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Positive solutions for coupled Schrödinger system with critical exponent in \(\mathbb{R}^{N}\) (\(N\geq5\))
Boundary Value Problems volume 2017, Article number: 104 (2017)
Abstract
In this paper, we study the following coupled Schrödinger system:
where \(N\geq5\) and \(2^{*}=\frac{2N}{N-2}\). Note that the nonlinearity and the coupling terms are both of critical growth. Using the mountain pass theorem, Ekeland’s variational principle and the concentration-compactness principle, we show that this system has at least one positive least energy solution for each \(\beta\in(-1,0)\cup (0,+\infty)\).
1 Introduction
In this paper, we consider the following coupled nonlinear Schrödinger system:
where \(N\geq3\), \(2^{*}=\frac{2N}{N-2}\) and \(\beta\in(-1,+\infty )\setminus \{0\}\). The functions \(f,g\) satisfy the following conditions:
- (F1):
-
\(f,g\in C^{2}(\mathbb{R})\), \(\lim_{t\rightarrow 0^{+}}\frac{f(t)}{t}=0\), \(\lim_{t\rightarrow\infty}\frac {f(t)}{t^{2^{*}-1}}=0\) for \(t\geq0\),
$$\lim_{t\rightarrow0^{+}}\frac{g(t)}{t}=0,\qquad \lim_{t\rightarrow\infty} \frac{g(t)}{t^{2^{*}-1}}=0\quad \text{for }t\geq0; $$ - (F2):
-
There exist \(\theta_{1},\theta_{2}>0\) small enough such that
$$tf'(t)\geq(1+\theta_{1})f(t)>0, \qquad tg'(t)\geq(1+\theta _{2})g(t)>0\quad \text{for } t>0; $$ - (F3):
-
\(f(t)\), \(g(t)\) are odd.
In recent years, there have been a lot of studies on the following coupled system of nonlinear Schrödinger equations:
where \(\Omega=\mathbb{R}^{N}\) (\(N\geq3\)) or Ω is a smooth bounded domain in \(\mathbb{R}^{N}\), \(1< p\leq\frac{2^{*}}{2}\), \(\mu_{1},\mu_{2}>0\) and \(\beta\neq0\). For the case \(p=2\), system (1.2) arises in the Hartree-Fock theory for a binary mixture of Bose-Einstein condensates in two different hyperfine states, see more details in [1–3]. The sign of β determines whether the interactions of the two states are attractive if \(\beta>0\) or repulsive if \(\beta<0\).
For problem (1.2), we are interested in the existence of a nontrivial solution \((u,v)\), i.e. \(u\not\equiv0\) and \(v\not\equiv 0\). However, one easily sees that (1.2) may admit a semitrivial solution of the form \((u,0)\) or \((0,v)\), which may cause some difficulties. When \(2<2p<2^{*}\), system (1.2) is a problem of subcritical growth. The existence and multiplicity of nontrivial solutions have been extensively studied, see [4–13] and the references therein.
For the critical case \(2p=2^{*}\), when Ω is a smooth bounded domain, there exist papers [14–17] studying this case. In [14], Chen and Zou studied problem (1.2) with \(N=4\). In [15], Chen and Zou studied problem (1.2) with \(N\geq5\) and they showed that if \(-\lambda_{1}(\Omega)<\lambda_{1}\leq\lambda _{2}<0\), then (1.2) has a positive least energy solution for any \(\beta\neq0\), where \(\lambda_{1}(\Omega)\) is the first eigenvalue of −Δ with the Dirichlet boundary condition. In [17], by a minimization method, Ye and Peng showed the existence of positive least energy solution for the special case \(\lambda_{1}=\lambda_{2}\). When \(2p=2^{*}\) and \(\Omega=\mathbb{R}^{N}\), by the Pohozaev identity, we see that problem (1.2) has only a trivial solution if \(\lambda_{1}\lambda _{2}>0\). To get nontrivial solutions, one usually adds lower order perturbation terms to the right-hand side of system (1.2), i.e. considering problem (1.1). Problem (1.1) can be seen as a counterpart of the following single equation:
or
Deng in [18] proved that if \(N\geq4\) and (F1)-(F3) hold, then (1.3) (or (1.4)) has at least one positive least energy radial solution, denoted by \(u_{1}\) (or \(v_{1}\)) and the corresponding energy denoted by \(B_{1}\) (or \(B_{2}\)). Hence we deduce that \((u_{1},0)\) and \((0,v_{1})\) are semitrivial solutions to problem (1.1), which may be an interference in the process of searching for nontrivial solutions. Recently, the author proves the special case \(N=4\) in [19]. In this paper, we consider (1.1) with higher dimensions \(N\geq5\). To state our main results, we denote \(H:= H^{1}(\mathbb{R} ^{N})\times H^{1}(\mathbb{R}^{N})\) with the norm defined as \(\Vert (u,v) \Vert _{H}= [\int_{\mathbb{R}^{N}}( \vert \nabla u \vert ^{2}+ \vert u \vert ^{2})+\int _{\mathbb{R} ^{N}}( \vert \nabla v \vert ^{2}+ \vert v \vert ^{2}) ]^{\frac{1}{2}}\), \(\forall (u,v)\in H\). It is well known that weak solutions of (1.1) correspond to critical points of the functional \(I:H\rightarrow\mathbb{R}\) defined as follows:
for any \((u,v)\in H\), where \(F(s)=\int_{0}^{s}f(t)\,dt\), \(G(s)=\int _{0}^{s}g(t)\,dt\). We say \((u,v)\in H\) is a positive least energy solution of (1.1) if \((u,v)\) is a nontrivial solution of (1.1) with \(u>0\), \(v>0\) and
Our main result is as follows.
Theorem 1.1
Suppose that (F1)-(F3) hold and \(N\geq5\).
-
(1)
For any \(\beta>0\), problem (1.1) has at least one positive least energy solution.
-
(2)
For any \(\beta\in(-1,0)\), problem (1.1) has at least one radial and positive least energy solution.
Remark 1.2
-
(1)
For \(\beta<0\), we do not know whether the solution obtained in Theorem 1.1 is a least energy solution of (1.1) in H or not.
-
(2)
When \(N=4\), it is proved in [19] that (1.1) has a radially positive solution for any \(\beta>0\) and \(\beta\neq1\). Comparing this with Theorem 1.1(1), we see that the case \(N\geq5\) is completely different from the case \(N=4\). In the proof of Theorem 1.1(1), we should point out that \(2^{*}<4\) is an essential condition, which makes the method not applicable to the case \(N=4\).
-
(3)
The method to prove Theorem 1.1(2) can be similarly used to show that when \(N=4\) and \(-1<\beta<0\), (1.1) has at least one radially positive least energy solution.
By (F1), (F3), for any \(\varepsilon>0\), there exists \(C_{\varepsilon }>0\) such that
By (F2), (F3), we have
(see Remark 1.3 in [18]), then \(\frac{F(t)}{t^{2}}\) and \(\frac{f(t)}{ \vert t \vert }\) is nondecreasing on \(t\in \mathbb{R}\setminus \{0\}\).
Since the nonlinearity and the coupling terms in problem (1.1) are both critical, the existence of nontrivial solutions to (1.1) depends heavily on the least energy solutions of the corresponding limit problem
where \(D^{1,2}(\mathbb{R}^{N})=\{u\in L^{2}(\mathbb{R}^{N}) | | \nabla u| \in L^{2}(\mathbb{R}^{N})\}\).
Recall that S is the best constant of \(D^{1,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{*}}(\mathbb{R}^{N})\), i.e.
For any \(\varepsilon>0\) and \(y\in\mathbb{R}^{N}\), S is achieved by the Aubin-Talenti instanton (see [20, 21])
and
As showed in [15], the following manifold
contains all nontrivial solutions to problem (1.8). Set \(A:=\inf_{(u,v)\in P}J(u,v)\), where \(J(u,v)\) is the corresponding energy functional. It is proved in [15] that when \(N\geq5\), A is attained and \(A<\frac{1}{N}S^{\frac{N}{2}}\) for each \(\beta>0\); and while \(\beta<0\), \(A=\frac{2}{N}S^{\frac{N}{2}}\) is not attained. This fact brings about the difference of the existence result in Theorem 1.1 between \(\beta>0\) and \(\beta<0\). To prove Theorem 1.1, we easily see that the functional I possesses a mountain pass geometry and then a \((PS)\) sequence exists. For \(\beta >0\), we could pull the mountain pass energy down below \(\min\{A,B_{1},B_{2}\}\), then the \((PS)\) condition holds for I. However, the above energy estimate cannot be directly applied to the case \(\beta<0\) since A is not attained when \(\beta<0\). We overcome this difficulty by working in the radially symmetric Sobolev subspace \(H_{r}=H^{1}_{r}(\mathbb{R} ^{N})\times H^{1}_{r}(\mathbb{R}^{N})\), where \(H^{1}_{r}(\mathbb{R}^{N})=\{u\in H^{1}(\mathbb{R}^{N})\vert u(x)=u( \vert x \vert )\}\) and using the constrained minimization on the following manifold defined similarly to (1.11):
Then Theorem 1.1 is proved. It is necessary to point out that due to the existence of the perturbation terms in I, we need the assumption \(\beta>-1\) to show that the manifold M is a suitable one for our problem, i.e. a minimizer of I constrained on M is a nontrivial solution of (1.1).
Throughout this paper, we use standard notations. For simplicity, we write \(\int_{\Omega} h\) to mean the Lebesgue integral of \(h(x)\) over a domain \(\Omega\subset\mathbb{R}^{N}\). \(L^{p}:= L^{p}(\mathbb {R}^{N})\) (\(1\leq p<+\infty\)) is the usual Lebesgue space with the standard norm \(\vert \cdot \vert _{p}\). We use ‘→’ and ‘⇀’ to denote the strong and weak convergence in the related function space, respectively. C, \(\{C_{i}\}_{i=1}^{+\infty}\) will denote a positive constant unless specified. We use ‘:=’ to denote definitions. \(B_{r}(x):=\{y\in\mathbb{R}^{N} || y-x| < r\}\). We use ‘\(X^{-1}\)’ to denote the dual space of X. We denote a subsequence of a sequence \(\{u_{n}\}\) as \(\{u_{n}\}\) to simplify the notation unless specified.
The paper is organized as follows. In Section 2, we prove Theorem 1.1 with \(\beta>0\); in Section 3, we give the proof of Theorem 1.1 with \(\beta\in(-1,0)\).
2 Proof of Theorem 1.1 with \(\beta>0\)
In this section, we consider the case \(\beta>0\). We first give some preliminary results.
Lemma 2.1
Suppose that (F1)-(F3) hold and \(\beta>0\), then I possesses a mountain pass geometry around \((0,0)\):
-
(1)
There exist \(\rho,\sigma>0\) such that \(\inf_{ \Vert (u,v) \Vert _{H}=\rho}I(u,v)\geq\sigma\);
-
(2)
There exists \((u_{0},v_{0})\in H\) such that \(\Vert (u_{0},v_{0}) \Vert _{H}>\rho\) and \(I(u_{0},v_{0})<0\).
Proof
For any \((u,v)\in H\setminus \{(0,0)\}\), by (1.6) and the Sobolev embedding inequality, there is a constant \(C>0\) such that
then there exist \(\sigma,\rho>0\) such that \(I(u,v)\geq\sigma\) for all \(\Vert (u,v) \Vert _{H}=\rho\).
By (1.7), there exists \(C>0\) such that
For any \(t\geq0\), we see that \(I(tu,tv)\rightarrow-\infty\) as \(t\rightarrow+\infty\). Then there exists \(t_{0}>0\) such that \(I(t_{0}u,t_{0}v)<0\) and \(\Vert (t_{0}u,t_{0}v)\Vert _{H}>\rho\). □
By the mountain pass theorem (see, e.g., Theorem 2.10 in [22]), there exists a \((PS)_{\mathcal{B}}\) sequence \(\{(u_{n},v_{n})\}\subset H\) such that
where
and \(\Gamma=\{\gamma\in C([0,1],H)\vert \gamma(0)=0,I(\gamma(t))<0\}\).
Lemma 2.2
Suppose that (F1)-(F3) hold and \(\beta>0\), then for any \((u,v)\in H\setminus \{(0,0)\}\), there exists a unique \(\tilde {t}=\tilde{t}_{(u,v)}>0\) such that \(\Psi(\tilde{t}u,\tilde{t}v)=0\) and \(I(\tilde{t}u,\tilde{t}v)=\max_{t\geq0}I(tu,tv)\), where
Proof
For any \((u,v)\in H\setminus \{(0,0)\}\) and any \(t\geq0\), by (F1)-(F3), we see that
has a unique critical point \(\tilde{t}>0\) corresponding to its maximum. Then \(h(\tilde{t})=\max_{t\geq0}h(t)\) and \(h'(\tilde{t})=0\). So \(\Psi(\tilde{t}u,\tilde{t}v)=0\). □
Set
By Lemma 2.2, \(\mathcal{M}\neq\varnothing\). Indeed, \(\mathcal{M}\) contains all nontrivial and semitrivial solutions of (1.1). For any \((u,v)\in\mathcal{M}\), by \(\beta>0\) and (1.7), we have \(I(u,v)=I(u,v)-\frac{1}{2} \Psi(u,v)\geq0\), i.e. \(I(u,v)\) is bounded from below on \(\mathcal{M}\). Moreover, it is easy to check that
For each \(\beta>0\), as showed in [17], we set
Then \(S_{\beta}\geq0\) is well defined.
Lemma 2.3
For each \(\beta>0\),
and \(S_{\beta}\) is attained by \((\tau_{0} U_{\varepsilon,y},U_{\varepsilon,y})\), where \(\tau_{0}\) is the unique positive zero point of \(\varphi (\tau)=1+\beta\tau^{\frac{N}{N-2}}-\beta\tau^{\frac {4-N}{N-2}}-\tau^{\frac{4}{N-2}}\).
For \(\rho>0\), let \(\psi\in C_{0}^{\infty}(B_{2\rho}(0))\) be a cut-off function with \(0\leq\psi\leq1\) and \(\psi\equiv1 \) for \(\vert x \vert \leq \rho\). For \(\varepsilon>0\), denote
Then, by [23], we have
where \(C_{1}>0\) is a constant independent of ε.
Lemma 2.4
Suppose that (F 2) holds. For any sequences \(\{t_{\varepsilon}\}\), \(\{s_{\varepsilon}\}\) satisfying that there exist two constants \(0< C_{2}< C_{3}<+\infty\) independent of ε such that \(C_{2}\leq t_{\varepsilon},s_{\varepsilon}\leq C_{3}\) for ε small enough, then
As mentioned in Section 1, \(u_{1}\), \(v_{1}\in H^{1}(\mathbb{R}^{N})\) are respectively radially positive least energy solutions of (1.3) and (1.4) with the corresponding energy denoted by \(B_{1}\), \(B_{2}\), i.e.
By standard regularity arguments, \(u_{1},v_{1}\in C(\mathbb{R}^{N})\) and \(u_{1}(0)=\max_{x\in\mathbb{R}^{N}}u_{1}(x)\), \(v_{1}(0)=\max_{x\in\mathbb{R} ^{N}}v_{1}(x)\).
Lemma 2.5
Suppose that (F1)-(F3) hold and \(\beta>0\), then
Proof
The proof consists of two steps.
Step 1: We prove that \(\mathcal{B}<\frac{1}{N}S_{\beta }^{\frac{N}{2}}\).
For \(\varepsilon>0\), denote \((u_{\varepsilon},v_{\varepsilon}):= (\tau_{0}w_{\varepsilon}, w_{\varepsilon})\), where \(\tau_{0}\) is given in Lemma 2.3. By Lemma 2.2, there exists a unique \(t_{\varepsilon}>0\) such that
and \(I(t_{\varepsilon}u_{\varepsilon},t_{\varepsilon}v_{\varepsilon})=\max_{t\geq0}I(tu_{\varepsilon},tv_{\varepsilon})\).
We claim that \(\{t_{\varepsilon}\}_{\varepsilon>0}\) is bounded from below by a positive constant. Otherwise, there exists a sequence \(\{\varepsilon_{n}\}\subset\mathbb{R}_{+}\) satisfying \(\lim_{n\rightarrow+\infty}t_{\varepsilon_{n}}=0\) and \(I(t_{\varepsilon_{n}}u_{\varepsilon_{n}},t_{\varepsilon _{n}}v_{\varepsilon_{n}})=\max_{t\geq0} I(tu_{\varepsilon_{n}},tv_{\varepsilon_{n}})\), then by (2.2)-(2.5) and (F1), (F3), we have \(0< \mathcal{B}\leq\lim_{n\rightarrow+\infty}I(t_{\varepsilon _{n}}u_{\varepsilon_{n}}, t_{\varepsilon_{n}}v_{\varepsilon_{n}})=0\), which is impossible. So there exist \(0< C_{4}< C_{5}\) independent of ε satisfying that
Then, by (2.2)-(2.6) and Lemmas 2.3 and 2.4, we see that
Step 2: We prove that \(\mathcal{B}< B_{1}\) and \(\mathcal{B}< B_{2}\).
The idea of this proof follows from Lemma 2.7 in [15].
We define a function \(H:\mathbb{R}^{2}\rightarrow\mathbb{R}\) by
It is easy to check that \(H(1,0)=0\) and \(H_{t}(1,0)\neq0\). Then, by the implicit function theorem, there exist \(\delta>0\) and a function \(t(s)\in C^{1}(-\delta,\delta)\) such that
which implies that
Since \(\frac{2^{*}}{2}<2<2^{*}\) and \(\beta>0\), by direct calculation and (F1), (F2), we have
i.e.
So
Then
By (F2), we see that the function \(\widetilde{F}(t):=\frac{1}{2}f(t)t-F(t)\) is nondecreasing on \((0,+\infty)\). By (2.8), we may assume that \(0< t(s)\leq1\) for \(\vert s \vert \) small enough. So
By (F1), (F3), we have \(\lim_{s\rightarrow0}\frac {g(t(s)sv_{1})t(s)sv_{1}}{ \vert s \vert ^{\frac{2^{*}}{2}}}=\lim_{s\rightarrow 0}\frac{G(t(s)sv_{1})}{ \vert s \vert ^{\frac{2^{*}}{2}}}=0\). Thus, by (2.4), (2.7)-(2.10) and \(\beta>0\), we see that for \(\forall s\in(-\delta,\delta)\),
Hence \(\mathcal{B}< B_{1}\). Similarly, we have \(\mathcal{B}< B_{2}\). Therefore the proof of the lemma is completed. □
Lemma 2.6
[22], Vanishing lemma
Let \(r>0\) and \(2\leq q<2^{*}\). If \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb{R}^{N})\) and
then \(u_{n}\rightarrow0\) in \(L^{p}(\mathbb{R}^{N})\) for \(2< p<2^{*}\).
Proof of Theorem 1.1 with \(\beta>0\)
By Lemma 2.1, there exists a sequence \(\{(u_{n},v_{n})\}\subset H\) such that
By (1.7), we easily see that \(\{(u_{n},v_{n})\}\) is uniformly bounded in H.
Let
then \(\delta_{1},\delta_{2}\in[0,+\infty)\). If \(\delta_{1}=\delta_{2}=0\), then by the vanishing lemma 2.6, we have \(u_{n}\rightarrow0\) in \(L^{p}(\mathbb{R}^{N})\) and \(v_{n}\rightarrow0\) in \(L^{p}(\mathbb {R}^{N})\), \(\forall 2< p<2^{*}\). By (F1), (F3), for any \(\varepsilon>0\), there exists \(C_{\varepsilon}>0\) such that
Then
which shows that \(\lim_{n\rightarrow+\infty}\int_{\mathbb{R} ^{N}}F(u_{n})=0\) since \(\varepsilon>0\) is arbitrary. Similarly,
By (2.11) and the boundedness of \(\{(u_{n},v_{n})\}\), we see that
For n large, we may assume that \(u_{n},v_{n}\in D^{1,2}(\mathbb {R}^{N})\setminus \{0\}\). Hence, by (2.14), we see that
i.e. \(\mathcal{B}\geq\frac{1}{N}S_{\beta}^{\frac{N}{2}}\), which contradicts Lemma 2.5.
So we deduce that at least one of the following two inequalities \(\delta_{1}>0\) and \(\delta_{2}>0\) holds. Without loss of generality, we may assume that \(\delta_{1}>0\). There exists a sequence \(\{y_{n}\}\subset\mathbb{R}^{N}\) such that
Set
Then \(\{(\tilde{u}_{n},\tilde{v}_{n})\}\) is also a bounded \((PS)_{\mathcal{B}}\) sequence for I. Up to a subsequence, we may assume that there exists \((\tilde{u},\tilde{v})\in H\) such that \((\tilde{u}_{n},\tilde{v}_{n})\rightharpoonup(\tilde{u},\tilde{v})\) in H, then \(I'(\tilde{u},\tilde{v})=0\). Moreover, by the Sobolev embedding theorem, we have
which and (2.15) imply that \(\tilde{u}\not\equiv0\).
If \(\tilde{v}\equiv0\), then ũ is a nontrivial solution of \(-\Delta u+u= \vert u \vert ^{2^{*}-2}u+f(u)\) in \(\mathbb{R}^{N}\). Then \(I(\tilde {u},0)\geq B_{1}\). Hence, by \(\beta>0\), (1.7) and Fatou’s lemma, we have
which contradicts Lemma 2.5. So \(\tilde{v}\not\equiv0\). Then \((\tilde{u},\tilde{v})\) is a nontrivial solution to (1.1). Thus \((\tilde{u},\tilde{v})\in\mathcal{M}\) and
which shows that \(I(\tilde{u},\tilde{v})=\mathcal{B}\). Moreover,
Indeed, since \((\tilde{u},\tilde{v})\) is a nontrivial solution to (1.1), \(I(\tilde{u},\tilde{v})\geq m\). On the other hand, for any nontrivial solution \((u,v)\) to (1.1), then \((u,v)\in\mathcal{M}\), which shows that \(I(u,v)\geq\mathcal {B}\). Hence \(m\geq\mathcal{B}=I(\tilde{u},\tilde{v})\) since \((u,v)\) is arbitrary. So \(I(\tilde{u},\tilde{v})=m\).
Since the functional I and the manifold \(\mathcal{M}\) are symmetric, we see that \(( \vert \tilde{u} \vert , \vert \tilde{v} \vert )\) is also a nontrivial solution to (1.1) and \(I( \vert \tilde{u} \vert , \vert \tilde{v} \vert )=\mathcal{B}\). By regularity and the maximum principle, we obtain that \(\vert \tilde {u} \vert , \vert \tilde{v}\vert >0\) in \(\mathbb{R}^{N}\). □
3 Proof of Theorem 1.1 with \(\beta\in(-1,0)\)
In this section, we study the existence of radially positive least energy solutions to (1.1) when \(-1<\beta<0\). Denote \(H_{r}:=H^{1}_{r}(\mathbb{R}^{N})\times H^{1}_{r}(\mathbb{R}^{N})\). To prove the theorem, we set
Then \(M\neq\varnothing\). In fact, taking \(u,v\in C_{0}^{\infty }(\mathbb{R} ^{N})\), \(u,v\in H^{1}_{r}(\mathbb{R}^{N})\) with \(u,v\not\equiv0\) and \(\operatorname{supp}(u)\cap \operatorname{supp}(v)=\varnothing\), then by (F1)-(F3), there exist \(t_{1},t_{2}>0\) such that \((t_{1}u,t_{2}v)\in M\). For any \((u,v)\in H_{r}\), by \(\beta\in (-1,0)\), the Hölder inequality and the Cauchy inequality, we have
Then the minimization problem
is well defined and \(B\geq0\).
Lemma 3.1
Suppose that (F1)-(F3) hold and \(\beta\in(-1,0)\), then \(B>0\) and I is coercive on M. Moreover, there exists \(C_{0}>0\) such that \(\int_{\mathbb{R}^{N}} \vert u \vert ^{2^{*}}, \int_{\mathbb{R}^{N}} \vert v \vert ^{2^{*}}\geq C_{0} \) for any \((u,v)\in M\).
Proof
Since \(\beta\in(-1,0)\), for each \((u,v)\in M\), by (3.1) and (1.7), we have
which implies that \(B>0\) and I is coercive on M.
For any \((u,v)\in M\), by \(\beta<0\), (1.5) and the Sobolev embedding inequality, we have
which implies that \(\Vert u \Vert ^{2}\geq C_{1}\) for some \(C_{1}>0\) and then \(\int _{\mathbb{R}^{N}} \vert u \vert ^{2^{*}}\geq\frac{1}{2C} \Vert u \Vert ^{2}\geq C_{2}\) for some \(C_{2}>0\). Similarly, \(\int_{\mathbb{R}^{N}} \vert v \vert ^{2^{*}}\geq C_{3}\) for some \(C_{3}>0\). Set \(C_{0}=\min\{C_{2},C_{3}\}\), then the lemma is proved. □
Lemma 3.2
Suppose that (F1)-(F3) hold and \(\beta\in(-1,0)\), then \(B<\min \{B_{1}+\frac{1}{N} S^{\frac{N}{2}},B_{2}+\frac{1}{N} S^{\frac{N}{2}} \}\).
Proof
We first prove that there exist \((t_{\varepsilon }u_{1},sw_{\varepsilon})\in M\), where \(u_{1}\), \(w_{\varepsilon}\) are defined in (2.4) and (2.2). It is enough to prove that there exist \(t_{\varepsilon},s_{\varepsilon}>0\) solving the following system:
Since \(\int_{\mathbb{R}^{N}}( \vert \nabla u_{1} \vert ^{2}+u_{1}^{2})=\int_{\mathbb{R} ^{N}}( \vert u_{1} \vert ^{2^{*}}+f(u_{1})u_{1})\) and \(\frac{2^{*}}{2}<2\), by (F2) and the second equation of (3.2), we have \(0< s^{\frac{2^{*}}{2}}=h(t)\), \(t>1\), where
Moreover, \(h(1)=0\) and \(\lim_{t\rightarrow+\infty }h(t)=+\infty\). Then (3.2) is equivalent to
We see that \(\lim_{t\rightarrow1^{+}}\tilde{h}_{\varepsilon}(t)=+\infty>0\) and \(\lim_{t\rightarrow+\infty}\tilde {h}_{\varepsilon}(t)=-\infty\), so there exists \(t_{\varepsilon}>1\) such that \(\tilde{h}_{\varepsilon}(t_{\varepsilon})=0\). Set \(s_{\varepsilon }=[h(t_{\varepsilon})]^{\frac{2}{2^{*}}}>0\). Then (3.2) has a solution \((t_{\varepsilon},s_{\varepsilon})\).
If \(\lim_{\varepsilon\rightarrow0^{+}}s_{\varepsilon}=0\), then by (2.3) and (1.5), we have
We deduce from the second equation of system (3.2) that
which is impossible. So there exists \(s_{0}>0\) independent of ε such that \(s_{\varepsilon}\geq s_{0}\) for ε small. If \(\lim_{\varepsilon\rightarrow0^{+}}t_{\varepsilon }=+\infty\), then \(\lim_{\varepsilon\rightarrow 0^{+}}s_{\varepsilon}=\lim_{\varepsilon\rightarrow 0^{+}}[h(t_{\varepsilon})]^{\frac{2}{2^{*}}}=+\infty\). Note that
where we have used the fact that \(\int_{\mathbb{R}^{N}}\vert w_{\varepsilon}\vert ^{\frac {2^{*}}{2}}\leq o(\varepsilon^{2})\), which is given in Lemma 3.1 of [15]. Then, by the second equation of (3.2) and (F1), we have
which is a contradiction. So there exist \(t_{1}\), \(s_{1}\) independent of ε such that \(1\leq t_{\varepsilon}\leq t_{1}\) and \(s_{0}\leq s_{\varepsilon}\leq s_{1}\) for ε small. Then we have
Therefore,
Similarly, we can also prove that \(B< B_{2}+\frac{1}{N} S^{\frac{N}{2}}\). □
Lemma 3.3
Suppose that (F1)-(F3) hold and \(\beta\in(-1,0)\), then there exists a bounded \((PS)_{B}\) sequence \(\{(u_{n},v_{n})\}\subset M\) for I.
Proof
By Lemma 3.1 and Ekeland’s variational principle (see [22]), there exists a minimizing sequence \(\{(u_{n},v_{n})\}\subset M\) satisfying that
Lemma 3.1 shows that \(\{(u_{n},v_{n})\}\) is uniformly bounded in \(H_{r}\). For any \((\varphi,\phi)\in H_{r}\) with \(\Vert \varphi \Vert \), \(\Vert \phi \Vert \leq1\) and each \(n\in\mathbb{N}\), define \(h_{n}\), \(j_{n}:\mathbb{R}^{3}\rightarrow \mathbb{R}\) by
Let \(\mathbf{0}=(0,0,0)\). Then \(h_{n},j_{n}\in C^{1}(\mathbb{R}^{3},\mathbb {R})\) satisfy that \(h_{n}(\mathbf{0})=j_{n}(\mathbf{0})=0\) and
Define the matrix
We see that (F2), (F3) show that \(f(u_{n})u_{n}-f'(u_{n})u_{n}^{2}<0\) and \(g(v_{n})v_{n}-g'(v_{n})v_{n}^{2}<0\). Then, by \(-1<\beta<0\), (3.1) and Lemma 3.2, we have
By the implicit function theorem, there exist \(\delta_{n}>0\) and functions \(s_{n}(t)\), \(l_{n}(t)\in C^{1}(-\delta_{n},\delta_{n})\) such that \(s_{n}(0)=l_{n}(0)=0\),
and
Since \(\{(u_{n},v_{n})\}\) is uniformly bounded in H, we see that
where \(C>0\) is independent of n. Denote
then \((\varphi_{n,t},\phi_{n,t})\in M\) for \(\forall t\in(-\delta _{n},\delta_{n})\). It follows from (3.6) that
By \((u_{n},v_{n})\in M\) and the Taylor expansion, we have
where \(r(n,t)=o( \Vert (t\varphi+s_{n}(t)u_{n},t\phi+l_{n}(t)v_{n}) \Vert _{H})\) as \(t\rightarrow0\). By (3.7), we see that
where C is independent of n. Hence \(r(n,t)=o(t)\). By (3.8)-(3.10) and letting \(t\rightarrow0\), we have
where C is independent of n. Hence \(I'(u_{n},v_{n})\rightarrow0\), i.e. \(\{(u_{n},v_{n})\}\) is a bounded \((PS)_{B}\) sequence for I. □
Proof of Theorem 1.1 with \(\beta\in(-1,0)\)
By Lemmas 3.3 and 3.1, there exists a bounded \((PS)_{B}\) sequence \(\{(u_{n},v_{n})\}\subset M\) satisfying that
where \(C_{0}>0\) is given in Lemma 3.1. Up to a subsequence, there exists \((u,v)\in H_{r}\) such that \((u_{n},v_{n})\rightharpoonup(u,v)\) in \(H_{r}\). Then \(I'(u,v)=0\). Moreover, by (F1), (F3) we see that
If \(u\equiv0\) and \(v\equiv0\), then
where \(o_{n}(1)\rightarrow0\) as \(n\rightarrow+\infty\). Similarly to the proof of (3.2) in Lemma 3.2, there exist \(t_{n},s_{n}>0\) such that \((t_{n}u_{n},s_{n}v_{n})\in P\), i.e.
So \(J(t_{n}u_{n},s_{n}v_{n})=\frac{1}{N}\int_{\mathbb{R}^{N}}(\vert \nabla (t_{n}u_{n}) \vert ^{2}+ \vert \nabla(s_{n}v_{n})\vert ^{2})\geq A\). Set
By (3.13), we see that \(c_{1}+e\leq d_{1}\) and \(c_{2}+e\leq d_{2}\). By (3.1) we have \(e^{2}< d_{1}d_{2}\). If \(t_{n}\rightarrow+\infty\) as \(n\rightarrow+\infty\), then the first equation of (3.14) implies that \(s_{n}\rightarrow+\infty\). Hence, by the second equation of (3.14), we show that
which is a contradiction. So we may assume that \(t_{n}\rightarrow t_{\infty}\geq0\) and \(s_{n}\rightarrow s_{\infty}\geq0\).
If \(e=0\), then (3.13) and (3.14) imply that \(t_{\infty}, s_{\infty}\leq1\). If \(e>0\), we assume that \(t_{\infty}>1\). Then, by the first equation of (3.14), we have \(s_{\infty}>1\). Similarly to the proof of (3.15), we see that
which is a contradiction. Therefore, \(t_{\infty}\leq1\). Similarly, \(s_{\infty}\leq1\). So we have
By Lemma 3.2 and Section 1, we have \(\frac{2}{N}S^{\frac{N}{2}}=A\leq B< B_{1}+\frac{1}{N}S^{\frac{N}{2}}\), which implies that \(\frac{1}{N}S^{\frac{N}{2}}\leq B_{1}\). It contradicts (2.4). Therefore the case \(u,v\equiv0\) does not occur.
If \(u\not\equiv0\) and \(v\equiv0\), then u is a nontrivial solution of \(-\Delta u+u= \vert u \vert ^{2^{*}-2}u+f(u)\) in \(\mathbb{R}^{N}\). Then \(I(u,0)\geq B_{1}\). By \((u_{n},v_{n})\in M\) and \(\beta<0\), we have
where \(o_{n}(1)\rightarrow0\) as \(n\rightarrow+\infty\). By (3.11) we have \(\Vert v_{n} \Vert ^{2}\geq S^{\frac{N}{2}}\). So by (3.12) we see that
which is a contradiction to Lemma 3.2. So \(u\not\equiv0\) and \(v\equiv0\) do not occur. Similarly, \(u\equiv0\) and \(v\not\equiv0\) do not occur. Therefore, \(u\not\equiv0\) and \(v\not\equiv0\), i.e. \((u,v)\) is a nontrivial solution to (1.1). Similarly to the proof in Section 2, we get that \((u,v)\) is a radial and positive least energy solution of (1.1) with \(I(u,v)=B\). □
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The authors are partially supported by NSFC (11501428, 11501143). The first author is partially supported by the PhD launch scientific research projects of Guizhou Normal University No. 2014.
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Peng, YF., Ye, HY. Positive solutions for coupled Schrödinger system with critical exponent in \(\mathbb{R}^{N}\) (\(N\geq5\)). Bound Value Probl 2017, 104 (2017). https://doi.org/10.1186/s13661-017-0834-5
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DOI: https://doi.org/10.1186/s13661-017-0834-5