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Sign-changing solutions for coupled Schrödinger system
Boundary Value Problems volume 2024, Article number: 69 (2024)
Abstract
In this paper we study the following nonlinear Schrödinger system:
where \(3\leq p, q<5\), α, β are positive parameters. We show that there exists \(\lambda _{k}>0\) such that the equation has at least k radially symmetric sign-changing solutions and at least k seminodal solutions for each \(k\in \mathbb{N}\) and \(\lambda \in (0, \lambda _{k})\). Moreover, we show the existence of a least energy radially symmetric sign-changing solution for each \(\lambda \in (0, \lambda _{0})\) where \(\lambda _{0}\in (0, \lambda _{1}]\).
1 Background and main results
Consider the following nonlinear coupled Schrödinger system:
Here \(\Omega =\mathbb{R}^{N}\) or Ω is a smooth bounded domain in \(\mathbb{R}^{N}\), α, β are positive parameters and \(\lambda \neq 0\) is a coupling constant.
In the case \(p=q=3\), system (1.1) becomes the cubic system:
which arises in the study of many physical phenomena like nonlinear optics and Bose–Einstein condensation (cf. [15, 17]). Therefore, in the last decades, system (1.2) has received great interest from mathematicians. When Ω is the entire space \(\mathbb{R}^{N}\), the existence of least energy and other finite energy solutions of (1.2) was studied in [2, 11, 12, 18, 21, 22, 27] and the references therein. In particular, when \(\lambda >0\) is sufficiently large, infinitely many radially symmetric sign-changing solutions of (1.2) were obtained in [23]. Liu and Wang [20] studied a general m-coupled system (\(m\ge 2\)) and proved that system (1.2) has infinitely many nontrivial solutions, but whether solutions obtained in [20] are positive or sign-changing cannot be determined there (see also [21]). When \(\Omega \subset \mathbb{R}^{N}\) (\(N=2, 3\)) is a smooth bounded domain, there are also many papers studying (1.2). Lin and Wei [18] proved that a least energy solution of (1.2) exists within an appropriate range of λ. Dancer, Wei, and Weth [14] and Noris and Ramos [24] proved the existence of infinitely many positive solutions of (1.2). When Ω is a ball, a multiplicity result on positive radially symmetric solutions was given in [29]. Later, by using a global bifurcation approach, the result of [29] was reproved by [4] without requiring the symmetric condition. Under some more general assumptions, Sato and Wang [26] proved that system (1.2) has infinitely many semipositive solutions (i.e., at least one component is positive). In [14], the authors proved the existence of unbounded sequence solutions for \(N\leq 3\) and \(\lambda \leq -1\). As pointed out above, for \(\lambda \leq -1\), Wei and Weth [29] proved that (1.2) has a radially symmetric solution, which turns out to be a positive solution.
We remark that the existence of infinitely many sign-changing solutions or seminodal solutions to (1.2) was solved by Chen, Lin, and Zou [10] and Liu, Liu, and Wang [19] independently, where \(N\leq 3\) and \(\lambda <0\).
To the best of our knowledge, the existence of sign-changing solutions to (1.1) has not ever been studied in the literature when \(\Omega =\mathbb{R}^{3}\) and \(3\leq p, q<5\). The main goal of this paper is to study the existence of sign-changing solutions, seminodal solutions, and least energy sign-changing solutions to problem (1.1) when \(\lambda >0\) is small. This will complement the study made in [14, 19, 21, 22, 29].
Definition 1.1
A solution \((u, v)\) is called nontrivial if \(u\not \equiv 0\) and \(v\not \equiv 0\), a solution \((u, v)\) is semitrivial if \((u, v)\) is type of \((u, 0)\) or \((0, v)\). We call a solution \((u, v)\) positive if \(u>0\) and \(v>0\) in \(\mathbb{R}^{N}\), a solution \((u, v)\) sign-changing if both u and v change sign, a solution \((u, v)\) seminodal if one changes sign and the other one is positive.
The first main result of the current paper is as follows.
Theorem 1.1
Assume \(\alpha , \beta >0\). Then for any \(k\in \mathbb{N}\) there exists \(\lambda _{k}>0\) such that system (1.1) possesses at least k radially symmetric sign-changing solutions for each fixed \(\lambda \in (0, \lambda _{k})\).
We can also study some further properties of the sign-changing solutions obtained in Theorem 1.1. It is well known that a nontrivial solution \((u, v)\in H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})\) is called a least energy solution if its energy is minimal among the energy of all nontrivial solutions. A sign-changing solution is called a least energy sign-changing solution if it has the least energy among all sign-changing solutions. Precisely, we have the following theorem.
Theorem 1.2
Assume \(\alpha , \beta >0\). Then there exists \(\lambda _{0}\in (0, \lambda _{1}]\) such that system (1.1) possesses a least energy radially symmetric sign-changing solution for each fixed \(\lambda \in (0, \lambda _{0})\).
Theorem 1.3
Assume \(\alpha , \beta >0\). Then for any \(k\in \mathbb{N}\) there exists \(\lambda _{k}>0\) such that system (1.1) possesses at least k seminodal solutions for each fixed \(\lambda \in (0, \lambda _{k})\).
Remark 1.1
We can prove that system (1.1) possesses at least k seminodal solutions with the first component positive and the second component radially symmetric sign-changing or the first component radially symmetric sign-changing and the second component positive.
The structure of this paper is as follows. In Sect. 2 we prove the existence of at least k radially symmetric sign-changing solutions. The main tool will be the use of a new notion of vector genus by [28] and a new constrained problem by [10], which will be used to construct minimax values. Remark that the ideas in [10, 28] cannot be used directly, and here we will give some new ideas. The crucial idea in this paper is turning to study a new problem with two constraints to obtain sign-changing solutions of (1.1). This idea has never been used for (1.1) in the literature up to our knowledge. We will give all the necessary details of the proof. Section 3 is then dedicated to the proof of Theorem 1.2 by using a minimizing argument. Finally in Sect. 4 we will present the proof of Theorem 1.3 applying the arguments in Sect. 2 and Sect. 3.
We give some notations here. Throughout this paper, we denote the norm of \(L^{p}(\mathbb{R}^{N})\) by \(|u|_{p}= (\int _{\mathbb{R}^{N}} |u|^{p} \,dx )^{\frac{1}{p}}\), the norm of \(H^{1}(\mathbb{R}^{N})\) by \(\|u\|^{2}=\int _{\mathbb{R}^{N}}(|\nabla u|^{2}+ |u|^{2}) \,dx \), and positive constants (possibly different in different places) by C. Define \(H_{r}:= H_{r}^{1}(\mathbb{R}^{N})\times H_{r}^{1}(\mathbb{R}^{N})\) as a subspace of \(H:= H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})\) with norm \(\|(u, v)\|_{H_{r}}^{2}:=\|u\|_{\alpha}^{2}+\|v\|_{\beta}^{2}\) where
2 Proof of Theorem 1.1
In this section, we assume that \(N=3\), \(3\leq p, q<2^{*}-1=5\) and \(\alpha , \beta >0\). Without loss of generality, we assume \(p\leq q\). Let \(\lambda \in (0, 1)\). For any \(k\in \mathbb{N}\), let \(X_{k+1}\subset H_{r}^{1}(\mathbb{R}^{3})\), \(\operatorname{dim}X_{k+1}=k+1\), and there exists \(u_{0}\in X_{k+1}\) and \(u_{0}>0\). Then there exists \(m>0\) such that for any \((u, v)\in X_{k+1}\times X_{k+1}\) satisfying \(|u|_{p+1}^{p+1}, |v|_{q+1}^{q+1}<2\), we have
Without loss of generality, we can assume \(m>1\). Obviously, the sign-changing solutions of system (1.1) are the critical points of the \(C^{2}\) functional \(\Phi _{\lambda}: H_{r}\rightarrow \mathbb{R}\) given by
We will look for solutions of Eq. (1.1) as critical points of the functional \(\Phi _{\lambda}\) restricted to the sphere
To obtain at least k sign-changing critical points, we need to define several minimax energy levels using a new definition of vector genus introduced by [28]. As in [28], we recall vector genus and take the transformations
Consider the class of sets
and for each \(A\in \mathcal{F}\) and \(k_{1}, k_{2}\in \mathbb{N}\), the class of functions
where \(\mathbb{R}^{0}:=\{0\}\).
Definition 2.1
(Vector genus, see [28]) For every nonempty and closed set \(A\subset H_{0}^{1}(\Omega )\) such that \(-A=A\), we define
and \(\gamma (A):=\infty \) if no such k exists.
Let \(A\in \mathcal{F}\) and take any \(k_{1}, k_{2}\in \mathbb{N}\). We say that \(\gamma (A)\geq (k_{1}, k_{2})\) if for every \(f\in F_{(k_{1}, k_{2})}(A)\) there exists \((u, v)\in A\) such that \(f(u, v)=(f_{1}(u, v), f_{2}(u, v))=(0, 0)\). We denote
Remark 2.1
Note that Definition 2.1 does not actually define the quantity \(\gamma (A)\) but gives the meaning of \(\gamma (A)\geq (k_{1}, k_{2})\) only. A different notation of genus was introduced by Chang, Wang, and Zhang in [8].
Lemma 2.1
(see [28])
Let \(f=(f_{1}, f_{2}): \prod_{i=1}^{2} S^{k_{i}}\rightarrow \prod_{i=1}^{2} \mathbb{R}^{k_{i}}\) be a continuous function such that \(f_{i}(\sigma _{i}(u, v))=-f_{i}(u, v)\), \(f_{i}(\sigma _{j}(u, v))=f_{i}(u, v)\) for any \(i, j=1, 2\), \(i\neq j\), then there exists \((u_{0}, v_{0})\in \prod_{i=1}^{2} S^{k_{i}}\) such that \(f(u_{0}, v_{0})=(0, \ldots ,0)\).
Lemma 2.2
(see [28])
The following properties hold.
-
(1)
Take \(A_{1}\times A_{2}\subset \mathcal{A}\) and let \(\eta _{i}: S^{k_{i}-1}\rightarrow A_{i}\) be a homeomorphism such that \(\eta _{i}(-x)=-\eta _{i}(x)\) for every \(x\in S^{k_{i}-1}\), \(i=1, 2\). Then \(A_{1}\times A_{2} \in \Gamma ^{(k_{1}, k_{2})}\), where \(S^{k_{i}-1}=\{x\in \mathbb{R}^{k_{i}}: |x|=1\}\).
-
(2)
We have \(\overline{\eta (A)}\in \Gamma ^{(k_{1}, k_{2})}\) whenever \(A \in \Gamma ^{(k_{1}, k_{2})}\) and a continuous map \(\eta : A\rightarrow \mathcal{A}\) is such that \(\eta \circ \sigma _{i}=\sigma _{i}\circ \eta \), \(\forall i=1, 2\).
Together with the notation of vector genus, to obtain sign-changing solutions, we will use cones of positive or negative functions based on the works such as [5, 13, 30]. We define the cone
and take \(\mathcal{P}:=\bigcup_{i=1}^{2} (\mathcal{P}_{i}\cup - \mathcal{P}_{i} )\). Moreover, for any \(\delta >0\), we define
where
where \(u^{\pm}:=\max \{0, \pm u\}\).
Lemma 2.3
For any \(0<\delta <2^{-\frac{1}{p+1}}\), there holds \(A\backslash \mathcal{P}_{\delta }\neq \emptyset \) whenever \(A \in \Gamma ^{(k_{1}, k_{2})}\) with \(k_{1}, k_{2}\geq 2\).
Proof
For any \(A \in \Gamma ^{(k_{1}, k_{2})}\), define \(f=(f_{1}, f_{2})\) by
then \(f\in F_{(k_{1}, k_{2})}(A)\), so by Definition 2.1, there exists \((u_{0}, v_{0})\in A\) such that \(f(u_{0}, v_{0})=(0, \ldots ,0)\). By \(A\in \mathcal{A}\), we deduce that
therefore, \(\operatorname{dist} ((u_{0}, v_{0}), \mathcal{P})=2^{-\frac{1}{p+1}}\), and so \((u_{0}, v_{0})\in A \backslash \mathcal{P}_{\delta}\) for any \(0<\delta <2^{-\frac{1}{p+1}}\). □
For technical reasons, we will work on the neighborhood of \(\mathcal{A}\) in \(H_{r}^{1}(\mathbb{R}^{3})\),
when \(u\in \mathcal{A}^{*}\), \((u, v)\not \equiv (0, 0)\). Define
Let \(S_{p}\) and \(S_{q}\) be the sharp constants of the Sobolev embedding \(H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow L^{p+1}(\mathbb{R}^{3})\) and \(H_{r}^{1}(\mathbb{R}^{3})\hookrightarrow L^{q+1}(\mathbb{R}^{3})\), respectively,
For any \((u, v)\in H_{r}\backslash \{(0, 0)\}\), we have
where \(t_{u,v,\lambda}, s_{u,v,\lambda}\geq 0\) satisfy
Note that for \(t, s\geq 0\),
Define
and
which implies
Since \(F_{1}(u, v, \lambda ; t, s)\) and \(G_{1}(u, v, \lambda ; t, s)\) are decreasing with respect to \(t>0\) and \(s>0\), respectively, \(F_{1}(u, v, \lambda ; 0, 0)>0\), \(G_{1}(u, v, \lambda ; 0, 0)>0\), so \(t_{u,v,\lambda}\), \(s_{u,v,\lambda}\) are unique. Note that for \(t, s\geq 0\), \(3\leq p, q<5\), by (2.9), we can choose some positive constant T such that \(\Phi _{\lambda}(tu, sv)<0\) for any \(t, s>T\), therefore, \(t_{u,v,\lambda}, s_{u,v,\lambda}\in [0, T]\).
Define
Then \(B_{m}\subset B_{\widetilde{m}}\), \(B_{m}^{*}\subset B_{\widetilde{m}}^{*}\).
Lemma 2.4
For any \(k\in \mathbb{N}\), there exist \(\widetilde{\lambda}\in (0, 1)\) and \(T_{1}>T_{2}>0\) such that for any \(\lambda \in (0, \widetilde{\lambda})\) and \((u, v)\in B_{\widetilde{m}}^{*}\), we have
Furthermore, there exist \(\lambda _{k}\in (0, \widetilde{\lambda}]\) and \(c_{k}>0\) such that for any \(\lambda \in (0, \lambda _{k})\), we have
Proof
We see from (2.9) and (2.10) that
Firstly, we claim that there exist \(\widetilde{\lambda}\in (0, 1)\) and \(T_{1}>T_{2}>0\) such that for any \(\lambda \in (0, \widetilde{\lambda})\) and \((u, v)\in B_{\widetilde{m}}^{*}\), we have
By (2.10),
Thus, we obtain that
Define
We see from (2.11) that \(\widetilde{\lambda}\in (0, 1)\). Moreover, by (2.7) and (2.10), for any \(\lambda \in (0, \widetilde{\lambda})\), we have
Then we get \(t_{u,v,\lambda}> (\frac{S_{p}}{8} )^{\frac{1}{p-1}}\). Similarly, we have \(s_{u,v,\lambda}> (\frac{S_{q}}{8} )^{\frac{1}{q-1}}\). Thus, we get
This completes \(T_{2}\leq t_{u,v,\lambda}\leq T_{1}\).
Now we prove the existence of \(\lambda _{k}\) and \(c_{k}\). For any \((u, v)\in \overline{B}_{\widetilde{m}}\) and \(\lambda \in (0, \widetilde{\lambda}]\), by (2.14), there holds
Hence,
and
then by (2.11), we can choose
such that \(c_{k}>0\) for any \(0<\lambda <\lambda _{k}\) the conclusion holds. □
For any \((u, v)\in B_{\widetilde{m}}^{*}\), the following linear problem
has a unique solution \((\varphi , \psi )\in H_{r}\setminus \{ (0, 0)\}\). Then we can choose \(\lambda _{k}\) small enough such that for any \(\varphi , \psi \in H_{r}^{1}(\mathbb{R}^{3})\),
and
Define
then \(\mu >0\), \(\nu >0\) and \((\widetilde{\varphi}, \widetilde{\psi}):=(\mu \varphi , \nu \psi )\) is the unique solution of
Fixed any \(k\in \mathbb{N}\), we define
There is an odd homeomorphism from \(S^{k}\) to \(A_{1}\) and \(A_{2}\). By Lemma \(2.2 (1)\), \(A : =A_{1}\times A_{2}\in \Gamma ^{(k+1, k+1)}\). Observe that from (2.1) we deduce that \(A\subset B_{m}\), and so by (2.13),
Define
Observe that \(\Gamma _{\lambda}^{(k_{1}, k_{2})}\neq \emptyset \), \(\Gamma _{\lambda}^{(k_{1}, k_{2})}\subset \Gamma _{\lambda}^{(k_{1}', k_{2}')}\) when \(k_{1}\geq k_{1}'\) and \(k_{2}\geq k_{2}'\). We are now ready to define a sequence of minimax energy levels which will turn out to be critical levels for \(\Phi _{\lambda}\) over \(\mathcal{A}\). For every \(k_{1}, k_{2}\in [2, k+1]\) and \(0<\delta <2^{-\frac{1}{p+1}}\), define
It is easy to see that
As a step towards to the proof of Theorem 1.1, we will prove that \(d_{\lambda ,\delta}^{k_{1},k_{2}}\) is indeed a critical level of \(\Phi _{\lambda}\) for δ sufficiently small. To prove Theorem 1.1, it is necessary to find a pseudogradient for \(\Phi _{\lambda}\) over \(\mathcal{A}\) for which \(\mathcal{P}_{\delta}\) is positively invariant for the associated flow. We can now define the operator
that is, for any \((u, v)\in B_{\widetilde{m}}^{*}\), \(K(u, v)=(\widetilde{\varphi}, \widetilde{\psi})\) is the unique solution of (2.16). It is easy to prove that \(K(\sigma _{i}(u, v))=\sigma _{i}(K(u, v))\), \(i=1, 2\).
Now, we give some property of the operator K. We can now prove that K is a compact \(C^{1}\) operator.
Lemma 2.5
The operator K is of class \(C^{1}\).
Proof
Define \(C^{1}\) maps \(J_{i}: B_{\widetilde{m}}^{*}\times H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}\rightarrow H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}\), \(i=1, 2\), by
and
then by (2.16), \(J_{1} ((u, v), \widetilde{\varphi}, \mu )=J_{2} ((u, v), \widetilde{\psi}, \nu )=0\). Moreover, the derivatives of \(J_{1}\) and \(J_{2}\) with respect to \((\omega , \gamma )\) at the point \(((u, v), \widetilde{\varphi}, \mu )\) and \(((u, v), \widetilde{\psi}, \nu )\) in the direction \((\omega _{0}, \gamma _{0})\), respectively, are
and
We claim that \(D_{\omega ,\gamma}J_{1} ((u, v), \widetilde{\varphi}, \mu )\) and \(D_{\omega ,\gamma}J_{2} ((u, v), \widetilde{\psi}, \nu )\) are bijective maps. In fact, for any \((\omega , \gamma )\in H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}\), the following linear problems
have unique solutions \(\omega _{1}, \omega _{2}\in H_{r}^{1}(\mathbb{R}^{3})\), \(\omega _{2}\neq 0\) by \(u\in B_{\widetilde{m}}^{*}\) and (2.12), then we define
we have
that is, \(D_{\omega ,\gamma}J_{1} ((u, v), \widetilde{\varphi}, \mu )\) is surjective. Similarly, \(D_{\omega ,\gamma}J_{2} ((u, v), \widetilde{\psi}, \nu )\) is surjective.
If \(D_{\omega ,\gamma}J_{1} ((u, v), \widetilde{\varphi}, \mu )(\omega _{0}, \gamma _{0})=(0, 0)\), then
so \(\omega _{0}\equiv 0\), \(\gamma _{0} t_{u,v,\lambda}^{p-1}|u|^{p-1}u\equiv 0\), by \(t_{u,v,\lambda}>0\), \(u\in B_{\widetilde{m}}^{*}\), we have \(\gamma _{0}=0\), this implies \(D_{\omega ,\gamma}J_{1} ((u, v), \widetilde{\varphi}, \mu )\) is injective. Therefore, \(D_{\omega ,\gamma}J_{1} ((u, v), \widetilde{\varphi}, \mu )\) is bijective. Similarly, \(D_{\omega ,\gamma}J_{2} ((u, v), \widetilde{\psi}, \nu )\) is a bijective map. Then we can apply the implicit theorem to the \(C^{1}\) maps \(D_{\omega ,\gamma}J_{1} ((u, v), \widetilde{\varphi}, \mu )\) and \(D_{\omega ,\gamma}J_{2} ((u, v), \widetilde{\psi}, \nu )\), we have the conclusions. □
Lemma 2.6
Let \(\{(u_{n}, v_{n})\}_{n\geq 1}\subset B_{\widetilde{m}}\). For any \(0<\lambda <\lambda _{k}\), there exists \((\widetilde{\varphi}_{0}, \widetilde{\psi}_{0})\in H_{r}\) such that, up to a subsequence,
Proof
Since \(\{(u_{n}, v_{n})\}_{n\geq 1}\subset B_{\widetilde{m}}\), we have
and \(|u_{0}|_{p+1}=|v_{0}|_{q+1}=1\). By (2.12), we also have
Then by (2.3), (2.7), (2.12), and (2.15),
Similar estimates hold for \(\psi _{n}\), we get \(\|\psi _{n}\|^{2}_{\beta}\leq C \| \psi _{n}\|_{\beta}\), so \(\{(\varphi _{n}, \psi _{n})\}_{n\geq 1}\subset H_{r}\) are bounded. Thus
Then by (2.15) and Hölder’s inequality,
Hence,
Similarly, we have \(\|\psi _{n}\|^{2}_{\beta}=\|\psi _{0}\|^{2}_{\beta}+o(1)\). Therefore, we have \((\varphi _{n}, \psi _{n})\rightarrow (\varphi _{0}, \psi _{0})\) strongly in \(H_{r}\) and \((\varphi _{0}, \psi _{0})\) satisfies
since \(|u_{0}|_{p+1}=|v_{0}|_{q+1}=1\), so \(\varphi _{0}\neq 0\), \(\psi _{0}\neq 0\) and
We see that
This completes the proof. □
Define
then by (2.13) we obtain \(B_{m}\subset B_{\widetilde{m}, \lambda}\).
Lemma 2.7
For any \(0<\delta <2^{-\frac{1}{p+1}}\) sufficiently small, we have that
Proof
Suppose by contradiction that there exist \(\delta _{n}\rightarrow 0\) and \((u_{n}, v_{n})\in B_{\widetilde{m}, \lambda}\) satisfying \(\operatorname{dist} ((u_{n}, v_{n}), \mathcal{P} )<\delta _{n}\) and \(\operatorname{dist} (K(u_{n}, v_{n}), \mathcal{P} )\geq \frac{\delta _{n}}{2}\). We suppose that \(\operatorname{dist} ((u_{n}, v_{n}), \mathcal{P} )=|u_{n}^{-}|_{p+1}\) without loss of generality. Let \((\widetilde{\varphi}_{n}, \widetilde{\psi}_{n})=K(u_{n}, v_{n})\) and \(\widetilde{\varphi}_{n}=\mu _{n}\varphi _{n}\), \(\widetilde{\psi}_{n}=\nu _{n} \psi _{n}\). By a similar proof as in Lemma 2.6, we have that \(\mu _{n}\) and \(\nu _{n}\) are uniformly bounded. By (2.12), we can take \(\lambda _{k}\) smaller if necessary such that for any \(\lambda \in (0, \lambda _{k})\) and \((u, v)\in B_{\widetilde{m}}^{*}\), we get
This together with (2.7) and (2.16) allows us to get
and hence \(\operatorname{dist} (K(u_{n}, v_{n}), \mathcal{P} )\leq | \widetilde{\varphi}_{n}^{-}|_{p+1}\leq C\delta _{n}^{p} < \frac{\delta _{n}}{2}\) for n sufficiently large, which is a contradiction. This completes the proof. □
Now define a map
It is easy to prove that \(V(\sigma _{i}(u, v))=\sigma _{i}(V(u, v))\), \(i=1, 2\). We will prove that if \((u, v)\in B_{\widetilde{m}}\backslash \mathcal{P}\), \(V(u, v)=0\), then \((t_{u,v,\lambda}u, s_{u,v,\lambda}v)\) is a sign-changing solution of Eq. (1.1). Firstly, we prove that V satisfies the Palais–Smale type condition and V is a pseudogradient for \(\sup_{t,s\geq 0} \Phi _{\lambda}(tu, sv)\) over \(B_{\widetilde{m}}\). Denote \(\Psi _{\lambda}(u, v):=\sup_{t,s\geq 0} \Phi _{\lambda}(tu, sv)\).
Lemma 2.8
(Palais–Smale type condition) Let \((u_{n}, v_{n})\in B_{\widetilde{m}}\) be such that
Then there exists \((u_{0}, v_{0})\in B_{\widetilde{m}}\) such that \((u_{n}, v_{n})\rightarrow (u_{0}, v_{0})\) strongly in \(H_{r}\), up to a subsequence, and \(V(u_{0}, v_{0})=0\). We also have
Proof
Similar as Lemma 2.6, we have, up to a subsequence,
Then we have, as \(n\rightarrow \infty \),
whence
Then \((u_{n}, v_{n})\rightarrow (u_{0}, v_{0})\) strongly in \(H_{r}\) and \((u_{0}, v_{0})\in \overline{B}_{\widetilde{m}}\),
then by (2.13), we have \((u_{0}, v_{0})\in B_{\widetilde{m}}\), \(V(u_{0}, v_{0})=\lim_{n\rightarrow \infty}V(u_{n}, v_{n})=0\).
Finally, we prove that V is a pseudogradient for \(\Psi _{\lambda}(u, v)\) over \(B_{\widetilde{m}}\). By (2.9) and (2.10) we can prove that
hold for any \((u, v)\in \mathcal{B}_{\widetilde{m}}\) and \(\omega \in H_{r}^{1}(\mathbb{R}^{3})\). We can take \(\lambda _{k}\) smaller if necessary such that for any \(\lambda \in (0, \lambda _{k})\) by (2.19), (2.20), (2.12), and (2.16)
This completes the proof. □
Lemma 2.9
There exists a unique global solution \(\eta =(\eta _{1}, \eta _{2}): \mathbb{R}^{+}\times B_{\widetilde{m}, \lambda}\rightarrow H_{r}\) for the initial value problem
Moreover,
-
(1)
For any \(t>0\) and \((u, v)\in B_{\widetilde{m},\lambda}\), there holds \(\eta (t, (u, v))\in B_{\widetilde{m},\lambda}\);
-
(2)
For any \(t>0\), \((u, v)\in B_{\widetilde{m},\lambda}\), there holds \(\eta (t, \sigma _{i}(u, v))=\sigma _{i} ( \eta (t, (u, v)) )\), \(i=1, 2\);
-
(3)
For any \((u, v)\in B_{\widetilde{m},\lambda}\), \(\Psi _{\lambda} ( \eta (t, (u, v)) )\) is nonincreasing in t;
-
(4)
There exists \(\delta _{0}\in (0, 2^{-\frac{1}{p+1}})\) such that, for any \(0<\delta <\delta _{0}\), \((u, v)\in B_{\widetilde{m},\lambda}\cap \mathcal{P}_{\delta}\) and \(t>0\), there holds \(\eta (t, (u, v))\in \mathcal{P}_{\delta}\).
Proof
It follows from Lemma 2.5 that \(V\in C^{1}(B_{\widetilde{m}}^{*}, H_{r})\). As \(B_{\widetilde{m},\lambda}\subset B_{\widetilde{m}}\subset B_{ \widetilde{m}}^{*}\), we get that \(V\in C^{1}(B_{\widetilde{m},\lambda}, H_{r})\). Then there exists a solution \(\eta : [0, T_{\max})\times B_{\widetilde{m},\lambda}\rightarrow H_{r}\), where \(T_{\max}\) is the maximal time such that (2.21) has a solution \(\eta \in B_{\widetilde{m}}^{*}\).
For any \((u, v)\in B_{\widetilde{m},\lambda}\) and \(t\in (0, T_{\max})\), there holds
so we have
Then
Similarly, there holds
we deduce that for any \((u, v)\in B_{\widetilde{m},\lambda}\) and \(t\in [0, T_{\max})\),
Thus, for any \(t\in [0, T_{\max})\), \((u, v)\in B_{\widetilde{m}}\), we have \(\eta (t, (u, v))\in B_{\widetilde{m}}^{*}\cap \mathcal{A}=B_{ \widetilde{m}}\). If \(T_{\max}<+\infty \), then \(\eta (T_{\max}, (u, v))\in \mathcal{C}_{\widetilde{m}}\). There holds \(\Psi _{\lambda} (\eta (T_{\max}, (u, v)) )\geq c_{k}\) by (2.13). Moreover,
On the other hand, we see from \((u, v)\in B_{\widetilde{m},\lambda}\) and (2.22),
it yields a contradiction, so \(T_{\max}=+\infty \), \(\eta (t, (u, v))\in B_{\widetilde{m},\lambda}\) and \((1)(3)\) hold.
Since \(V(\sigma _{i}(u, v))=\sigma _{i}(V(u, v))\), \(i=1, 2\), then \((2)\) holds.
Take \(\delta _{0}>0\) as in Lemma 2.7, note that as \(t\rightarrow 0\),
hence for any \(0<\delta <\delta _{0}\), \((u, v)\in B_{\widetilde{m},\lambda}\cap \mathcal{P}_{\delta}\), we have
for sufficiently small \(t>0\), and \((4)\) holds. This completes the proof. □
To prove Theorem 1.1, we will give that \(d_{\lambda ,\delta}^{k_{1},k_{2}}\) is indeed critical energy level for \(\delta >0\) sufficiently small.
Lemma 2.10
For any \(k\in \mathbb{N}\), \(k_{1}, k_{2}\in [2, k+1]\), \(0<\delta <\delta _{0}\), and \(0<\lambda <\lambda _{k}\), there exists \((\widetilde{u}_{0}, \widetilde{v}_{0})\in H_{r}\) such that \((\widetilde{u}_{0}, \widetilde{v}_{0})\) is a sign-changing solution of Eq. (1.1) and \(\Phi _{\lambda}(\widetilde{u}_{0}, \widetilde{v}_{0})=d_{\lambda , \delta}^{k_{1},k_{2}}\).
Proof
By (2.18) we see that \(d_{\lambda ,\delta}^{k_{1},k_{2}}< c_{k}\). Assume that there is small \(0<\varepsilon <1\) such that for any \((u, v)\in B_{\widetilde{m},\lambda}\), \(|\Psi _{\lambda}(u, v)-d_{\lambda ,\delta}^{k_{1},k_{2}}|\leq 2 \varepsilon \), \(\operatorname{dist} ((u, v), \mathcal{P} )\geq \delta \), there holds \(\|V(u, v)\|_{H_{r}}^{2}\geq \varepsilon \). By (2.17), there exists \(A\in \Gamma _{\lambda}^{(k_{1},k_{2})}\) such that
then \(\sup_{A} \Psi _{\lambda}(u, v)< c_{k}\), \(A\subset B_{\widetilde{m},\lambda}\). Thus we consider the set \(A_{0}=\eta (\frac{4}{T_{2}^{2}}, A)\), \(A_{0}\in B_{\widetilde{m},\lambda}\) by Lemma 2.9(1). From Lemma 2.2(2), Lemma 2.3, and Lemma 2.9(3), we get
so \(A_{0}\in \Gamma _{\lambda}^{(k_{1},k_{2})}\) and \(A_{0}\backslash{\mathcal{P}_{\delta}}\neq \emptyset \). Then, by (2.15), (2.19), and Lemma 2.9(3), for the \(\varepsilon >0\), \(t\in [0, \frac{4}{T_{2}^{2}}]\), there exists \((u, v)\in A\) such that \(\eta (\frac{4}{T_{2}^{2}}, (u, v))\in A_{0}\backslash {\mathcal{P}_{ \delta}}\) satisfying
We conclude that \(\|V (\eta (t, (u, v)) )\|^{2}_{H_{r}}\geq \varepsilon \) for any \(t\in [0, \frac{4}{T_{2}^{2}}]\) and
Therefore, by integrating over 0 to \(\frac{4}{T_{2}^{2}}\) and (2.24), we have
it yields a contradiction, and therefore, for any \(\varepsilon =\frac{1}{n}>0\), there exists \((u_{n}, v_{n})\in B_{\widetilde{m},\lambda}\) such that
By Lemma 2.8, there exists \((u_{0}, v_{0})\in B_{\widetilde{m},\lambda}\) such that \((u_{n}, v_{n})\rightarrow (u_{0}, v_{0})\) strongly in \(H_{r}\), up to a subsequence. Hence, we have
We conclude that \((u_{0}, v_{0})\) is sign-changing and \((u_{0}, v_{0})=K(u_{0}, v_{0})=(\widetilde{\varphi}_{0}, \widetilde{\psi}_{0})\). It follows from (2.16) that \((u_{0}, v_{0})\) satisfies
On the other hand, \(t_{u_{0},v_{0},\lambda}\) and \(s_{u_{0},v_{0},\lambda}\) satisfy
then we have \(\mu =\nu =1\). Hence, we have that \((t_{u_{0},v_{0},\lambda} u_{0}, s_{u_{0},v_{0},\lambda} v_{0})\) is a sign-changing solution of Eq. (1.1) by problem (2.25) and
This completes the proof. □
Proof of Theorem 1.1
Observe that from Lemma 2.10 we know that for any \(k\in \mathbb{N}\), \(k_{1}, k_{2}\in [2, k+1]\), \(0<\delta <\delta _{0}\), and \(0<\lambda <\lambda _{k}\), there exists a sign-changing solution \((\widetilde{u}_{0}, \widetilde{v}_{0})\) with \(\Phi _{\lambda}(\widetilde{u}_{0}, \widetilde{v}_{0})=d_{\lambda , \delta}^{k_{1},k_{2}}\). For any fixed \(k_{1}\in [2, k+1]\), we have
Suppose that problem (1.1) has at most \(k-1\) sign-changing solutions by contradiction, then there exists \(k_{2}\in [2, k]\) satisfying
Now define
then \(\mathcal{M}\subset \mathcal{F}\) is finite. So there exist \(N\in [1, k-1]\) and \(\{(u_{n}, v_{n})\}_{1\leq n\leq N}\subset \mathcal{M}\) such that
For any \(1\leq n\leq N\), there exist open neighborhoods \(\Omega _{n}^{1}\), \(\Omega _{n}^{2}\), \(\Omega _{n}^{3}\), \(\Omega _{n}^{4}\) of \(\{(u_{n}, v_{n})\}\), \(\{(-u_{n}, v_{n})\}\), \(\{(u_{n}, -v_{n})\}\), \(\{(-u_{n}, -v_{n})\}\), respectively, such that
Define
we can choose \(\rho >0\) small enough such that \(\mathcal{M}_{2\rho}\subset \Omega \). Since \(\mathcal{M}\) is finite, then there is \(\varepsilon _{0}\in (0, \frac{c_{k}-d}{2})\) such that for any \((u, v)\in B_{\widetilde{m}}\backslash (\mathcal{P}_{\delta }\cup \mathcal{M}_{\rho})\), \(|\Psi _{\lambda}(u, v)-d|\leq 2 \varepsilon _{0}\), we have
In fact, if for any \(\varepsilon =\frac{1}{n}>0\) there exists \((u_{n}, v_{n})\in B_{\widetilde{m}}\backslash (\mathcal{P}_{\delta } \cup \mathcal{M}_{\rho})\) satisfying \(|\Psi _{\lambda}(u_{n}, v_{n})-d|\leq 2 \varepsilon \), then there holds \(\|V(u_{n}, v_{n})\|_{H_{r}}^{2}\leq \varepsilon \). Then, by Lemma 2.8, there exists \((u_{0}, v_{0})\in B_{\widetilde{m}}\backslash (\mathcal{P}_{\delta } \cup \mathcal{M}_{\rho})\) such that \((u_{n}, v_{n})\rightarrow (u_{0}, v_{0})\) strongly in \(H_{r}\), up to a subsequence, \(\Psi _{\lambda}(u_{0}, v_{0})=d\) and \(V(u_{0}, v_{0})=0\). Therefore, \((u_{0}, v_{0})\in \mathcal{M}_{\rho}\). It yields a contradiction.
Moreover, for \((u, v)\in \mathcal{M}\), \(V(u, v)=0\), then for \(\rho >0\) small enough, there exists \(T_{0}>0\) such that for any \((u, v)\in \overline{\mathcal{M}}_{2\rho}\),
Let
By (2.17), for \(\varepsilon _{0}>0\), there exists \(A\in \Gamma _{\lambda}^{(k_{1},k_{2}+1)}\) such that
Let \(B:=A\backslash {\mathcal{M}_{2\rho}}\), then \(B\subset \mathcal{F}\).
We claim that \(\gamma (B)\geq (k_{1}, k_{2})\). In view of a contradiction, suppose that \(\gamma (B)<(k_{1}, k_{2})\). From Definition 2.1, we know that there exists \(f\in F_{(k_{1},k_{2})}(B)\) such that \(f(u, v)= (f_{1}(u, v), f_{2}(u, v) )\neq (0, 0)\) for any \((u, v)\in B\). Take \(\widetilde{f}=(\widetilde{f}_{1}, \widetilde{f}_{2})\in C (H_{r}, \mathbb{R}^{k_{1}-1}\times \mathbb{R}^{k_{2}-1} )\) such that \(\widetilde{f}|_{B}=f\) by Tietze’s extension theorem. Define
then \(F:=(F_{1}, F_{2})\in C (H_{r}, \mathbb{R}^{k_{1}-1}\times \mathbb{R}^{k_{2}-1} )\), \(F|_{B}=4\widetilde{f}\), \(F_{i}(\sigma _{i}(u, v))=-4\widetilde{f}_{i}(u, v)=-F_{i}(u, v)\) and \(F_{i}(\sigma _{j}(u, v))=4\widetilde{f}_{i}(u, v)=F_{i}(u, v)\), \(i\neq j\), \(i, j=1, 2\).
Define the continuous function
and \(g(\sigma _{1}(u, v))=g(u, v)\), \(g(\sigma _{2}(u, v))=-g(u, v)\). Take \(\widetilde{g}\in C(H_{r}, \mathbb{R})\) such that \(\widetilde{g}|_{\Omega}=g\) by Tietze’s extension theorem. Define
then \(G\in C(H_{r}, \mathbb{R})\), \(G|_{\Omega}=4\widetilde{g}\), \(G(\sigma _{1}(u, v))=G(u, v)\), and \(G(\sigma _{2}(u, v))=-G(u, v)\). Therefore, we can define
then \(H:=(H_{1}, H_{2})\in C (A, \mathbb{R}^{k_{1}-1}\times \mathbb{R}^{k_{2}} )\) and \(H\in F_{(k_{1},k_{2}+1)}(A)\). Since \(A\in \Gamma _{\lambda}^{(k_{1},k_{2}+1)}\), \(\gamma (A)\geq (k_{1}, k_{2}+1)\), so there exists \((u, v)\in A\) such that \(H(u, v)=(0, 0)\). If \((u, v)\in B=A\backslash \mathcal{M}_{2\rho}\), then
a contradiction. Thus \((u, v)\in \mathcal{M}_{2\rho}\), then
a contradiction. Therefore, \(\gamma (B)\geq (k_{1}, k_{2})\).
Since \(B\subset A\subset B_{\widetilde{m}}\), \(\sup_{B} \Psi _{\lambda}(u, v)\leq \sup_{A} \Psi _{ \lambda}(u, v)< c_{k}\), then we have \(B\subset B_{\widetilde{m},\lambda}\) and \(B\in \Gamma _{\lambda}^{(k_{1},k_{2})}\). Define \(B_{0}:=\eta (\frac{\rho}{2T_{0}}, B)\), then \(B_{0}\subset B_{\widetilde{m},\lambda}\), \(B_{0}\in \Gamma ^{(k_{1},k_{2})}\), \(B_{0}\backslash P_{\delta}\neq \emptyset \), and \(\sup_{B_{0}} \Psi _{\lambda}(u, v)\leq \sup_{B} \Psi _{\lambda}(u, v)< c_{k}\) by Lemma 2.2(2) and Lemma 2.3, so \(B_{0}\in \Gamma _{\lambda}^{(k_{1},k_{2})}\). Thus \(\sup_{B_{0}\backslash \mathcal{P}_{\delta}} \Psi _{\lambda}(u, v)\geq d_{\lambda ,\delta}^{k_{1},k_{2}}\) by (2.17).
We claim that \(\eta (t, (u, v))\notin \mathcal{M}_{\rho}\) for any \(t\in (0, \frac{\rho}{2T_{0}})\), \((u, v)\in B\). In view of a contradiction, if there exists \(t_{0}\in (0, \frac{\rho}{2T_{0}})\) such that \(\eta (t_{0}, (u, v))\in \mathcal{M}_{\rho}\), for \((u, v)\in B=A\backslash \mathcal{M}_{2\rho}\), by the continuity of η, there exists \(0\leq t_{1}< t_{2}\leq t_{0}\) satisfying \(\eta (t_{1}, (u, v))\in \partial \mathcal{M}_{2\rho}\), \(\eta (t_{2}, (u, v))\in \partial \mathcal{M}_{\rho}\), and \(\eta (t, (u, v))\in \mathcal{M}_{2\rho}\backslash \mathcal{M}_{ \rho} \) for any \(t\in (t_{1}, t_{2})\). Then by (2.27) we have
so \(\frac{\rho}{2T_{0}}\leq t_{2}-t_{1}\leq t_{0}-0<\frac{\rho}{2T_{0}}\), this yields a contradiction.
For \(\varepsilon _{0}>0\), there exists \((u, v)\in B\) such that \(\eta (\frac{\rho}{2T_{0}}, (u, v))\in B_{0}\backslash \mathcal{P}_{ \delta}\) satisfies
Moreover, \(\eta (t, (u, v))\in B_{\widetilde{m},\lambda}\) for any \(t\geq 0\), then by Lemma 2.9(4), \(\eta (t, (u, v))\notin \mathcal{P}_{\delta}\) for any \(t\in [0, \frac{\rho}{2T_{0}}]\). Therefore,
In particular, \((u, v)\notin P_{\delta}\). Moreover, by (2.29) and Lemma 2.9\((3)\), we get
that is,
So we see from (2.26) and Lemma 2.8 that
Finally, we deduce from (2.28), (2.31), and (2.32) that
this yields a contradiction. This completes the proof. □
3 Proof of Theorem 1.2
Using Theorem 1.1, for \(k=1\), there exists \(\lambda _{1}>0\) such that system (1.1) has a radially symmetric sign-changing solution \((u_{1}, v_{1})\) for any \(\lambda \in (0, \lambda _{1})\) and for \(k_{1}=k_{2}=2\),
Let
then \(U_{\lambda}\neq \emptyset \) by Theorem 1.1, we can define
and \(d_{\lambda}< c_{1}\). Let \((u_{n}, v_{n})\in U_{\lambda}\) be a minimizing sequence of \(d_{\lambda}\) with \(\Phi _{\lambda}(u_{n}, v_{n})\rightarrow d_{\lambda}\), \(\Phi _{\lambda}(u_{n}, v_{n})< c_{1}\) and \(\Phi _{\lambda}'(u_{n}, v_{n})=0\). Then
Observe that \(\{(u_{n}, v_{n})\}_{n\geq 1}\) is bounded in \(H_{r}\), we may assume that, up to a subsequence,
Since \(\Phi _{\lambda}'(u_{n}, v_{n})=0\), it is standard to prove that
and \(\Phi _{\lambda}'(u_{0}, v_{0})=0\), \(\Phi _{\lambda}(u_{0}, v_{0})=d_{\lambda}\).
Moreover, \(\Phi _{\lambda}'(u_{n}, v_{n})(u_{n}^{\pm}, 0)=0\) and \(\Phi _{\lambda}'(u_{n}, v_{n})(0, v_{n}^{\pm})=0\), we deduce from (2.7) and (3.1) that
We can choose \(0<\lambda _{0}<\lambda _{1}\) small enough such that for any \(\lambda \in (0, \lambda _{0})\) we have
which implies \(|u_{n}^{\pm}|_{p+1}\geq \xi _{1}>0\) for any \(n\geq 1\). Similarly, \(|v_{n}^{\pm}|_{q+1}\geq \xi _{2}>0\) for any \(n\geq 1\). Therefore, \(|u_{0}^{\pm}|_{p+1}\geq \xi _{1}>0\), \(|v_{0}^{\pm}|_{q+1}\geq \xi _{2}>0\), and so Eq. (1.1) has a least energy sign-changing solution \((u_{0}, v_{0})\). This completes the proof. □
4 The proof of Theorem 1.3
In this section, we obtain seminodal solutions \((u, v)\) such that u is positive, v is sign-changing and use the same notations as in Sect. 2 for convenience. Define the \(C^{1}\) functional
where \((u, v)\in \widetilde{H}_{r}:=\{(u, v)\in H_{r}: u^{+}\neq 0, v \neq 0\}\),
As in Sect. 2, for any \((u, v)\in \mathcal{A}\), we define
It is easy to prove that Lemma 2.4 also holds in this section by trivial modifications. Then define
For any \((u, v)\in \mathcal{B}_{\widetilde{m}}^{*}\), \(\lambda \in (0, \lambda _{k})\), we consider the following linear problem:
then (4.2) has a unique solution \((\varphi , \psi )\in H_{r}\backslash \{(0, 0)\}\). Define
Then \((\widetilde{\varphi}, \widetilde{\psi}):=(\mu \varphi , \nu \psi )\) is the unique solution of the following problem:
We can now also define the operator
Then, by similar proofs as in Lemma 2.5 and Lemma 2.6, we have that \(K\in C^{1}(B_{\widetilde{m}}^{*}, H_{r})\) and K satisfies the Palais–Smale type condition. Define the map
Consider the class of sets
for each \(A\in \mathcal{F}\) and \(k_{2}\geq 2\), the class of functions
To obtain seminodal solutions, we should also define a cone of positive functions, that is,
thus, v is sign-changing if \(\operatorname{dist}_{q+1} ((u, v), \mathcal{P})>0\).
Under the new definitions (4.4)–(4.6), we define vector genus, slightly different from Definition 2.1.
Definition 4.1
Let \(A\in \mathcal{F}\) and take any \(k_{2}\in \mathbb{N}\) with \(k_{2}\geq 2\). We say that \(\gamma (A)\geq (1, k_{2})\) if for every \(f\in F_{(1, k_{2})}(A)\) there exists \((u, v)\in A\) such that \(f(u, v)=0\). We denote
Lemma 4.1
-
(1)
Take \(A:=A_{1}\times A_{2}\subset \mathcal{A}\) and let \(\eta : S^{k_{2}-1}\rightarrow A_{2}\) be a homeomorphism such that \(\eta (-x)=-\eta (x)\) for every \(x\in S^{k_{2}-1}\). Then \(A\in \Gamma ^{(1, k_{2})}\);
-
(2)
We have \(\overline{\eta (A)}\in \Gamma ^{(1, k_{2})}\) whenever \(A \in \Gamma ^{(1, k_{2})}\) and a continuous map \(\eta : A\rightarrow \mathcal{A}\) is such that \(\eta \circ \sigma _{2}=\sigma _{2}\circ \eta \).
Proof
\((1)\) For every \(f\in F_{(1, k_{2})}(A)\) and \(u\in A_{1}\), we define a map
then by (4.6) it is easy to see that h is continuous and
Then Borsuk–Ulam theorem yields \(x_{0}\in S^{k_{2}-1}\) such that \(h(x_{0})=f(u, \eta (x_{0}))=0\). By Definition 4.1, we have \(A\in \Gamma ^{(1, k_{2})}\).
\((2)\) Fix any \(f\in F_{(1, k_{2})}(\overline{\eta (A)})\), then by (4.6) we have \(f\circ \eta \in F_{(1, k_{2})}(A)\). Since \(A\in \Gamma ^{(1, k_{2})}\), there exists \((u_{0}, v_{0})\in A\) such that \(f\circ \eta (u_{0}, v_{0})=0\). Then by \(\eta (u_{0}, v_{0})\in \overline{\eta (A)}\) we have \(\gamma (\overline{\eta (A)})\geq (1, k_{2})\), that is, \(\overline{\eta (A)}\in \Gamma ^{(1, k_{2})}\). This completes the proof. □
Lemma 4.2
Assume \(k_{2}\geq 2\). Then, for any \(0<\delta <2^{-\frac{1}{q+1}}\) and \(A\in \Gamma ^{(1, k_{2})}\), we have \(A\backslash \mathcal{P}_{\delta }\neq \emptyset \).
Proof
For any \(A\in \Gamma ^{(1, k_{2})}\), define f by
then \(f\in F_{(1, k_{2})}(A)\), so by Definition 4.1, there exists \((u_{0}, v_{0})\in A\) such that \(f(u_{0}, v_{0})=0\). We deduce from \(A\in \mathcal{A}\) that
Therefore, \(\operatorname{dist}_{q+1} ((u_{0}, v_{0}), \mathcal{P})=2^{-\frac{1}{q+1}}\), and so \((u_{0}, v_{0})\in A\backslash \mathcal{P}_{\delta}\) for any \(0<\delta <2^{-\frac{1}{q+1}}\). This completes the proof. □
Fixed any \(k\in \mathbb{N}\), we define
By Lemma 4.1(1), \(A:=A_{1}\times A_{2}\in \Gamma ^{(1, k+1)}\), \(A\subset B_{\widetilde{m}}\), and \(\sup_{A} \Psi _{\lambda}(u, v)< c_{k}\). Then we can define
For any \(k_{2}\in [2, k+1]\) and \(0<\delta <2^{-\frac{1}{q+1}}\), we define a sequence of minimax energy level:
It is easy to see that
Lemma 2.7 and Lemma 2.8 also hold in Sect. 4.
Lemma 4.3
There exists a unique global solution \(\eta : \mathbb{R}^{+}\times B_{\widetilde{m},\lambda}\rightarrow H_{r}\) for the initial value problem
Moreover, (1), (3), (4) of Lemma 2.9hold and
-
(2)
For any \(t>0\), \((u, v)\in B_{\widetilde{m},\lambda}\), \(\eta (t, \sigma _{2}(u, v))=\sigma _{2} ( \eta (t, (u, v)) )\).
Proof
From the above discussion, we see that \(V\in C^{1}(B_{\widetilde{m}}^{*}, H_{r})\). As \(B_{\widetilde{m},\lambda}\subset B_{\widetilde{m}}\subset B_{ \widetilde{m}}^{*}\), we get that \(V\in C^{1}(B_{\widetilde{m},\lambda}, H_{r})\), then there exists a solution \(\eta : [0, T_{\max})\times B_{\widetilde{m},\lambda}\rightarrow H_{r}\), where \(T_{\max}\) is the maximal time such that (4.8) has s solution \(\eta \in B_{\widetilde{m}}^{*}\).
For any \((u, v)\in B_{\widetilde{m},\lambda}\) and \(t\in (0, T_{\max})\), there holds
so we have
Since \(\int _{\mathbb{R}^{3}}(\eta _{1}^{+}(0, (u, v)))^{p+1} \,dx =\int _{ \mathbb{R}^{3}}(u^{+})^{p+1} \,dx =1\), then for any \(t\in [0, T_{\max})\),
The rest of the proof is the same as Lemma 2.9. This completes the proof. □
Proof of Theorem 1.2
Observe that from Lemma 2.10, for any \(k_{2}\in [2, k+1]\), \(0<\delta <\delta _{0}\) small, there exists \((u_{0}, v_{0})\in B_{\widetilde{m}}\) such that
We conclude that \(v_{0}\) is sign-changing and \((u_{0}, v_{0})=K(u_{0}, v_{0})=(\widetilde{\varphi}_{0}, \widetilde{\psi}_{0})\). It follows from (4.3) that \((u_{0}, v_{0})\) satisfies
and \(|u_{0}^{+}|_{p+1}=|v_{0}|_{q+1}=1\), then by (4.1) we have \(\mu =\nu =1\). Moreover, (4.9) yields
We can take \(\lambda _{k}\) small enough if necessary such that for any \(\lambda \in (0, \lambda _{k})\) and \((u_{0}, v_{0})\in \mathcal{B}_{\widetilde{m}}^{*}\),
then \(\|u_{0}^{-}\|_{\alpha}^{2}=0\), so \(u_{0}\geq 0\). By the strong maximum principle, \(u_{0}>0\). Hence we have that \((t_{u_{0},v_{0},\lambda}u_{0}, s_{u_{0},v_{0},\lambda}v_{0})\) is a seminodal solution of (1.1) with \(t_{u_{0},v_{0},\lambda}u_{0}\) positive and \(s_{u_{0},v_{0},\lambda}v_{0}\) sign-changing,
By similar proof as Theorem 1.1, we complete the proof. □
Data Availability
No datasets were generated or analysed during the current study.
References
Akhmediev, N., Ankiewicz, A.: Novel soliton states and bifurcation phenomena in nonlinear fiber couplers. Phys. Rev. Lett. 70, 2395–2398 (1993)
Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. 75, 67–82 (2007)
Atkinson, F.V., Brézis, H., Peletier, L.A.: Nodal solutions of elliptic equations with critical Sobolev exponents. J. Differ. Equ. 85, 151–170 (1990)
Bartsch, T., Dancer, N., Wang, Z.Q.: A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differ. Equ. 37, 345–361 (2010)
Bartsch, T., Liu, Z., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Commun. Partial Differ. Equ. 29, 25–42 (2004)
Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)
Cerami, G., Solimini, S., Struwe, M.: Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69, 289–306 (1986)
Chang, K.C., Wang, Z.Q., Zhang, T.: On a new index theory and non semi-trivial solutions for elliptic systems. Discrete Contin. Dyn. Syst. 28, 809–826 (2010)
Chen, Z., Lin, C., Zou, W.: Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations. J. Differ. Equ. 255, 4289–4311 (2013)
Chen, Z., Lin, C., Zou, W.: Infinitely many sign-changing and seminodal solutions for a nonlinear Schrödinger system. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 15, 859–897 (2016)
Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Ration. Mech. Anal. 205, 515–551 (2012)
Chen, Z., Zou, W.: An optimal constant for the existence of least energy solutions of a coupled Schrödinger system. Calc. Var. Partial Differ. Equ. (2012). https://doi.org/10.1007/s00526-012-0568-2
Conti, M., Merizzi, L., Terracini, S.: Remarks on variational methods and lower-upper solutions. Nonlinear Differ. Equ. Appl. 6, 371–393 (1999)
Dancer, N., Wei, J., Weth, T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, 953–969 (2010)
Frantzeskakis, D.J.: Dark solitons in atomic Bose-Einstein condensates: from theory to experiments. J. Phys. A 43, 213001 (2010)
Kim, S.: On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Commun. Pure Appl. Anal. 12, 1259–1277 (2013)
Kivshar, Y.S., Luther-Davies, B.: Dark optical solitons: physics and applications. Phys. Rep. 298, 81–197 (1998)
Lin, T., Wei, J.: Ground state of N coupled nonlinear Schrodinger equations in \(\mathbb{R}^{3}\), \(n\leq 3\). Commun. Math. Phys. 255, 629–653 (2005)
Liu, J., Liu, X., Wang, Z.Q.: Multiple mixed states of nodal solutions for nonlinear Schrödinger systems. Calc. Var. Partial Differ. Equ. 52, 565–586 (2015)
Liu, Z., Wang, Z.-Q.: Ground states and bound states of a nonlinear Schrödinger system. Adv. Nonlinear Stud. 10, 175–193 (2010)
Liu, Z., Wang, Z.Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282, 721–731 (2008)
Maia, L., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger systems. J. Differ. Equ. 229, 743–767 (2006)
Maia, L., Montefusco, E., Pellacci, B.: Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system. Commun. Contemp. Math. 10, 651–669 (2008)
Noris, B., Ramos, M.: Existence and bounds of positive solutions for a nonlinear Schrödinger system. Proc. Am. Math. Soc. 138, 1681–1692 (2010)
Quittner, P., Souplet, P.: Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications. Commun. Math. Phys. 311, 1–19 (2012)
Sato, Y., Wang, Z.: On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, 1–22 (2013)
Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in \(\mathbb{R}^{3}\). Commun. Math. Phys. 271, 199–221 (2007)
Tavares, H., Terracini, S.: Sign-changing solutions of competition diffusion elliptic systems and optimal partition problems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, 279–300 (2012)
Wei, J., Weth, T.: Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Ration. Mech. Anal. 190, 83–106 (2008)
Zou, W.: Sign-Changing Critical Points Theory. Springer, New York (2008)
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Zhang, J. Sign-changing solutions for coupled Schrödinger system. Bound Value Probl 2024, 69 (2024). https://doi.org/10.1186/s13661-024-01881-z
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Published:
DOI: https://doi.org/10.1186/s13661-024-01881-z