reduce to a Schrödinger system with nonlinear couplings. In [5], the authors studied a Schrödinger system with nonlinear couplings
where \(a>0\) and \(2<2p+2<2^{*}\). In the autonomous case, they proved that if \(b>0\) is large enough, Eqs. (1.2) have a positive ground state solution with both nontrivial components. Similar systems were also studied in [6–9]. When \(b(x)=0\), Eqs. (1.1) reduce to a Schrödinger system with linear couplings. In [10], the authors studied a Schrödinger system with linear couplings. Applying the classical Nehari manifold approach, they proved the existence of ground state solution and multiplicity results. For the other works about linearly coupled system, we refer the readers to [11, 12] and the references therein. When \(\lambda b(x) \neq 0\), Eqs. (1.1) are a Schrödinger system with linear and nonlinear couplings. There are few papers concerning this class of system. The authors in [13–15] proved the existence results of (1.1) with \(f_{1}(u)=f_{2}(u)=u^{3}\), \(q=2\) and \(b(x)=b\). To the best of our knowledge, there is almost no research concerning the system with general nonlinearities.
When \(\lambda =0\) and \(b(x)=0\), Eqs. (1.1) reduce to two scalar equations. The Schrödinger equation with different potentials and nonlinearities is actively studied, see for instance [16–21]. We just mention some results about asymptotic behaviour of ground state solution. Guo and Mederski in [16] studied a Schrödinger equation with sum of periodic and inverse-square potentials as follows:
where \(V(x)\) is periodic. The superlinear and subcritical term f satisfies a weak monotonicity condition. They proved the existence of ground state solution and the asymptotic behaviour of ground state solution in the limit \(\mu \rightarrow 0\). Later in [17, Theorem 1.3], Bieganowski studied the Schrödinger equation
where \(2< q< p<2^{*}\) and the potential functions \(V(x)\) and \(K(x)\) are \(\mathbb{Z}^{N}\)-periodic. The author studied the asymptotic behaviour of ground state solution as \(K(x)\rightarrow 0\) in \(L^{\infty }( \mathbb{R}^{N})\) by using variational methods.
In the presence of general nonlinearities, periodic potentials and nonlinear couplings, we study the asymptotic behaviour of ground state solutions of (1.1) in the limit \(b(x)\rightarrow 0\) in \(L^{\infty }(\mathbb{R}^{N})\). We assume that
- (B):
\(0\leq b(x)\in L^{\infty }(\mathbb{R}^{N})\) is \(\mathbb{Z}^{N}\)-periodic.
The nonlinearities \(f_{i},i=1,2\), satisfy:
- (\(F_{1}\)):
\(f_{i}\in \mathcal{C}^{1}(\mathbb{R})\) and there exist \(c_{1},c_{2}>0\) such that
$$ \bigl\vert f'_{i}(u) \bigr\vert \leq c_{1}\bigl(1+ \vert u \vert ^{p-2} \bigr)\quad \text{and}\quad \bigl\vert f_{i}(u) \bigr\vert \leq c_{2}\bigl(1+ \vert u \vert ^{p-1}\bigr)\quad \text{for all } u\in \mathbb{R}. $$- (\(F_{2}\)):
\(\lim_{|u|\rightarrow 0^{+}} \frac{f_{i}(u)}{|u|}=0\), \(f_{i}(-u)=-f_{i}(u)\) for all \(u\in \mathbb{R}\).
- (\(F_{3}\)):
\(\lim_{|u|\rightarrow +\infty } \frac{F_{i}(u)}{|u|^{2}}\rightarrow +\infty \), where \(F_{i}(u)=\int _{0}^{u}f_{i}(s)\,ds\).
- (\(F_{4}\)):
\(u\mapsto \frac{f_{i}(u)}{|u|^{2q-1}}\) is nondecreasing on \((-\infty ,0)\cup (0,+\infty )\).
To study asymptotic behaviour of ground state solution of (1.1), we introduce the following condition from [17]:
- (\(F_{5}\)):
There exist \(d>0\) and \(2< t\leq p\) such that
$$ f_{i}(u)u-2F_{i}(u)\geq d \vert u \vert ^{t}. $$
We state our main results in what follows.
Theorem 1.1
Suppose that (\(F_{1}\))–(\(F_{4}\)) and (B) are satisfied.
- (i)
Then Eqs. (1.1) have a ground state solutionω, where
$$ \omega =(u,v) \textstyle\begin{cases} u>0 \quad \textit{and}\quad v< 0\quad & \textit{as } {-}\sqrt{a _{1}a_{2}}< \lambda < 0, \\ u>0 \quad \textit{and}\quad v>0& \textit{as } 0< \lambda < \sqrt{a_{1}a_{2}}. \end{cases} $$ - (ii)
Moreover, (\(F_{5}\)) holds, every function in the sequence\((b_{n})\)satisfies (B) and\(b_{n}\rightarrow 0\)in\(L^{\infty }( \mathbb{R}^{N})\)as\(n\rightarrow +\infty \). If\((u_{n},v_{n})\)is a ground state solution of (1.1) with\(b(x)=b_{n}(x)\), then there is a sequence\((z_{n})\subset \mathbb{Z}^{N}\)such that
$$ \bigl(u_{n}(\cdot + z_{n}),v_{n}(\cdot + z_{n})\bigr)\rightarrow (u,v) \quad \textit{strongly in }E, $$where\((u,v)\)is a ground state solution of (1.1) with\(b(x)=0\).
In Theorem 1.1, since we are concerned with (1.1) involving general nonlinearities and nonlinear couplings, moreover \(f_{1}\) and \(f_{2}\) are independent with each other, the problem becomes complicated in applying variational methods. To prove the existence of ground state solution of (1.1), we find a Palais–Smale sequence on Nehari manifold and use concentration compactness argument to deal with the lack of compactness of the sequence in \(\mathbb{R}^{N}\). The proof of (ii) is mainly based on the Nehari manifold method and takes inspiration from [17]. By concentration compactness argument and periodicity of energy functional, we find that there exists a sequence \((z_{n})\subset \mathbb{Z}^{N}\) such that the weak limit of \((u_{n}(\cdot + z_{n}),v_{n}(\cdot + z _{n}))\) is nontrivial and is a ground state solution of (1.1) with \(b(x)=0\). Then, a further evaluation of the least energy functional allows us to get the convergence in (ii).
We also study the existence and nonexistence of ground state solution of (1.1) in the presence of non-periodic couplings. In what follows \(b(x)\) satisfies:
- (\(B_{1}\)):
\(0\leq b(x)\in L^{\infty }(\mathbb{R}^{N})\) and \(b(x)=b_{ \mathrm{per}}(x)+b_{\mathrm{loc}}(x)\), where \(0\leq b_{\mathrm{per}}(x) \in L^{\infty }(\mathbb{R}^{N})\) is \(\mathbb{Z}^{N}\)-periodic and \(b_{\mathrm{loc}}(x)\in L^{\infty }(\mathbb{R}^{N})\cap L^{ \frac{p}{p-2q}}(\mathbb{R}^{N})\) satisfies \(\lim_{|x|\rightarrow \infty }b_{\mathrm{loc}}(x)=0\).
Theorem 1.2
Suppose that (\(F_{1}\))–(\(F_{4}\)) and (\(B_{1}\)) are satisfied.
- (i)
If\(b_{\mathrm{loc}}(x)\geq 0\)for a.e. \(x\in \mathbb{R}^{N}\)and\(b_{\mathrm{loc}}(x)>0\)on a positive measure set, then (1.1) has a ground state solutionω, where
$$ \omega =(u,v) \textstyle\begin{cases} u>0 \quad \textit{and}\quad v< 0 & \textit{as } {-}\sqrt{a _{1}a_{2}}< \lambda < 0, \\ u>0\quad \textit{and}\quad v>0& \textit{as } 0< \lambda < \sqrt{a_{1}a_{2}}. \end{cases} $$ - (ii)
If\(b_{\mathrm{loc}}(x)\leq 0\)for a.e. \(x\in \mathbb{R}^{N}\)and\(b_{\mathrm{loc}}(x)<0\)on a positive measure set, then (1.1) has no ground state solution.
In Theorem 1.2, \(b(x)\) is non-periodic, which brings some difficulties to prove that the weak limit of the obtained PS sequence is nontrivial since the translation of energy functional is not invariant. By comparing its least energy with that in the periodic case, we can deduce that the weak limit is nontrivial. Finally, with concentration compactness argument and direct energy estimation, the existence and nonexistence results are proved under suitable assumptions on the sign of \(b_{\mathrm{loc}}(x)\).
The paper is organized in the following way. In Sect. 2 we present several technical results which will be used throughout this paper. In Sect. 3 we study PS sequences on Nehari manifold. We prove Theorem 1.1 in Sect. 4 and Theorem 1.2 in Sect. 5.
2 Preliminaries
We denote the Hilbert space \(E:=H^{s}(\mathbb{R}^{N})\times H ^{s}(\mathbb{R}^{N})\) endowed with the norm (see [1]) \(\|\omega \|^{2}:=\|(u,v)\|^{2}=\|u\|^{2}+\|v\|^{2}\), where
\(|\cdot |_{p}\) stands for the norm of \(L^{p}(\mathbb{R}^{N})\) and \(|(\cdot ,\cdot )|_{p}=(|\cdot |_{p}^{p}+|\cdot |_{p}^{p})^{ \frac{1}{p}}\) stands for the norm of \(L^{p}(\mathbb{R}^{N})\times L ^{p}(\mathbb{R}^{N})\). It is well known that weak solutions of (1.1) are critical points of functional \(\mathcal{J}(\omega )= \mathcal{J}(u,v): E\rightarrow \mathbb{R}\)
Denote
and Nehari manifold
Assumptions (\(F_{2}\)) and (\(F_{4}\)) imply that
The following lemma is standard and follows from (\(F_{1}\))–(\(F_{2}\)).
Lemma 2.1
For\(\varepsilon >0\), there exists\(C_{\varepsilon }>0\)such that
We need several lemmas for our proof.
Lemma 2.2
For\(\lambda \in (-\sqrt{a_{1}a_{2}},\sqrt{a_{1}a_{2}})\setminus \{0\}\), there holds
Proof
Since \(\lambda \in (-\sqrt{a_{1}a_{2}},\sqrt{a_{1}a_{2}})\setminus \{0\}\), then \(0<\frac{|\lambda |}{\sqrt{a_{1}a_{2}}}<1\) and
It follows that \(\|\omega \|^{2}-2\lambda \int _{\mathbb{R}^{N}}uv\,dx \geq (1-\frac{|\lambda |}{\sqrt{a_{1}a_{2}}})\|\omega \|^{2}\). The proof of \(\|\omega \|^{2}-2\lambda \int _{\mathbb{R}^{N}}uv\,dx\leq (1+\frac{| \lambda |}{\sqrt{a_{1}a_{2}}})\|\omega \|^{2}\) is analogous. □
Lemma 2.3
Suppose that (\(F_{1}\))–(\(F_{4}\)) are satisfied and a potential function\(b(x)\)satisfies (B) or (\(B_{1}\)), one has\(\beta :=\inf_{\omega \in \mathcal{N}}{\|\omega \|}>0\).
Proof
Let \(\omega _{n}\in \mathcal{N}\) be such that \(\|\omega _{n}\|\rightarrow 0\), then
which implies that
for a constant \(C>0\). Let \(\varepsilon >0\) be such that \(1-\varepsilon C>0\), then
It is a contradiction. Hence \(\inf_{\omega \in \mathcal{N}} {\|\omega \|}>0\). □
Lemma 2.4
Suppose that (\(F_{1}\))–(\(F_{4}\)) are satisfied, and a potential function\(b(x)\)satisfies (B) or (\(B_{1}\)), then:
- (\(A_{1}\)):
There exists\(r>0\)such that\(a:=\inf_{\|\omega \|=r} \mathcal{J}(\omega )>\mathcal{J}(0)=0\);
- (\(A_{2}\)):
For any\(\omega \in E\setminus \{(0,0)\}\), there exists\(t>0\)such that\(\mathcal{J}(t\omega )<0\);
- (\(A_{3}\)):
For\(t\in (0,\infty )\setminus \{1\}\)and\(\omega \in \mathcal{N}\), there holds
$$ \varphi (t):=\frac{t^{2}-1}{2}\mathcal{I}'(\omega )\omega - \mathcal{I}(t\omega )+\mathcal{I}(\omega )< 0; $$- (\(A_{4}\)):
For any\(\omega \in E\setminus \{(0,0)\}\), there exists a unique number\(t>0\)such that\(t\omega \in \mathcal{N}\)and\(\mathcal{J}(t\omega )=\max_{r\geq 0}\mathcal{J}(r \omega )\).
Proof
(\(A_{1}\)) Applying the fractional Sobolev embedding theorem [1] and Lemma 2.1, there exists \(C>0\) such that
Hölder’s inequality implies that
Let \(r,C_{1}>0\) for \(\|\omega \|\leq r\) and r be sufficiently small, we have
For \(\|\omega \|=r\), it suffices to show that
(\(A_{2}\)) For any \(\omega \in E\setminus \{(0,0)\}\) and \(t>0\), by using Fatou’s lemma and (\(F_{3}\)), we have
which implies that
Hence \(\mathcal{J}(t\omega )\rightarrow -\infty \) as \(t\rightarrow + \infty \).
(\(A_{3}\)) For \(\omega \in \mathcal{N}\) and \(t>0\), let
obviously, \(\varphi '(t)=t\mathcal{I}'(\omega )\omega -\mathcal{I}'(t \omega )\omega \). It follows from Lemma 2.3 that
We have
In view of (\(F_{4}\)), if \(t<1\), then
While for \(t>1\), we have \(\varphi '(t)=t\mathcal{I}'(\omega )\omega - \mathcal{I}'(t\omega )\omega <0\). Hence \(\varphi (t)<\varphi (1)=0\) for \(t\in (0,+\infty )\setminus \{1\}\).
(\(A_{4}\)) In view of (\(A_{1}\)) and (\(A_{2}\)), for any \(\omega \in E\setminus \{(0,0)\}\), there exists a maximum point \(t_{\mathrm{max}}\) of \(t\mapsto \mathcal{J}(t\omega )\) such that \(\mathcal{J}'(t_{\mathrm{max}}\omega )\omega =0\) and \(t_{\mathrm{max}} \omega \in \mathcal{N}\).
For any \(\omega \in \mathcal{N}\) and \(t\in (0,+\infty )\setminus \{1\}\), we have
□
Lemma 2.5
\(\mathcal{J}\)is coercive on\(\mathcal{N}\), i.e. there is a sequence\((\omega _{n})\subset \mathcal{N}\)such that\(\mathcal{J}(\omega _{n}) \rightarrow +\infty \)as\(\|\omega _{n}\|\rightarrow +\infty \).
Proof
Let \((\omega _{n})\subset \mathcal{N}\) be a sequence such that \(\|\omega _{n}\|\rightarrow +\infty \) as \(n\rightarrow +\infty \). From (2.2), we find
□
The Nehari manifold \(\mathcal{N}\) has the following properties.
Proposition 2.6
-
(i)
\(\mathcal{N}\subset E\)is a\(\mathcal{C}^{1}\)-manifold;
-
(ii)
ωis a nonzero free critical point of\(\mathcal{J}\)if and only ifωis a critical point of\(\mathcal{J}\)constrained on\(\mathcal{N}\);
-
(iii)
If\((\omega _{n})\)is a\((PS)\)sequence for\(\mathcal{J}|_{ \mathcal{N}}\), then\(\omega _{n}\)is a\((PS)\)sequence for\(\mathcal{J}\).
Proof
(i) For \(\omega \in \mathcal{N}\), we denote
Let \(\varphi _{i}(s):=\frac{f_{i}(s)}{s^{2q-1}}\) for \(s>0\). In view of (\(F_{4}\)), we have \(\frac{d\varphi _{i}(s)}{ds}\geq 0\), i.e. \(f_{i}'(s)s^{2q-1}-(2q-1)f_{i}(s)s^{2q-2}\geq 0\) for \(s>0\), which implies
Assume \(s<0\), then \(-s>0\) and \(-f_{i}(-s)s-f_{i}'(-s)s^{2}\leq (2q-2)f _{i}(-s)s\) for \(s<0\), in view of (\(F_{2}\)), we find
It is clear that
It follows from Lemma 2.3 that
then
Hence \(\mathcal{N}\subset E\) is a \(\mathcal{C}^{1}\)-manifold.
(ii) If \(\omega \neq (0,0)\) is a critical point of \(\mathcal{J}\), then \(\mathcal{J}'(\omega )=0\) and \(\omega \in \mathcal{N}\). If \(\omega \in \mathcal{N}\) is a critical point of \(\mathcal{J}\) on \(\mathcal{N}\), by applying the Lagrange multiplier theorem, one has \(\mathcal{J}'(\omega )=\delta \xi '(\omega )\) and \(\mathcal{J}'(\omega )\omega =\delta \xi '(\omega )\omega \) for \(\delta \in \mathbb{R}\). From (2.5) we deduce that \(\delta =0\) and \(\mathcal{J}'(\omega )=0\).
(iii) Let \((\omega _{n})\subset \mathcal{N}\) be a \((PS)\) sequence of \(\mathcal{J|_{N}}\), then
which implies \((\omega _{n})\) is bounded in E. For some \(\delta _{n} \in \mathbb{R}\), we have
thus \(\delta _{n}\xi '(\omega _{n})\omega _{n}+\circ (1)=\mathcal{J}'( \omega _{n})\omega _{n}=0\). From (2.5) we deduce that \(\delta _{n}\rightarrow 0\). In view of (2.6), we get \(\mathcal{J}'( \omega _{n})\rightarrow 0\). □
3 Palais–Smale sequences on Nehari manifold
In this section, \(f_{i}\) satisfies (\(F_{1}\))–(\(F_{4}\)), a potential function \(b(x)\) satisfies (B) or (\(B_{1}\)).
Lemma 3.1
There exists a bounded sequence\((u_{n},v_{n})\subset \mathcal{N}\)such that\(\mathcal{J}(u_{n},v_{n})\rightarrow c\)and\(\mathcal{J}'(u_{n},v _{n})\rightarrow 0\)as\(n\rightarrow +\infty \).
Proof
It follows from Lemma 2.3 and Lemma 2.5 that \(\mathcal{J}\) is bounded from below on \(\mathcal{N}\). By using Ekeland’s variational principle [22], there exists a sequence \((u_{n},v_{n})\subset \mathcal{N}\) such that
Hence \(\mathcal{J}(u_{n},v_{n})\rightarrow \inf_{\mathcal{N}}\mathcal{J}(u,v)=c\) as \(n\rightarrow +\infty \). It follows that
and
For a fixed \((y,z)\in E\) and \(\|(y,z)\|\leq 1\), we denote
Obviously, \(G_{n}(0,0)=\mathcal{J}'(u_{n}, v_{n})(u_{n}, v_{n})=0\). In view of (2.5), we have
By implicit function theorem, there exist \(C^{1}\) functions \(t_{n}(s):(-\delta _{n},\delta _{n})\rightarrow \mathbb{R}\) such that \(t_{n}(0)=0\) and
Differentiating \(G_{n}(s,t_{n}(s))\) in s at \(s=0\), we have
Combining Lemma 2.3 and (2.5), we get
It is clear that
By using Hölder’s inequality, embedding theorem and (3.3), we find
In view of (\(F_{1}\)), Lemma 2.1 and (3.3), we have
Moreover, we deduce that
It follows from (3.8)–(3.11) that
Combining (3.6), (3.7) and (3.12), we get
Denote
From (3.4) and (3.5), we find \((y,z)_{n,s}\in \mathcal{N}\) for \(s\in (-\delta _{n},\delta _{n})\). It follows from (3.1) that
Applying Taylor’s expansion on the left-hand side of (3.15), we get
where \(r(n,s)=o(\|(\bar{y},\bar{z})_{n,s}\|)\) as \(|s|\rightarrow 0\). Combining (3.3), (3.13), (3.14) and \(t_{n}(0)=0\), we find
where \(C_{6}\) is independent of n. It follows that \(r(n,s)=o(|s|)\) as \(|s|\rightarrow 0\). From (3.15), (3.16) and (3.17), we get
Hence \(\mathcal{J}'(u_{n},v_{n})\rightarrow 0\) as \(n\rightarrow + \infty \). □
From (iii) of Proposition 2.6 and Lemma 3.1, we get that \((\omega _{n})\) is bounded in E and \(\mathcal{J}'(\omega _{n})\rightarrow 0\). Hence there exists a subsequence of \((\omega _{n})\) such that \((u_{n},v_{n})\rightharpoonup (u_{0},v_{0})\) in E. Then we have the following result.
Lemma 3.2
Suppose\(\omega _{n}\rightharpoonup \omega _{0}\)inEand\(\mathcal{J}'(\omega _{n})\rightarrow 0\), then\(\mathcal{J}'(\omega _{0})=0\).
Proof
For any \(\phi =(\varphi ,\psi )\), φ, \(\psi \in C_{0} ^{\infty }(\mathbb{R}^{N})\), we have
Up to a subsequence, we have
The weak convergence \(\omega _{n}\rightharpoonup \omega _{0}\) implies that \(\langle (u_{n},v_{n}),(\varphi ,\psi )\rangle \rightarrow \langle (u _{0},v_{0}),(\varphi ,\psi )\rangle \), \(\int _{\mathbb{R}^{N}}u_{n} \psi \,dx\rightarrow \int _{\mathbb{R}^{N}}u_{0}\psi \,dx\) and \(\int _{\mathbb{R}^{N}}v_{n}\varphi \,dx\rightarrow \int _{\mathbb{R}^{N}}v _{0}\varphi \,dx\).
Let \(K\subset \mathbb{R}^{N}\) be a compact set containing supports of φ, ψ, then \((u_{n},v_{n})\rightarrow (u_{0},v _{0})\) in \(L^{t}(K)\times L^{t}(K)\) for \(1\leq t<2^{*}\). By [23, Theorem 4.9], there exist \(l_{K}(x)\in L^{2q}(K)\) and \(m_{K}(x)\in L^{2q}(K)\) such that \(|u_{n}(x)|\leq l_{K}(x)\) and \(|v_{n}(x)|\leq m_{K}(x)\) for a.e. \(x\in K\). Let \(h_{K}(x):=l_{K}(x)+m _{K}(x)\) for \(x\in K\), then \(h_{K}(x)\in L^{2q}(K)\) and
Hence \(b(x)|u_{n}|^{q-2}u_{n}|v_{n}|^{q}\varphi \leq b(x)h_{K}^{2q-1}| \varphi |\) for a.e. \(x\in K\), and
Applying Lebesgue’s dominated convergence theorem, we deduce that
By similar arguments as above and Lemma 2.1, we deduce
It follows from (3.19) that
Hence \(\mathcal{J}'(u_{0},v_{0})=0\). □
We introduce the vanishing lemma from [24].
Lemma 3.3
([24, Lemma 2.4])
Assume that\(\{u_{k}\}\)is a bounded sequence in\(H^{s}(\mathbb{R}^{N})\), which satisfies
Then\(u_{k}\rightarrow 0\)strongly in\(L^{r}(\mathbb{R}^{N})\)for every\(2< r<\frac{2N}{N-2s}\).
Lemma 3.4
Assume that\(\{\omega _{n}\}\)is a PS sequence constrained on\(\mathcal{N}\), which satisfies
then\(\|\omega _{n}\|\rightarrow 0\).
Proof
Combining Lemma 3.3 and (3.21), we get \(u_{n},v_{n} \rightarrow 0\) in \(L^{r}(\mathbb{R}^{N})\) for \(2< r<2^{*}\). For \(\omega _{n}\subset \mathcal{N}\), we have
It is clear that
Let \(\varepsilon \rightarrow 0\), we have \(|\int _{\mathbb{R}^{N}}f_{1}(u _{n})u_{n}\,dx|\rightarrow 0\). Moreover,
It follows from (3.22) that \(\|\omega _{n}\|\rightarrow 0\). □
4 Ground states of a Schrödinger system with periodic couplings
We prove (i) and (ii) of Theorem 1.1 in Sects. 4.1–4.2, respectively.
4.1 Existence
Step 1: We find\((u_{0},v_{0})\in E\)such that\(\mathcal{J}'(u_{0},v _{0})=0\).
In view of Lemma 3.1, there exists a bounded \((PS)_{c}\)-sequence of \(\mathcal{J}\) constrained on \(\mathcal{N}\), i.e. a sequence \({\omega _{n}}\subset \mathcal{N}\) such that \(\mathcal{J}( \omega _{n})\rightarrow c\) and \((\mathcal{J}|_{\mathcal{N}})'(\omega _{n})\rightarrow 0\). It follows from (iii) of Proposition 2.6 that \(\mathcal{J}'(\omega _{n})\rightarrow 0\). In view of Lemma 3.2, up to a subsequence, then
and \(\mathcal{J}'(u_{0},v_{0})=0\).
Step 2: We check whether\((u_{0},v_{0})\neq (0,0)\).
Suppose
It follows from Lemma 3.4 that \(\|(u_{n},v_{n})\|\rightarrow 0\). We get a contradiction with respect to Lemma 2.3. By Lions’ lemma [25] there exists \((y_{n})\subset \mathbb{R}^{N}\) such that
We assume, without loss of generality, that
For each \(y_{n}\in \mathbb{R}^{N}\), we will find \(z_{n}\in \mathbb{Z} ^{N}\) such that \(B(y_{n},1)\subset B(z_{n},1+\sqrt{N})\), then
Since \(\mathcal{J}\) and \(\mathcal{N}\) are invariant under translations of the form \(\omega \mapsto \omega (\cdot -k)\) with \(k\in \mathbb{Z} ^{N}\), we may assume that \((z_{n})\) is bounded in \(\mathbb{Z}^{N}\). It is clear that \(u_{0}\neq 0\) by \(u_{n}\rightarrow u_{0}\) in \(L_{ \mathrm{loc}}^{2}(\mathbb{R}^{N})\). Hence \(\omega _{0}=(u_{0},v_{0}) \neq (0,0)\), \((u_{0},v_{0})\in \mathcal{N}\) and \(\mathcal{J}(u_{0},v _{0})\geq c\).
Step 3: We find\((u',v')\)such that\(\mathcal{J}'(u',v')=0\)and\(\mathcal{J}(u',v')=c\), where\(u'>0\)and\(v'<0\)as\(\lambda \in (-\sqrt{a_{1}a_{2}},0)\),\(u'>0\)and\(v'>0\)as\(\lambda \in (0,\sqrt{a_{1}a_{2}})\).
Applying Fatou’s lemma, we get
From the above computations, we find that \(\mathcal{J}(u_{0},v_{0})=c\). Hence \((u_{0},v_{0})\neq (0,0)\) is a ground state solution of (1.1).
Case 1. \(\lambda \in (-\sqrt{a_{1}a_{2}},0)\).
It is clear that \(\|(|u_{0}|,-|v_{0}|)\|\leq \|(u_{0},v_{0}) \|\). By (\(A_{4}\)) of Lemma 2.4, there exists \(t>0\) such that \((t|u_{0}|,-t|v_{0}|)\in \mathcal{N}\) and \(\mathcal{J}(t|u_{0}|,-t|v _{0}|)\geq c\), then
Let \((u',v'):=(t|u_{0}|,-t|v_{0}|)\), \(u'\geq 0\) and \(v'\leq 0\), we get that \((u',v')\) is a ground state solution of (1.1) by (ii) of Proposition 2.6. It follows from (1.1) that
and
In view of (1.1), if \(u'=0\), then \(v'=0\). Hence \(u',v'\neq 0\) by \((u',v')\neq (0,0)\). Applying the strong maximum principle [26] to each equality of (1.1), we get that \((u',v')\), \(u'>0\) and \(v'<0\) is a ground state solution of (1.1).
Case 2. \(\lambda \in (0,\sqrt{a_{1}a_{2}})\).
There exists \(t'>0\) such that \((t'|u_{0}|,t'|v_{0}|)\in \mathcal{N}\) and \(\mathcal{J}(t'|u_{0}|,t'|v_{0}|)\geq c\). We deduce that
By similar arguments in Case 1, we get that \((u',v')\), \(u'>0\) and \(v'>0\) is a ground state solution of (1.1). This completes the proof of (i) of Theorem 1.1.
4.2 Asymptotic behaviour of ground states as \(b_{n}\rightarrow 0\) in \(L^{\infty }(\mathbb{R}^{N})\)
Denote
From (2.2), it suffices to show that \(G_{i}(u)\geq 0\). The following version of Brezis–Lieb lemma [16] is crucial to proving the asymptotic behaviour of ground states.
Lemma 4.1
(Brezis–Lieb lemma)
Assume that (\(F_{1}\))–(\(F_{4}\)) are satisfied, let\(\{u_{n}\}\)be a bounded sequence such that\(u_{n}\rightharpoonup u\)weakly in\(H^{s}(\mathbb{R}^{N})\). Then
Proof
It is clear that
where \(g_{i}(u):=\frac{d}{du}G_{i}(u)\), and \(g_{i}(u)=\frac{1}{2}f _{i}'(u)u-\frac{1}{2}f_{i}(u)\). From (\(F_{1}\)), we find
Since \((u_{n}-u+tu)\) is bounded in \(H^{s}(\mathbb{R}^{N})\), by using Hölder’s inequality, (\(F_{1}\)) and Lemma 2.1, we get \(\int _{\mathbb{R}^{N}}g_{i}(u_{n}-u+tu)u\,dx\) is bounded. For every \(\varepsilon >0\), there is \(\sigma >0\) such that
for any \(n\in \mathbb{N}\) and every measurable subset \(\varOmega \subset \mathbb{R}^{N}\) such that \(|\varOmega |<\sigma \). Thus \((g_{i}(u_{n}-u+tu)u)\) is uniformly integrable. Moreover, for any \(\varepsilon >0\), there exists a measurable subset \(\varOmega \subset \mathbb{R}^{N}\) of finite measure \(|\varOmega |<+\infty \) such that, for any \(n\geq 1\),
Hence \((g_{i}(u_{n}-u+tu)u)\) is tight over \(\mathbb{R}^{N}\). Since \(g_{i}(u_{n}-u+tu)u\rightarrow g_{i}(tu)u\) a.e. in \(\mathbb{R}^{N}\), in view of the Vitali convergence theorem, \(g_{i}(tu)u\) is integrable and
From (4.2), we deduce
This completes the proof of Lemma 4.1. □
We denote that \(\mathcal{J}_{n}\) is the corresponding functional of (1.1) with \(b(x)=b_{n}(x)\), \(\mathcal{J}_{0}\) is the corresponding functional of (1.1) with \(b(x)=0\). \(\mathcal{N} _{n}\) and \(\mathcal{N}_{0}\) are well defined in a similar way. Denote
From (i) of Theorem 1.1, there exist \(\omega _{n}\in \mathcal{N}_{n}\) such that \(\mathcal{J}_{n}(\omega _{n})=c_{n}\) and \(\omega _{0}\in \mathcal{N}_{0}\) such that \(\mathcal{J}_{0}(\omega _{0})=c _{0}\). We need several lemmas for the proof.
Lemma 4.2
Suppose that (\(F_{1}\))–(\(F_{4}\)) and (B) are satisfied, then\(\omega _{n}\)is bounded inE. Moreover, one has
Proof
Let \(t_{n}>0\) be such that \(t_{n}\omega _{n}\in \mathcal{N}_{0}\), we have
Let \(t_{n}'>0\) be such that \(t_{n}'\omega _{0}\in \mathcal{N}_{n}\), then
Combining (4.3) and (4.4), we have
Since \(t_{n}\omega _{n}\in \mathcal{N}_{0}\), we have
Suppose \(t_{n}\rightarrow +\infty \), in view of (\(F_{3}\)) and (2.2), we get that
It is a contradiction. Hence \((t_{n})\) is bounded. It follows from (2.2) and (4.4) that
Hence \(\omega _{n}\) is bounded in E, then
In view of (4.5), we deduce that \(c_{n}\rightarrow c_{0}\) as \(n\rightarrow +\infty \). This completes the proof of Lemma 4.2. □
Lemma 4.3
For each ground state solution\(\omega _{n}\)of\(\mathcal{J}_{n}\), there exist\(\omega \neq (0,0)\)and\((z_{n})\subset \mathbb{Z}^{N}\)such that\(\omega _{n}(\cdot +z_{n})\rightharpoonup \omega \)inE. Moreover, ωis a ground state solution of\(\mathcal{J}_{0}\), i.e. \(\mathcal{J}_{0}'(\omega )=0\)and\(\mathcal{J}_{0}(\omega )=c_{0}\).
Proof
In view of Lemma 4.2, \(\omega _{n}\) is bounded in E. Suppose
Applying similar arguments in Lemma 3.4, we get \(\|\omega _{n} \|\rightarrow 0\). Since
we have
On the other hand, from (\(A_{1}\)) and (\(A_{4}\)) in Lemma 2.4, we have
It is a contradiction. By Lions’ lemma [25] there exists \((y_{n})\subset \mathbb{R}^{N}\) such that
We assume, without loss of generality, that
For each \(y_{n}\in \mathbb{R}^{N}\), we will find \(z_{n}\in \mathbb{Z} ^{N}\) such that \(B(y_{n},1)\subset B(z_{n},1+\sqrt{N})\), then
Let \(\bar{\omega }_{n}:=\omega _{n}(\cdot +z_{n})\), \(\bar{u}_{n}:=u _{n}(\cdot +z_{n})\) and \(\bar{v}_{n}:=v_{n}(\cdot +z_{n})\), up to a subsequence, there exists \(\omega \in E\) such that
We have
Hence \(u\neq 0\) and \(\omega \neq (0,0)\).
For any \(\phi =(\varphi ,\psi )\), \(\varphi , \psi \in C_{0} ^{\infty }(\mathbb{R}^{N})\), it is clear that
By using Hölder’s inequality, we deduce that
Combining (4.7) and (4.8), we find that \(\mathcal{J}'_{0}(\bar{ \omega }_{n})\phi \rightarrow 0\). It follows from Lemma 3.2 that \(\mathcal{J}'_{0}(\bar{\omega }_{n})\phi \rightarrow \mathcal{J}'_{0}( \omega )\phi \) and \(\mathcal{J}'_{0}(\omega )=0\).
Since \(c_{n}\rightarrow c_{0}\), using Fatou’s lemma, we have
Thus ω is a ground state solution of \(\mathcal{J}_{0}\). This completes the proof of Lemma 4.3. □
Proof of (ii) of Theorem 1.1
We find that
Since \(\bar{\omega }_{n}\rightharpoonup \omega \) in E, we have \(\langle \omega ,\bar{\omega }_{n}-\omega \rangle \rightarrow 0\), \(\lambda \int _{\mathbb{R}^{N}}u(\bar{v}_{n}-v)\,dx\rightarrow 0\) and \(\lambda \int _{\mathbb{R}^{N}}v(\bar{u}_{n}-u)\,dx\rightarrow 0\). It is suffices to show that
From Lemma 4.1, we get
In view of (4.9), we have
It follows from (4.11) and (4.12) that
Since \(G_{i}\geq 0\), \(i=1,2\), then
From (\(F_{5}\)), we deduce that
and \(|\bar{v}_{n}-v|^{t}_{t}\rightarrow 0\). By fractional embedding theorem [1], we get that \(\bar{u}_{n}\) and \(\bar{v}_{n}\) are bounded in \(L^{r}(\mathbb{R}^{N})\) for \(2\leq r\leq 2^{*}\). Using (4.13) and the interpolation inequality, we get \(\bar{u}_{n} \rightarrow u\) and \(\bar{v}_{n}\rightarrow v\) in \(L^{r}(\mathbb{R} ^{N})\) for \(2< r<2^{*}\). For any \(\varepsilon >0\), there exists \(C_{\varepsilon }>0\) such that
Let \(\varepsilon \rightarrow 0\), we have
Moreover,
It follows from (4.10) that \(\|\bar{\omega }_{n}-\omega \| \rightarrow 0\). This completes the proof of (ii) of Theorem 1.1. □
5 Ground states of a Schrödinger system with non-periodic couplings
We prove (i) and (ii) of Theorem 1.2 in Sects. 5.1–5.2, respectively. We denote that \(\mathcal{J}_{\mathrm{per}}\) is the corresponding functional of (1.1) with \(b(x)=b_{ \mathrm{per}}(x)\). \(\mathcal{N}_{\mathrm{per}}\) and \(c_{\mathrm{per}}\) are well defined in a similar way.
5.1 Existence
We need the following lemma.
Lemma 5.1
Assume that\(b_{\mathrm{loc}}(x)\geq 0\)for a.e. \(x\in \mathbb{R}^{N}\)and\(b_{\mathrm{loc}}(x)>0\)on a positive measure set, then\(c< c_{\mathrm{per}}\).
Proof
From (i) of Theorem 1.1, we find a critical point \(\omega '\) of \(\mathcal{J}_{\mathrm{per}}\), where
\(\mathcal{J}_{\mathrm{per}}(u',v')= c_{\mathrm{per}}\) and \(\mathcal{J}'_{\mathrm{per}}(u',v')= 0\). We get that \(\int b_{ \mathrm{loc}}(x)|u'|^{q}|v'|^{q}\,dx>0\). Let \(t>0\) be such that \(t(u',v')\in \mathcal{N}\), then
□
Proof of (i) of Theorem 1.2
We divide the proof into three steps. Step 1 and Step 3 are similar with those in Sect. 4.1, we omit them here. By similar arguments as Step 1 in Sect. 4.1, we find \((u_{n},v_{n})\rightharpoonup (u_{0},v_{0})\) in E and \(\mathcal{J}'(u_{0},v_{0})=0\).
Step 2: We check whether\((u_{0},v_{0})\neq (0,0)\).
Similarly, from Step 2 in Sect. 4.1, there exists \(z_{n}\in \mathbb{Z}^{N}\) such that \(B(y_{n},1)\subset B(z_{n},1+ \sqrt{N})\) and
We claim that \((z_{n})\) is bounded, and hence \(u_{0}\neq 0\), \((u_{0},v_{0})\in \mathcal{N}\) and \(\mathcal{J}(u_{0},v_{0})\geq c\).
We check the claim. Suppose that \((z_{n})\) is unbounded, then we can choose a subsequence of \((z_{n})\) such that \(|z_{n}|\rightarrow \infty \) as \(n\rightarrow \infty \). Let \(\bar{u}_{n}:=u_{n}(\cdot +z _{n})\), \(\bar{v}_{n}:=v_{n}(\cdot +z_{n})\), up to a subsequence, then
We deduce that
by (5.2). We find that \(\bar{u}\neq 0\) by \(\bar{u}_{n}\rightarrow \bar{u}\) in \(L_{\mathrm{loc}}^{2}(\mathbb{R}^{N})\), thus \(\bar{\omega }=(\bar{u},\bar{v})\neq (0,0)\). For any \(\phi =(\varphi ,\psi )\), \(\varphi , \psi \in C_{0}^{\infty }(\mathbb{R}^{N})\), we have
Let \(K\subset \mathbb{R}^{N}\) be a compact set containing supports of φ, ψ, then
as \(|z_{n}|\rightarrow \infty \). Combining (5.3) and (5.4), we get \(\mathcal{J}'_{\mathrm{per}}(\bar{\omega }_{n}) \phi \rightarrow 0\). It follows from Lemma 3.2 that \(\mathcal{J}'_{\mathrm{per}}(\bar{\omega }_{n})\phi \rightarrow \mathcal{J}'_{\mathrm{per}}(\bar{\omega })\phi \) and \(\mathcal{J}'_{ \mathrm{per}}(\bar{\omega })=0\). Hence \((\bar{u},\bar{v})\in \mathcal{N}_{\mathrm{per}}\) and \(\mathcal{J}_{\mathrm{per}}(\bar{u}, \bar{v})\geq c_{\mathrm{per}}\). It is clear that
Applying Fatou’s lemma, we find
We get a contradiction with Lemma 5.1. Hence \((z_{n})\) is bounded. □
5.2 Nonexistence
Suppose by contradiction that there exists a ground state solution of (1.1), i.e. \(\omega _{0}=(u_{0},v_{0})\neq (0,0)\) such that \(\mathcal{J}(u_{0},v_{0})=c\) and \(\mathcal{J}'(u_{0},v_{0})=0\). By using similar arguments as Step 3 in Sect. 4.1, we find a critical point of \(\mathcal{J}\), where
\(\mathcal{J}(u',v')=c\) and \(\mathcal{J}'(u',v')=0\).
Lemma 5.2
Assume that\(b_{\mathrm{loc}}(x)\leq 0\)for a.e. \(x\in \mathbb{R}^{N}\)and\(b_{\mathrm{loc}}(x)<0\)on a positive measure set, then\(c>c_{\mathrm{per}}\).
Proof
It is clear that \(\int b_{\mathrm{loc}}(x)|u'|^{q}|v'|^{q}\,dx<0\). Let \(t>0\) be such that \(t(u',v')\in \mathcal{N}_{\mathrm{per}}\), then
□
Let \(\omega \in \mathcal{N}_{\mathrm{per}}\) be a ground state solution of \(\mathcal{J}_{\mathrm{per}}\), i.e. \(\mathcal{J}_{ \mathrm{per}}(u,v)=c_{\mathrm{per}}\) and \(\mathcal{J}'_{\mathrm{per}}(u,v)=0\). Denote that \(\bar{\omega }:= \omega (\cdot -y)\) for \(y\in \mathbb{Z}^{N}\), we find that \(\bar{ \omega }\in \mathcal{N}_{\mathrm{per}}\). There exists \(t>0\) such that \(t\bar{\omega }\in \mathcal{N}\). For any \(y\in \mathbb{Z}^{N}\), we have
Obviously, \(\int _{\mathbb{R}^{N}}b_{\mathrm{loc}}(x)t^{2q}|\bar{u}|^{q}| \bar{v}|^{q}\,dx=\int _{\mathbb{R}^{N}}b_{\mathrm{loc}}(x+y)t^{2q}|u|^{q}|v|^{q}\,dx\). Since \(\mathcal{J}_{\mathrm{per}}\) is coercive on \(\mathcal{N}_{ \mathrm{per}}\) and \(\mathcal{J}_{\mathrm{per}}(t\omega )=\mathcal{J} _{\mathrm{per}}(t\bar{\omega })\leq c_{\mathrm{per}}\), we find that tω is bounded in E. Furthermore, u, v are bounded in \(L^{2q}(\mathbb{R}^{N})\) and \(L^{p}(\mathbb{R}^{N})\) by embedding theorem. For any \(0<\varepsilon <1\), we choose \(R(\varepsilon )>0\) such that \(\int _{B_{R}^{c}}|u|^{2q}\,dx<\varepsilon ^{2}\), and choose \(y(\varepsilon )>0\) such that \(\int _{B_{R}}|b_{\mathrm{loc}}(x+y)|^{ \frac{p}{p-2q}}\,dx<\varepsilon ^{\frac{p}{p-2q}}\), then there exist \(C_{1},C_{2},C_{3}>0\) such that
where ε is arbitrary. In view of (5.7), let \(|y|\) be sufficiently large, we get \(c_{\mathrm{per}}\geq c\). It is contradictory to Lemma 5.2. This completes the proof of (ii) of Theorem 1.2.
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Duan, X., Wei, G. & Yang, H. Ground states for a coupled Schrödinger system with general nonlinearities. Bound Value Probl 2020, 22 (2020). https://doi.org/10.1186/s13661-020-01331-6
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DOI: https://doi.org/10.1186/s13661-020-01331-6