Existence of least energy nodal solution for Kirchhoff-type system with Hartree-type nonlinearity

1.   Introduction and main result

In this article, we are interested in the least energy nodal solution for the following Kirchhoff-type system

where $a, \mu > 0$, $b, l\geq0$, $\alpha\in(0, 3)$, $p\in[2, 3)$ and $\phi|u|^{p-2}u$ is a Hartree-type nonlinearity (in fact, $\phi = I\ast|u|^{p}$, where $I$ is the Riesz potential defined by (1.10)), $(-\Delta)^{\alpha/2}$ is the fractional Laplacian. The potential function $V\in C(\mathbb{R}^{3}, \mathbb{R}_{+})$ and function $f\in C^{1}(\mathbb{R}, \mathbb{R})$ satisfy the following hypotheses:

$(V)$ for every $M > 0$, the set $V_{M}: = \{x\in\mathbb{R}^{3}:V(x)\leq M\}$ has a finite Lebesgue measure, i.e.$m(V_{M}) < \infty$;

$(f_{1})\; \lim_{|t|\rightarrow0}\frac{f(t)}{|t|^{2p-1}} = 0;$

$(f_{2})$ there exist $q\in(2p, 6)$ and $C > 0$ such that $|f'(t)|\leq C(1 + |t|^{ q-2})$ for all $t\in\mathbb{R}$;

$(f_{3})\; \frac{f(t)}{t^{2p-1}}$ is strictly increasing for $t > 0$ and is strictly decreasing for $t < 0$.

In the past decades, many mathematicians pay their much attention to nonlocal problems. The appearance of nonlocal terms in the equations not only marks its importance in many physical applications but also causes some difficulties and challenges from a mathematical point of view. Certainly, this fact makes the study of nonlocal problems particularly interesting. The following Schrödinger-Poisson system is a typical nonlocal problem

Recently, many authors have been devoted to the study for system (1.2) or similar problems. Especially on nodal solutions to problems like (1.2), and indeed some interesting results were obtained, see for examples, ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14]} and the references therein. In fact, there are very few results about nodal solutions to Schrödinger-Poisson system with critical growth. In ^[14], Zhong and Tang ^[14] considered the existence of ground state nodal solution for following system with critical growth

where $k, f\geq0$, $0 < \lambda < \lambda_{1}$(where $\lambda_{1}$ is the first eigenvalue of the problem $-\Delta u+u = \lambda f(x)u$ in $H^{1}(\mathbb{R}^{3})$). However, if $k(x)\equiv1$, their methods seems not valid because their results depends on the case $k\in L^{p}(\mathbb{R}^{3})\cap L^{\infty}(\mathbb{R}^{3})$ for some $p\in [2, \infty)$.

In ^[11], Wang, Zhang ang Guan considered the existence of least energy nodal solution for following system with critical growth

Via the constraint variational method and quantitative deformation lemma, they obtained the existence and asymptotic behavior of least energy nodal solution for system (1.4).

As another typical nonlocal problem, the following Kirchhoff-type equation

has also aroused many mathematicians's wide concern. Especially, There are many papers about nodal solutions to problems like (1.5) ^{[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]}. However, to the best of our knowledge, the most results seem to be obtained in the subcritical case. It is noticed that the second author ^[27] considered the least energy nodal solution for following Kirchhoff equation with critical growth

where $\Omega\subset \mathbb{R}^{3}$ is a bounded domain with a smooth boundary $\partial\Omega$, $\lambda, a, b > 0$. By using the constraint variational method and quantitative deformation lemma, the author studied the existence and energy characteristics of least energy nodal solution to Eq 1.6.

In ^[32], Zhao and Liu studied the following Kirchhoff equation with critical growth

where $a, b, \mu > 0, 5 < q < 6$ are constants and $V$ is a radial function and is bounded from below by a positive constant. By using the truncation method, they proved that, for any given positive integer $k$, the problem has a radial solution with $k$ nodal domains exactly.

When $\alpha = 2$, system (1.1) is related to following system

where $a$, $b$ are positive constants. Since there are both nonlocal operator and nonlocal nonlinear term, the study of system (1.7) become more complicated. In recent years, there are some scholars began to show interest to problem like (1.7), see ^{[33,34,35,36,37,38,39,40,41,42,43,44]} and references therein. However, to our best knowledge, few papers considered nodal solutions to problem like (1.7). Via gluing the function methods, Deng and Yang ^[34] studied the nodal solutions for system (1.8) with $f(u) = |u|^{p-2}u, p\in(4, 6).$ In ^[39], Wang, Li and Hao studied the existence and the asymptotic behavior of least energy nodal solution for system (1.8) by using the constraint variation methods.

Recently, in order to research uniformly Kirchhoff-type equation and Schrödinger-Poisson system, Li, Gao and Zhu ^[45] considered the following Kirchhoff-type system

where $a > 0$, $b, l\geq0$, $\alpha\in(0, 3)$ and $p\in[2, 3+\alpha)$. More precisely, they studied the existence and asymptotic behavior of least energy nodal solution for system (1.9).

Inspired by the works mentioned above, especially by ^[11,27,45], in this paper, we investigate the existence of the least energy nodal solution to Kirchhoff-type system (1.1).

Before presenting our main results, we denote $L^{r}(\mathbb{R}^{3})$ the Lebesgue space with the norm $|u|_{r}: = (\int_{\mathbb{R}^{3}}|u|^{r}dx)^{\frac{1}{r}}$, $1\leq r < \infty.$ Let $D^{1, 2}(\mathbb{R}^{3}) = \{u\in L^{6}(\mathbb{R}^{3}): \nabla u\in L^{2}(\mathbb{R}^{3})\}$ be a Hilbert space with the inner product and corresponding norm

Denote $E: = H^{1}_{V}(\mathbb{R}^3)$ is given Hilbert space

by equipped with the inner product and norm

Under the condition $(V)$, according to Remark 3.5 of ^[46], the embedding $E\hookrightarrow L^{r}(\mathbb{R}^{3})$ is continuous for each $r\in[2, 6]$, and is compact for each $r\in[2, 6)$.

It follows from the second equation $(-\Delta)^{\alpha/2}\phi = l|u|^p$ of system (1.1) that the unique solution is $\phi = I\ast|u|^{p}$, where $I$ is the Riesz potential defined by

and $\ast$ is the convolution of two functions in $\mathbb{R}^{3}$. Hence, system (1.1) can also be rewritten as a Hartree-type equation

So, the energy functional associated with system (1.1) is defined by

for any $u\in E$.

Moreover, under our conditions, $J_{\mu}(u)$ belongs to $C^{1}$, and the Fréchet derivative of $J_{\mu}(u)$ is

for any $u, v\in E$.

The weak solution of system (1.1) is the critical point of the functional $J_{\mu}(u)$. Furthermore, if $u\in E$ is a weak solution of system (1.1) with $u^{\pm}\neq0$, then we say that $u$ is a nodal solution of system (1.1), where $u^{+} = \max\{u(x), 0\}, u^{-} = \min\{u(x), 0\}.$ In this paper, we borrow some ideas from ^{[11,24,26,27,45,47]} and seek a minimizer of the energy functional $J_{\mu}$ over the constraint $\mathcal{M}_{\mu} = \{u\in E, u^{\pm}\neq0\; \text{and}\; \langle J_{\mu}'(u), u^{\pm}\rangle = 0\},$ and then prove that the minimizer is a nodal solution of system (1.1).

The main results can be stated as follows.

Theorem 1.1. Suppose that $(V)$ and $(f_{1})-(f_{3})$ are satisfied. Then, there exists $\mu^{\star} > 0$ such that for all $\mu\geq\mu^{\star}$, the system (1.1) has a least energy nodal solution $u_{\mu}$.

Theorem 1.2. Suppose that $(V)$ and $(f_{1})-(f_{3})$ are satisfied. Then, there exists $\mu^{\star\star} > 0$ such that for all $\mu\geq\mu^{\star\star}$, then the $c^{\ast} > 0$ is achieved and

where $c^{\ast} = \inf_{u\in \mathcal{N}_{\mu}}J_{\mu}(u)$, $\mathcal{N}_{\mu} = \{u\in E\backslash\ \{0\}|\langle J_{\mu}'(u), u \rangle = 0 \}$, and $u_{\mu}$ is the least energy nodal solution obtained in Theorem 1.1. In particular, $c^{\ast} > 0$ is achieved either by a positive or a negative function.

2.   Technical lemmas

Lemma 2.1. (^[45]) Under the condition $(V)$, if $u_{n}\rightharpoonup u$ and $v_{n}\rightharpoonup v$ in $E$, then

In particular,

Now, fixed $u\in E$ with $u^{\pm}\neq0$, we define function $G_{u}:[0, \infty)\times[0, \infty)\rightarrow\mathbb{R}$ and mapping $H_{u}:[0, \infty)\times[0, \infty)\rightarrow\mathbb{R}^{2}$ by

Inspired by ^[1,11,24,27] and similar to that of in ^[1,11,24,27], we have following Lemmas 2.2–2.3. For reader convenient, we give the details of proof.

Lemma 2.2. Assume that $(f_{1})-(f_{3})$ hold, if $u\in E$ with $u^{\pm}\neq0$, then $G_{u}$ has the following properties:

(i) The pair $(s, t)$ is a critical point of $G_{u}$ with $s, t > 0$ if and only if $s u^{+}+t u^{-}\in \mathcal{M}_{\mu}$;

(ii) The function $G_{u}$ has a unique critical point $(s_{u}, t_{u})$ on $(0, \infty)\times(0, \infty)$, which is also the unique maximum point of $G_{u}$ on $[0, \infty)\times[0, \infty)$; Furthermore, if $\langle J_{\mu}'(u), u^{\pm}\rangle\leq0,$ then $0 < s_{u}, t_{u}\leq1.$

Proof. (i) It follows from definition of $G_{u}$ that

which implies that (i) holds.

In the following, we prove (ii). We shall proceed through several steps to complete the proof.

Step 1. We prove the existence of $s_{u}$ and $t_{u}$.

From $(f_{1})$ and $(f_{2})$, for any $\varepsilon > 0$, there is $C_{\varepsilon} > 0$ such that

So, together with Sobolev embedding theorem, one gets that

Choosing $\varepsilon > 0$ such that $(1-\mu\varepsilon C_{4}) > 0$, it follows from $2 < q < 6$ that

By similarly arguments, we have that

So, from (2.2) and (2.3), there exists $\gamma_{1} > 0$ such that

for all $s, t\geq0$.

Thanks to $(f_{1})$ and $(f_{3})$, we conclude that

for a.e. $x\in{\mathbb {R}^{3}}$.

Let $s = \gamma_{2}' > \gamma_{1}$ and $\gamma_{2}'$ large enough, by (2.5), we have that

for any $t\in[\gamma_{1}, \gamma_{2}'].$

Similarly, let $t = \gamma_{2}' > \gamma_{1}$ and $\gamma_{2}'$ large enough, we conclude that

for any $s\in[\gamma_{1}, \gamma_{2}'].$

Combining (2.6) and (2.7), choose $\gamma_{2} > \gamma_{2}'$ large enough, we have that

for all $s, t\in[\gamma_{1}, \gamma_{2}]$.

Thanks to (2.4) and (2.8), it follows from Miranda's Theorem ^[48] that there is $(s_{u}, t_{u})\in(0, \infty)\times(0, \infty)$ such that $H_{u}(s_{u}, t_{u}) = (0, 0)$, and then $s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\mu}$.

Step 2. We prove the uniqueness of $(s_{u}, t_{u})$.

By standard arguments, we only prove the uniqueness in case of $u\in\mathcal{M}_{\mu}$ here.

For any $u\in\mathcal{M}_{\mu}$, we have that

Suppose $(s_{0}, t_{0})$ be an other pair of numbers such that $s_{0}u^{+}+t_{0}u^{-}\in\mathcal{M}_{\mu}$ with $0 < s_{0}\leq t_{0}$. So, one has that

and

It follows from (2.9) and (2.11) that

Thanks to $(f_{3})$, we obtain that $t_{0}\leq 1$.

Similarly, by (2.9), (2.10) and $(f_{3})$, we conclude that $s_{0}\geq1$.

Consequently, $s_{0} = t_{0} = 1$.

Step 3. If $\langle J_{\mu}'(u), u^{\pm}\rangle\leq0,$ then $0 < s_{u}, t_{u}\leq1.$

Suppose $s_{u}\geq t_{u} > 0$. Combining $s_{u}u^{+}+t_{u}u^{-}\in\mathcal{M}_{\mu}$ and $\langle J_{\mu}'(u), u^{\pm}\rangle\leq0,$ one has

So, according to condition $(f_{3})$, we get $s_{u}\leq1$. Thus, we have that $0 < s_{u}, t_{u}\leq1$.

Step 4. $(s_{u}, t_{u})$ is the unique maximum point of $G_{u}$ on $[0, \infty)\times[0, \infty)$.

Obviously, it follows from (2.5) that

Hence, $(s_{u}, t_{u})$ is the unique critical point of $G_{u}$ in $[0, \infty)\times[0, \infty)$.

At same time, let $t_{0}\geq0$ be fixed, we infer that

That is, $G_{u}(s, t)$ is an increasing function with respect to $s$ if $s$ is small enough.

Similarly, we conclude that $G_{u}(s, t)$ is an increasing function with respect to $t$ if $t$ is small enough.

Therefore, we conclude that maximum point of $G_{u}$ cannot be achieved on the boundary of $[0, \infty)\times[0, \infty)$. That is, $(s_{u}, t_{u})$ is the unique maximum point of $G_{u}$ on $[0, \infty)\times[0, \infty)$.

Lemma 2.3. There exist $\rho > 0$ such that $\|u^{\pm}\|\geq\rho$ for all $u\in\mathcal{M}_{\mu}$.

Proof. For any $u\in\mathcal{M}_{\mu}$, we have

Thanks to (2.1), one has

So, $(1-\mu\varepsilon C_{1})\|u^{\pm}\|^{2} \leq\mu C_{2}\|u^{\pm}\|^{q}+C_{3}\|u^{\pm}\|^{6}.$ Choosing $\varepsilon$ small enough such that $(1-\lambda\varepsilon C_{1}) > 0$, we get the conclusion.

Lemma 2.4. Let $c_{\mu} = \inf_{u\in\mathcal{M}_{\mu}}J_{\mu}(u)$, then we have that $\lim_{\mu\rightarrow\infty} c_{\mu} = 0.$

Proof. For any $u\in\mathcal{M}_{\mu}$, $\langle J_{\mu}'(u), u\rangle = 0$. Thanks to $(f_{3})$, it is easy to obtain that

and is increasing when $t > 0$ and decreasing when $t < 0$. Then, one gets

So, by Lemma 2.3 we have that $J_{\mu}(u) > 0$, for all $u\in\mathcal{M}_{\mu}$. Hence, $c_{\mu} = \inf_{u\in{M}_{\mu}}J_{\mu}(u)$ is well-defined.

Let $u\in E$ with $u^{\pm}\neq0$ be fixed. According to Lemma 2.2, for each $\mu > 0$, there exist $s_{\mu}, t_{\mu} > 0$ such that $s_{\mu}u^{+}+t_{\mu}u^{-}\in\mathcal{M}_{\mu}$.

By using Lemma 2.2 again and Hardy-Littlewood-Sobolev inequality (Page 106 of ^[49]), we have that

To our end, we just prove that $s_{\mu}\rightarrow0$ and $t_{\mu}\rightarrow0$, as $\mu\rightarrow\infty$.

Let

By (2.5), we get

which implies that $B_{u}$ is bounded.

Let $\{\mu_{n}\}\subset (0, \infty)$ be such that $\mu_{n}\rightarrow\infty$ as $n\rightarrow\infty$. Then, there exist $s_{0}$ and $t_{0}$ such that

as $n\rightarrow\infty$ (in subsequence sense).

We claim $s_{0} = t_{0} = 0$.

Suppose, by contradiction, that $s_{0} > 0$ or $t_{0} > 0$. Thanks to $s_{\mu_{n}}u^{+}+t_{\mu_{n}}u^{-}\in \mathcal{M}_{\mu_{n}}$, for any $n \in \mathbb{N}$, we have

According to $s_{\mu_{n}}u^{+}\rightarrow s_{0}u^{+}$ and $t_{\mu_{n}}u^{-}\rightarrow t_{0}u^{-}$ in $E$, (2.3) and (2.5), we conclude that

as $n\rightarrow\infty$.

It follows from $\mu_{n}\rightarrow\infty$ as $n\rightarrow\infty$ and $\{s_{\mu_{n}}u^{+}+t_{\mu_{n}}u^{-}\}$ is bounded in $E$ that we have a contradiction with the equality (2.15). Hence, $s_{0} = t_{0} = 0$.

That is, $\lim_{\mu\rightarrow\infty} c_{\mu} = 0$.

Lemma 2.5. There exist $\mu^{\star} > 0$ such that for all $\mu\geq\mu^{\star}$, the infimum $c_{\mu}$ is achieved.

Proof. According to definition of $c_{\mu}$, there is a sequence $\{u_{n}\}\subset\mathcal{M}_{\mu}$ such that

Obviously, $\{u_{n}\}$ is a bounded in $E$. Then, up to a subsequence, still denoted by $\{u_{n}\}$, there exist $u\in E$ such that $u_{n}\rightharpoonup u.$

Since the embedding $E\hookrightarrow L^{r}(\mathbb{R}^3)$ is compact, for all $r\in[2, 6)$, we have

So,

Denote $\beta: = \frac{1}{3}S^{\frac{3}{2}}$, where

According to Lemma 2.4, there is $\mu^{\star} > 0$ such that $c_{\mu} < \beta$ for all $\mu\geq\mu^{\star}$.

Fix $\mu\geq\mu^{\star}$, it follows from Lemma 2.2 that

for all $s, t\geq0$.

By the weak lower semi-continuity of norm, Brezis-Lieb Lemma and Lemma 2.1, we have that

where

From above fact, one has that

for all $s \geq0$ and all $t \geq0$.

$\bf{Firstly, \; we\; prove\; that\; u^{\pm}\neq0.}$

Since the situation $u^{-}\neq0$ is analogous, we just prove $u^{+}\neq0$. By contradiction, we suppose $u^{+} = 0$.

${\bf{Case\; 1}}: \; B_{1} = 0$.

If $A_{1} = 0$, that is, $u^{+}_{n}\rightarrow u^{+} \; \text{in}\; E$. In view of Lemma 2.3, we obtain $\|u^{+}\| > 0$, which contradicts our supposition. If $A_{1} > 0$, let $t = 0$ in (2.16), one hase

for all $s\geq0$. Thanks to $c_{\mu} < \beta$, we have a contradiction.

${\bf{Case\; 2}}: \; B_{1} > 0$.

According to definition of $S$, we have that

According to (2.16), we have a contradiction.

From above discussions, we have that $u^{\pm}\neq0$.

$\bf{Secondly, \; we\; prove\; that\; B_{1} = B_{2} = 0}.$

Since the situation $B_{2} = 0$ is analogous, we only prove $B_{1} = 0$. By contradiction, we suppose that $B_{1} > 0$.

${\bf{Case\; 1}}: \; B_{2} > 0$.

Let $\widetilde{s}$ and $\widetilde{t}$ satisfy

Since $G_{u}$ is continuous, there exists $(s_{u}, t_{u})\in[0, \widetilde{s}]\times[0, \widetilde{t}]$ such that

In the following, we prove that $(s_{u}, t_{u})\in(0, \widetilde{s})\times(0, \widetilde{t})$.

Note that, if $t$ is small enough, we have that

for all $s\in[0, \widetilde{s}].$ That is, there exists $t_{0}\in[0, \widetilde{t}]$ such that $G_{u}(s, 0)\leq G_{u}(s, t_{0})$ for all $s\in[0, \widetilde{s}].$

Hence, we conclude that any point of $(s, 0)$ with $0\leq s\leq \widetilde{s}$ is not the maximizer of $G_{u}$, and then $(s_{u}, t_{u})\not\in[0, \widetilde{s}]\times\{0\}.$ Similarly, we have that $(s_{u}, t_{u})\not\in\{0\}\times[0, \widetilde{s}]$.

On the other hand, it is easy to see that

Then,

for all $s\in[0, \widetilde{s}]$ and all $t\in[0, \widetilde{t}]$.

Therefore, according to (2.16), we have that

for all $s\in[0, \widetilde{s}]$ and all $t\in[0, \widetilde{t}]$. So, $(s_{u}, t_{u})\not\in \{\widetilde{s}\}\times[0, \widetilde{t}]$ and $(s_{u}, t_{u})\not\in \times[0, \widetilde{s}]\times\{\widetilde{t}\}$.

At last, we get that $(s_{u}, t_{u})\in(0, \widetilde{s})\times(0, \widetilde{t})$. Hence, it follows that $(s_{u}, t_{u})$ is a critical point of $G_{u}$. By Lemma 2.1, we get $s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\mu}$.

Combining (2.16), (2.17) with (2.18), we infer that

which is a contradiction.

${\bf{Case\; 2}}: \; B_{2} = 0.$

In this case, we can maximize in $[0, \widetilde{s}]\times[0, \infty)$. Indeed, it is possible to show that there exist $t_{0}\in[0, \infty)$ such that $J_{\mu}(s_{u}u^{+}+t_{u}u^{-})\leq0,$ for all $(s, t)\in[0, \widetilde{s}]\times[t_{0}, \infty)$. Hence, there are $(s_{u}, t_{u})\in[0, \widetilde{s}]\times[0, \infty)$ satisfy

Following, we prove that $(s_{u}, t_{u})\in(0, \widetilde{s})\times(0, \infty)$.

It is noticed that $G_{u}(s, 0) < G_{u}(s, t)$ for $s\in [0, \widetilde{s}]$ and $t$ small enough, so we have $(s_{u}, t_{u})\not\in[0, \widetilde{s}]\times\{0\}.$

Meantime, $G_{u}(0, t) < G_{u}(s, t)$ for $t\in [0, \infty)$ and $s$ small enough, then we have $(s_{u}, t_{u})\not\in\{0\}\times[0, \infty).$

On the other hand, it is obvious that

for all $t\in[0, \infty)$.

Hence, we have that $G_{u}(\widetilde{s}, t)\leq0$ for all $t\in[0, \infty)$. Thus, $(s_{u}, t_{u})\not\in\{\widetilde{s}\}\times[0, \infty)$. And so $(s_{u}, t_{u})\in(0, \widetilde{s})\times(0, \infty)$. That is, $(s_{u}, t_{u})$ is an inner maximizer of $G_{u}$ in $[0, \widetilde{s})\times[0, \infty)$. So, $s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\mu}$.

Therefore, according to (2.17), we get

That is, we have a contradiction.

Therefore, from above discussion, we have that $B_{1} = B_{2} = 0$.

Lastly, we prove that c_µ is achieved.

For $u^{\pm}\neq0$, according to Lemma 2.2, there exist $s_{u}, t_{u} > 0$ such that $\overline{u}:s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\mu}$. Furthermore, it is easy to see that

So, we have that $0 < s_{u}, t_{u}\leq1$.

Since $u_{n}\in \mathcal{M}_{\mu}$, using Lemma 2.2 again, we get

Thanks to (2.14), $B_{1} = B_{2} = 0$ and the norm in $E$ is weak lower semicontinuous, we conclude that

Therefore, $s_{u} = t_{u} = 1$, and $c_{\mu}$ is achieved by $u_{\mu}: = u^{+}+u^{-}\in \mathcal{M}_{\mu}$.

3.   The proof of main results

Proof. $(\bf{Proof\; of\; Theorem\; 1.1})$ From Lemma 2.5, we just prove that the minimizer $u_{\mu}$ for $c_{\mu}$ is indeed a sign-changing solution of problem (1.1). That is, we need prove $J_{\mu}'(u_{\mu}) = 0$. Suppose, by contradiction, that $J_{\mu}'(u_{\mu})\neq0$. Then there exist $\delta > 0$ and $\theta > 0$ such that

Choose $\sigma\in(0, \min\{1/2, \frac{\delta}{\sqrt{2}\|u_{\mu}\|}\})$. Let $D: = (1-\sigma, 1+\sigma)\times(1-\sigma, 1+\sigma)$ and $g(s, t) = su_{\mu}^{+}+tu_{\mu}^{-}$, $(s, t)\in D$.

Thanks to Lemma 2.3, for $(s, t)\in (\mathbb{R}_{+}\times\mathbb{R}_{+})\backslash(1, 1)$, we have

So, it follows that

Let $\varepsilon: = min\{(c_{\mu}-\overline{c}_{\lambda}^{\mu})/2, \theta\delta/8\}$ and $S_{\delta}: = B(u_{\mu}, \delta)$, according to Lemma 2.3 in ^[50], there exists a deformation $\eta\in C([0, 1]\times E, E)$ satisfy

$(a) \; \eta(1, v) = v$ if $v\notin J_{\mu}^{-1}([c_{\mu}-2\varepsilon, c_{\mu}+2\varepsilon]\cap S_{2\delta})$;

$(b) \; \eta(1, J_{\mu}^{c_{\mu}+\varepsilon}\cap S_{\delta})\subset J_{\mu}^{c_{\mu}-\varepsilon}$, where $J_{\mu}^{c} = \{u\in E:J_{\mu}(u)\leq c\}$;

$(c) \; J_{\mu}(\eta(1, v))\leq J_{\mu}(v)$ for all $v\in E$.

From $(b)$ and Lemma 2.2, it is easy to see that

Next, we prove that $\eta(1, g(D))\cap\mathcal{M}_{\mu}\neq\varnothing$, which contradicts the definition of $c_{\mu}$.

Let $h(s, t): = \eta(1, g(s, t))$ and

and

Let

From condition $(f_{3})$, for $t\neq0$, we have

Therefore, by direct calculation, we can conclude that $\det M > 0$.

Since $\Psi_{0}(s, t)$ is a $C^{1}$ function and $(1, 1)$ is the unique isolated zero point of $\Psi_{0}$, by using the degree theory, we deduce that deg$(\Psi_{0}, D, 0) = 1$.

So, combining (3.1) with (a), we obtain $g(s, t) = h(s, t)$ on $\partial D$. Consequently, we obtain deg$(\Psi_{1}, D, 0) = 1$. Therefore, $\Psi_{1}(s_{0}, t_{0}) = 0$ for some $(s_{0}, t_{0})\in D$. By similar discussion as in ^[45], we can prove that $h(s, t)^{\pm}\neq0$. So, we obtain that $\eta(1, g(s_{0}, t_{0})) = h(s_{0}, t_{0})\in \mathcal{M}_{\mu}$, which is contradicted to (3.2).

Proof. $(\bf{Proof\; of\; Theorem\; 1.2})$ Similar as the proof of Lemma 2.5, there exists $\mu_{1}^{\star} > 0$ such that for all $\mu\geq\mu_{1}^{\star}$, there exists $v_{\mu}\in \mathcal{N_{\mu}}$ such that $J_{\mu}(v_{\mu}) = c^{\ast} > 0$. By standard arguments, the critical points of the functional $J_{\mu}$ on $\mathcal{N_{\mu}}$ are critical points of $J_{\mu}$ in $E$ and we obtain $J_{\mu}'(v_{\mu}) = 0$. That is, $v_{\mu}$ is a least energy solution of system (1.1). According to Theorem 1.1, we know that the system (1.1) has a least energy nodal solution $u_{\mu}$. Let $\mu^{\star\star} = \max\{\mu^{\star}, \mu^{\star}_{1}\}.$ As the proof of Lemma 2.2, there exist $s_{u^{+}}, t_{u^{-}}\in(0, 1)$ such that $s_{u^{+}}u^{+}\in \mathcal{N_{\mu}}, t_{u^{-}}u^{-}\in \mathcal{N_{\mu}}.$ Therefore, in view of Lemma 2.2 again, we infer that

$2c^{\ast}\leq J_{\mu}(s_{u^{+}}u^{+})+J_{\mu}(t_{u^{-}}u^{-})\leq J_{\mu}(s_{u^{+}}u^{+}+t_{u^{-}}u^{-}) < J_{\mu}(u^{+}+u^{-}) = c_{\mu}.$

4.   Conclusions

In this paper, by the minimization argument on the nodal Nehari manifold and the quantitative deformation lemma, we discussed the existence of least energy nodal solution for a class of Kirchhoff-type system with Hartree-type nonlinearity and critical growth. Our results improve and generalize some interesting results which were obtained in subcritical case.

Acknowledgments

This research was supported by the Natural Science Foundation of China (11561043, 11961043).

Conflict of interest

We declare that we have no conflict of interest.