Normalized solution for a kind of coupled Kirchhoff systems

1.   Introduction

In the present paper, we study the following Kirchhoff system with a coupled critical nonlinearity

having prescribed mass

where $m_1, m_2 > 0$, $\lambda_1, \lambda_2, \beta > 0$ and $N\geq3$, $\lambda_1, \lambda_2$ are unknown parameters that will appear as Lagrange multipliers.

Problem (1.1) originates from the steady-state analogy of the equation:

which was proposed by Kirchhoff in 1883 in ^[1] as the existence of the classical D'Alembert wave equation for the free vibration of elastic strings. The Kirchhoff model takes into consideration the changes in the length of the string that are caused by transverse vibrations.

In recent years, lots of interesting results on the normalized solutions for the Kirchhoff type problem that has been obtained. From a physical perspective, the mass $\int_{ \mathbb{R}^N }|u|^2dx = m$ may represent the number of the power supply in the framework of nonlinear optics or Bose-Einstein condensates. Alternatively, finding normalized solutions seems to be particularly meaningful because the $L^2$-norm of such solutions is a preserved quantity of the evolution, and their variational characterization can help to analyze the orbital stability or instability, e.g., see ^[2,3,4]. In Bose-Einstein condensates, the parameters $\mu_i$ and $\beta$ both describe the interactions between particles. When $\beta > 0$, the two components attract each other, while $\beta < 0$, the two components repel each other.

Based on the above important background, the problem like (1.1) has been studied in numerous papers. For example, Yang ^[5] has obtained a couple of positive solutions to the following equation:

where $a_i, b_i > 0 (i = 1, 2)$ and $2\leq N\leq4$. By proving that $(1.4)$ satisfies the mountain pass structure, they obtained a couple of positive solutions. In particular, as $\beta > 0$, Cao et al. ^[6] considered the $L^2$-subcritical case and $L^2$-critical case of the problem by the bifurcation method and showed the existence of normalized solutions when $N\leq3$. Eq (1.1) can also be formally transformed into the following fractional Kirchhoff equation

where $a_i, b_i(i = 1, 2), \lambda, \mu > 0$. When $s \in [\frac{3}{4}, 1)$ and $\gamma > 0$, by assuming that the nonlinear terms $f$ and $g$ satisfy Berestycki-Lions conditions, and combining with Pohožaev identity, Che and Chen in ^[7] proved problem $(1.5)$ has positive ground state solutions, and the asymptotic behavior of the solution was also studied when $\gamma \rightarrow 0^+$. When $s = 1$, Lü and Peng ^[8] proved that $(1.5)$ has vector solutions. We refer readers to ^[9,10] for multiplicity solutions. However, to our knowledge, there are few articles discussing the results regarding $N\geq5$ for the Kirchhoff-type system. This motivates us to consider the solution of the Kirchhoff system (1.1) for $N\geq3$ and with a coupled critical nonlinearity, where $2\leq p, q < 2+\frac{8}{N}$ and ${r_1}+{r_2} = 2^* = \frac{2N}{N-2}$.

Other forms of (1.1), such as the Schrödinger equation, have also been extensively studied. For example, Li and Zou ^[11] considered the case with $2 < p, {r_1}+{r_2} < {2^*}, q\leq{2^*}$ of the following equation:

When $2 < {r_1}+{r_2} < {2^*} = p = q$, Bartsch et al. in ^[12] have proved (1.6) has a normalized ground state solution and have also investigated the asymptotic behavior by the symmetric decreasing rearrangement and the Ekeland variational principle. When $2+\frac{4}{N} < p, q < {r_1}+{r_2} < {2^*}$ and $N\geq3$, Liu and Fang ^[13] obtained the existence of positive normalized solutions of $(1.6)$ by revealing the basic behavior of mountain-pass energy. Compared with Schrödinger equations, it is more challenging and interesting to study problem (1.1) due to the nonlocal term $\int_{ \mathbb{R}^N}|\nabla u|^2dx\Delta u$ and $\int_{ \mathbb{R}^N}|\nabla v|^2dx \Delta v$.

In order to study the solution of Eq (1.1) satisfying the normalized condition (1.2), it suffices to consider the critical points of the functional

on the constraint $S(m_1, m_2) = S(m_1)\times S(m_2)$, where $S(m) = \{u \in H^1(\mathbb{R}^N): \|u\|_2^2 = m\}$ for $m > 0$. In this paper, we employ the Pohožaev manifold, which is defined by (1.8) and plays a crucial role, encompassing all solutions that satisfy the condition $(u, v) \in S(m_1, m_2)$

where

where $\delta_t = \frac{N(t-2)}{2t}$. To accommodate the constraint $S(m)$, it becomes crucial to define dilation

Consider the following functionals $\mathcal{I}(u, v)$ and $\mathcal{L}_{u, v}(t)$

for any $(u, v)\in S(m_1, m_2)$.

Remark 1.1. As in ^[5], if (u, v) is a solution of (1.1), then $(u, v) \in P(m_1, m_2)$. We can also see that if $(u, v)\in S(m_1, m_2)$, then ($e^{\frac{Nt}{2}}u(e^tx), e^{\frac{Nt}{2}}v(e^tx))\in S(m_1, m_2)$. Furthermore, for fixed $(u, v)\in S(m_1, m_2)$, by performing a simple calculation, we can obtain $\left(\mathcal{L}_{u, v}\right)^\prime(0) = \mathcal{\vartheta}(u, v)$. Then we have that $(t\ast u, t\ast v)\in P(m_1, m_2)$ if and only if $t$ is a critical point of $\mathcal{L}_{u, v}(t)$. In addition, $(u, v)\in P(m_1, m_2)$ if $t = 0$ is a critical point of $\mathcal{L}_{u, v}(t)$.

To prove the existence of a normalized solution to (1.1), we use the following assumptions:

$(H_1)$ $N \in \{3, 4\}$, $2 < p, q < {2+\frac{8}{N}}$, ${r_1}+{r_2} = 2^*$.

$(H_2)$ $N\geq5$, $2 < p, q < {2+\frac{2}{N-2}}$, ${r_1}+{r_2} = 2^*$.

Here comes our main result:

Theorem 1.2. Assume that $(H_1)$ or $(H_2)$ is established. Then, there exist $\beta_\tau = \beta_\tau(m_1, m_2) > 0$ and $\rho_\tau = \rho_\tau(m_1, m_2) > 0$ such that for arbitrary $0 < \beta < \beta_\tau$, (1.1) has a positive ground state solution (u, v) for $\lambda_1, \lambda_2 < 0$, which satisfies

where

Remark 1.3. Due to the additional difficulties caused by the combined effect of the nonlocal term $\int_{ \mathbb{R}^N}|\nabla u|^2dx\Delta u$, $\int_{ \mathbb{R}^N}|\nabla v|^2dx\Delta v$ and multiple powers, the study is much more challenging; for example, the functional $\mathcal{I}(u, v)$ is composed of several distinct terms that exhibit varying scaling behavior with respect to the dilation $e^{\frac{Nt}{2}}u(e^tx)$. The intricate interplay among these terms makes it more difficult to ascertain the types of critical points for $\mathcal{I}(u, v)$ on $S(m_1, m_2)$. Furthermore, when proving $(\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v)$ in $D^{1, 2}(\mathbb{R}^N; \mathbb{R}^2)$, the inequalities that need to be estimated will also be more difficult.

Remark 1.4. From a variational point of view, besides the Sobolev critical exponent $2^*: = \frac{2N}{N-2}$ for $N \geq 3$ and $2^* = \infty$ for $N = 1, 2,$ a new $L^2$-critical exponent $P_N : = 2 + \frac{8}{N}$ arises that plays a pivotal role in the study of normalized solutions to (1.1). This threshold determines whether the constrained functional $\mathcal{I}(u, v)$ remains bounded from below on $S(m_1, m_2)$.

Definition 1.5. We say that ($\tilde{u}, \tilde{v}$) is a couple of ground state solutions to (1.1) on $S(m_1, m_2)$ if it is a couple of solutions to (1.1) having minimal energy among all the solutions, i.e., $d\mathcal{I}|_{S(m_1, m_2)}(\tilde{u}, \tilde{v}) = 0$ and

2.   Preliminary results

In this section, we recall some preliminary results that will be used later. Throughout this paper, we represent the norms on $L^t(\mathbb{R}^N)$ and $H^1(\mathbb{R}^N)$ with $\|\cdot\|_t$ and $\| \cdot \|$, respectively. Denote $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ by $\mathcal{V}$ with the norm

Let $L^t(\mathbb{R}^N; \mathbb{R}^2)$ be the space $L^t(\mathbb{R}^N \times \mathbb{R}^N)$ with the norm

$D^{1, 2}(\mathbb{R}^N)$ represents the closure of the $C_c^{\infty}(\mathbb{R}^N)$ with norm

For $N\geq3$, the best Sobolev constant is given by

For all $u \in H^1(\mathbb{R}^N)$, we consider the Gagliardo-Nirenberg-Sobolev inequality:

For any $u, v \in H^1(\mathbb{R}^N)$, by the Young's inequality, we can prove:

Furthermore, taking into consideration the existing results of the Kirchhoff equation as follows:

Solution $u$ of ($P_m$) can be found as critical points of the functional $\mathcal{I}_\mu(u)$ defined by

constrained to the $L^2$-sphere $S(m)$.

Similar to ^[14] and ^[6], we can get the following lemma.

Lemma 2.1 (^[6]). Assume that $p \in (2, 2+\frac{8}{N})$, $m > 0,$ and $\mu > 0$. Set

Then,

$(i)$ there exists a unique couple $(u_{m, \mu}, \lambda_m) \in \mathbb{R}^+\times H^1(\mathbb{R}^N)$ satisfying ($P_m$);

$(ii)$ $\mathcal{I}_\mu(u_{m, \mu}) = \zeta_p^\mu(m) < 0$;

$(iii)$ the map $m \mapsto \zeta_p^\mu(m)$ is strictly decreasing with respect to m, and $\zeta_p^\mu(m) \rightarrow -\infty$ as $m \rightarrow +\infty$.

3.   Proof of Theorem 1.2

To begin with, we set

and

Lemma 3.1. Let $m_1, m_2, \mu_1, \mu_2 > 0$ be given and assume $( H_1 )$ or $( H_2)$ holds. Then there exists $\beta_\tau$ = $\beta_\tau(m_1, m_2) > 0$ and $\rho_\tau = \rho_\tau(m_1, m_2) > (\|\nabla\gamma_1\|_2^2+\|\nabla\gamma_2\|_2^2)^{\frac{1}{2}}$ such that

Proof. For $(u, v) \in \mathcal{V}$, let $\rho = (\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2})^\frac{1}{2}$. From ($2.2$) and ($2.3$), we derive that

Recalling that $p\delta_{q} < 2$ and $q\delta_{q} < 2$, we can take a large enough

such that

Due to the fact that $2^*-2 > 0$, there exists a $\beta_\tau > 0$ such that

We conclude that $\mathcal{I}(u, v) > 0$ follows from (3.1)–(3.3). □

Define

where $\rho_\tau$ is defined in Lemma 3.1.

Lemma 3.2. Let $m_1, m_2, \mu_1, \mu_2 > 0$ be given, and ($H_1$) or ($H_2$) is true. Then for arbitrary $0 < \beta < \beta_\tau$, the following statements are true:

(i) $\mathcal{M}(m_1, m_2) < \zeta_1+\zeta_2 < 0$;

(ii) $\mathcal{M}(m_1, m_2) \leq \mathcal{M}(m_{\alpha1}, m_{\alpha2})$, for any $0 < m_{\alpha1} < m_1$ and $0 < m_{\alpha2} < m_2$.

Proof. $(i)$ From Lemma 3.1, we know that $(\gamma_1, \gamma_2) \in V(\rho_\tau)$. Moreover, we deduce that

$(ii)$ The proof is similar to that of ^[15]. We just need to prove that for arbitrary $\epsilon > 0$,

for any $0 < m_{\alpha1} < m_1$ and $0 < m_{\alpha2} < m_2$.

By Lemma 3.1 and the definition of $\mathcal{M}(m_{\alpha1}, m_{\alpha2})$, there exist $u, v \in S(m_{\alpha1}, m_{\alpha2})\cap V(\rho_\tau)$ such that

Define a cut-off function: $\omega \in C_m^\infty(\mathbb{R}^N)$ such that

For any $\imath > 0$, we define ($u_\imath(t), v_\imath(t)$) = ($u\omega(\imath t), v\omega(\imath t)$). Clearly, $(u_\imath, v_\imath)\rightarrow (u, v)$ in $\mathcal{V}$ as $\imath \rightarrow 0^+$. As a consequence, for $\eta > 0$ small enough, there exists a sufficiently small $\imath$ such that

Let $\chi(t) \in C_m^{\infty}(\mathbb{R}^N)$ such that supp$(\chi)\subset\{t \in \mathbb{R}^N:\frac{4}{\imath}\leq |t|\leq1+\frac{4}{\imath}\}$ and set

And observe that

for any $t\leq0$, hence,

Next, since

as $t \rightarrow -\infty$, we can obtain

It follows that

Using (3.5) and (3.6), we conclude

for $t\ll0$. □

Lemma 3.3. Let $m_1, m_2, \mu_1, \mu_2 > 0,$ and assume that either ($H_1$) is true or ($H_2$) is true. Then, for arbitrary $0 < \beta < \beta_\tau$ and $(u, v)\in S(m_1, m_2)$, $\mathcal{L}_{u, v}(t)$ has two critical points $\tau_{u_1v_1} < \tau_{u_2v_2} \in \mathbb{R}$ and two zero points $\varphi_1 < \varphi_2$ with $\tau_{u_1v_1} < \varphi_1 < \tau_{u_2v_2} < \varphi_2$. Moreover,

(i) if $(t\ast u, t\ast v) \in P(m_1, m_2)$, then $t = \tau_{u_1v_1}$ or $t = \tau_{u_2v_2}$;

(ii) $(\|\nabla t\ast u\|_{2}^{2}+\|\nabla t \ast v\|_{2}^{2})^\frac{1}{2}\leq \rho_\tau$ for all $t\leq \varphi_1$ and

where $\rho_\tau$ is given in Lemma 3.1;

(iii) $\mathcal{I}(\tau_{u_2v_2}\ast u, \tau_{u_2v_2}\ast v) = \max\left\{\mathcal{I}(t\ast u, t\ast v):t\in\mathbb{R}\right\}.$

Proof. $(i)$ Since $q\delta_q, p\delta_p < 2 < 2^*$, it can be seen that $\mathcal{L}_{u, v}(-\infty) = 0^-$ and $\mathcal{L}_{u, v}(+\infty) = -\infty$. According to Lemma 3.1, we obtain that $\mathcal{L}_{u, v}(t)$ has at least two critical points $\tau_{u_1v_1} < \tau_{u_2v_2}$, with $\tau_{u_1v_1}$ local minimum point of $\mathcal{L}_{u, v}(t)$ at a negative level and $\tau_{u_2v_2}$ global maximum point at a positive level. Secondly, similar to ^[5], it is not difficult to check that there are no other critical points. On the other hand,

Putting together all the considerations mentioned above, we conclude that $\mathcal{L}_{u, v}$ has exactly two critical points. By monotonicity and recalling the behavior at infinity, $\mathcal{L}_{u, v}$ has moreover exactly two zeros points $\varphi_1 < \varphi_2$ with $\tau_{u_1v_1} < \varphi_1 < \tau_{u_2v_2} < \varphi_2$. From Lemma 3.1 and $(i)$, we can deduce the $(ii)$ and $(iii)$. □

Corollary 3.4. Let $m_1, m_2, \mu_1, \mu_2 > 0,$ and assume that either ($H_1$) is true or ($H_2$) is true. Then, for arbitrary $0 < \beta < \beta_\tau$, the following inequality holds:

Next, we establish a necessary condition for the existence of a non-negative solution to (1.1). This Liouville-type result will be used to prove the existence of a positive solution.

Lemma 3.5.(^[16]) Suppose $0 < p \leq \frac{N}{N-2}$ when $N \geq 3$ and $0 < p < \infty$ when $N = 1, 2$. Let $u \in L^p(\mathbb{R}^N)$ be a smooth, nonnegative function and satisfy $-\Delta u \geq 0$ in $\mathbb{R}^N$. Then $u \equiv 0$ holds.

Lemma 3.6. Let $(u, v)\in S(m_1, m_2)$, $u, v\geq0,$ and $u, v\not\equiv0$, if (u, v) satisfies

then $\lambda_1, \lambda_2 < 0$.

Proof. Arguing by contradiction, we assume that $\lambda_1\geq0$. Since $u\geq0$, we have that all components on the right-hand side of

are nonnegative. Hence,

it is easy to see that

Moreover, modifying the standard elliptic regularity theorems, we can ensure that the smoothness of ($u, v$) is up to $C^2$. Hence, it follows from Lemma 3.5 that $u = 0$. This contradicts with $u\not\equiv0$; thus, $\lambda_1 < 0$. The proof of $\lambda_2 < 0$ is the same as that of $\lambda_1 < 0$. □

Lemma 3.7.(^[17]) Let $(u_n)_{n \geq 0} \subset H^1(\mathbb{R}^N)$ be a bounded sequence of spherically symmetric functions. If $N \geq 2$ or if $u_n(x)$ is a nonincreasing function of $|x|$ for every $n \geq 0$, then there exist a subsequence $(u_{n_k})_{k \geq 0}$ and $u \in H^1(\mathbb{R}^N)$ such that $u_{n_k} \rightarrow u$ as $k \rightarrow \infty$ in $L^p(\mathbb{R}^N)$ for every $2 < p < \frac{2N}{N-2}$.

Proof of Theorem 1.2. Let us consider a minimizing sequence $\{(u_n, v_n)\}$ for $\mathcal{I}|_{S(m_1, m_2)\cap V(2\rho_\tau)}$ and $\{(u_n, v_n)\}\subset \mathcal{V}\cap S(m_1, m_2)$. Without loss of generality, we can assume that $(u_n, v_n)\subset \mathcal{V}$ are nonnegative and radially decreasing for every $n$[Otherwise, we replace ($u_n, v_n$) with ($|u_n|^*, |v_n|^*$), which is the Schwarz rearrangement of ($|u_n|, |v_n|$)]. Furthermore, by Lemma 3.3 $(ii)$, $(\|\nabla s\ast u\|_{2}^{2}+\|\nabla s \ast v\|_{2}^{2})^\frac{1}{2}\leq \rho_\tau$, and $\{\tau_{u_{n}v_{n}}\ast u, \tau_{u_{n}v_{n}}\ast v\}$ is still a minimizing sequence for $\mathcal{I}|_{S(m_1, m_2)\cap V(2\rho_\tau)}$. And hence, by the Ekeland variational principle ^[18], it yields that there exists a new minimizing sequence $\{(\tilde{u}_n, \tilde{v}_n)\}$ satisfying

In the sequel, we divide the proof into three steps.

Step 1: $(\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v)$ in $L^t(\mathbb{R}^N; \mathbb{R}^2)$ for arbitrarily $t \in (2, 2^*)$.

In fact, from (3.8), we can know that $\mathcal{I}'|_{S(m_1, m_2)}(\tilde{u}_n, \tilde{v}_n)\to0$. By the Lagrange multipliers theorem, there exist two sequences $\{\lambda_{1, n}\} \subset \mathbb{R}$ and $\{\lambda_{2, n}\} \subset \mathbb{R}$ satisfying the following equation

for arbitrarily $(\phi, \psi)\in \mathcal{V}$. By substituting $(\tilde{u}_n, 0)$ and $(0, \tilde{v}_n)$ into (3.9), we can derive

and

Since $\{\tilde{u}_n, \tilde{v}_n\}\subset{S(m_1, m_2)\cap V(2\rho_\tau)}$, up to a subsequence, $(\lambda_{1, n}, \lambda_{2, n}) \rightarrow (\lambda_1, \lambda_2) \in \mathbb{R}^2$ and $(\tilde{u}_n, \tilde{v}_n) \rightharpoonup (u, v) \in \mathcal{V}$, where both $u$ and $v$ are non-negative. Combined with that, $\mathcal{\vartheta}(u, v) = 0$, then ($u, v$) is a weak solution of (1.1). By Lemma 3.7, we obtain that $(\tilde{u}_n, \tilde{v}_n) \rightarrow (u, v)$ in $L^t(\mathbb{R}^N, \mathbb{R}^2)$ for any $t \in (2, 2^*)$.

Step 2: $(\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v)$ in $D^{1, 2}(\mathbb{R}^N; \mathbb{R}^2)$.

Let $(u_n, v_n) = (\tilde{u}_n-u, \tilde{v}_n-v)$. Then $u_n\rightarrow0$ in $L^p(\mathbb{R}^N)$ and $v_n\rightarrow0$ in $L^q(\mathbb{R}^N)$. Moreover, from the Br$\acute{e}$zis-Lieb Lemma, we have

Since $\mathcal{\vartheta}(\tilde{u}_n, \tilde{v}_n)-\mathcal{\vartheta}(u, v)\rightarrow 0$, we can infer from (2.3) and (3.10) that

Up to a subsequence, we assume that $\|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\rightarrow R\geq0$. Then $R = 0$ or $R\geq \left(\frac{1}{\beta 2^*}\right)^{\frac{N-2}{2}}S^{\frac{N}{2}}$. If $R\geq \left(\frac{1}{\beta 2^*}\right)^{\frac{N-2}{2}}S^{\frac{N}{2}}$, from (3.8), (3.10), and (3.11), we have

This contradicts with Lemma 3.2 $(ii)$. Then $\|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\rightarrow 0$. Thus, we conclude $(\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v)$ in $D^{1, 2}(\mathbb{R}^N; \mathbb{R}^2)$.

Step 3: $(\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v)$ in $\mathcal{V}$.

From Step 1, then, as in ^[19], we know that there exists $(u, v) \in \mathcal{V}$ that is a weak solution of

with

We claim that $u\neq0$ and $v\neq0$. Indeed, if $v = 0$, then $u$ satisfies

By applying Lemma 2.1, we know that $\zeta_p^\mu(m)$ is strictly decreasing with respect to $m$. So

However,

which contradicts to Lemma 3.2 $(i)$. Hence, $v\neq0$. Similarly, we have $u\neq0$. Thus, from Lemma 3.6, we know $\lambda_1, \lambda_2 < 0$. Then, by substituting ($\tilde{u}_n, 0$) and ($u, 0$) into (3.9), we can derive

and

which implies that $\tilde{u}_n \rightarrow u$ in $H^1(\mathbb{R}^N)$ as $\lambda_1 < 0$. Similarly, we obtain $\tilde{v}_n \rightarrow v$ in $H^1(\mathbb{R}^N)$.

Therefore, we have $(\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v)$ in $\mathcal{V}$ and by Corollary 3.4, we have

Therefore, we deduce that ($u, v$) is a normalized solution. By the maximum principle, we conclude that ($u, v$) is a positive solution.

4.   Conclusions

In this paper, we establish the existence of a ground state solution for a nonlinear Kirchhoff-type system using the minimization of the energy functional over a combination of the mass-constrained and the closed balls. To the best of our knowledge, there are few articles that deal with a coupled critical nonlinearity of the Kirchhoff system. Especially, our assumptions on the parameters are different from the previous related works. Therefore, we need to use some new analytical tricks to estimate the critical value. Our results in this article improve and generalize the related ones in the literature. In addition, condition $2\leq p, q < 2+\frac{8}{N}$ means that our results are established in a critical setting. Therefore, a new research direction closely related to problem (1.1) is to replace $2\leq p, q < 2+\frac{8}{N}$ with the following $L^2$-supercritical condition: $2+\frac{8}{N}\leq p, q < 2^*$ $.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11961014) and Guangxi Natural Science Foundation (2021GXNSFAA196040).

Conflict of interest

The authors declare there is no conflicts of interest.