Existence of a ground-state solution for a quasilinear Schrödinger system

In this paper, we consider the following quasilinear Schrödinger system.

$\left\{\begin{array}{cc}-\mathrm{\Delta }u+u+\frac{k}{2}\left[\mathrm{\Delta }|u{|}^2\right]u=\frac{2\alpha }{\alpha +\beta }|u{|}^{\alpha -2}u|v{|}^{\beta },\hfill & \hfill x\in {\mathbb{R}}^N,\hfill \\ -\mathrm{\Delta }v+v+\frac{k}{2}\left[\mathrm{\Delta }|v{|}^2\right]v=\frac{2\beta }{\alpha +\beta }|u{|}^{\alpha }|v{|}^{\beta -2}v,\hfill & \hfill x\in {\mathbb{R}}^N,\hfill \end{array}\right.$

where k < 0 is a real constant, α > 1, β > 1, and α + β < 2*. We take advantage of the critical point theorem developed by Jeanjean (Proc. R. Soc. Edinburgh Sect A., 1999, 129: 787–809) and combine it with Pohožaev identity to obtain the existence of a ground-state solution, which is the non-trivial solution with the least possible energy.

1 Introduction

This article is concerned with the following quasilinear Schrödinger system:

$\left\{\begin{array}{cc}-\mathrm{\Delta }u+u+\frac{k}{2}\left[\mathrm{\Delta }|u{|}^2\right]u=\frac{2\alpha }{\alpha +\beta }|u{|}^{\alpha -2}u|v{|}^{\beta },\hfill & x\in {\mathbb{R}}^N,\hfill \\ -\mathrm{\Delta }v+v+\frac{k}{2}\left[\mathrm{\Delta }|v{|}^2\right]v=\frac{2\beta }{\alpha +\beta }|u{|}^{\alpha }|v{|}^{\beta -2}v,\hfill & x\in {\mathbb{R}}^N,\hfill \end{array}\right.(1.1)$

where k < 0 is a real constant.

Many scholars have made significant contributions to the study of the quasilinear Schrödinger system. Wang and Huang proved the existence of ground-state solutions for a class of systems by establishing a suitable Nehari–Pohožaev-type constraint set and considering related minimization problems in [2]. The existence of infinitely many solutions was established for the quasilinear Schrödinger system by the symmetric Mountain Pass Theorem; see [3]. The existence of positive solutions was obtained by using the monotonicity trick and Morse iteration in [4]. Chen and Zhang proved the existence of ground-state solutions by minimization under a convenient constraint and concentration compactness lemma in [5].

The quasilinear Schrödinger system (1.1) is in part motivated by the following quasilinear Schrödinger equation:

$iϵ\partial z=-ϵ\mathrm{\Delta }z+W\left(x\right)z-l\left(|z{|}^2\right)z-kϵ\mathrm{\Delta }h\left(|z{|}^2\right)h^{\prime }\left(|z{|}^2\right)z,\text{for}x\in {\mathbb{R}}^N,N>2,(1.2)$

where W(x) is a given potential, k is a real constant, and l and h are real functions that are essentially pure power forms. The quasilinear Schrödinger Equation 1.2 describes several physical phenomena with different h; see [6–8].

Let the case $h(s)=s,l(s)=\mu s^{\frac{p-1}{2}}$ and k > 0. Setting z(t, x) = exp(−iFt)u(x), one can obtain a corresponding equation of elliptic type which has the formal variational structure:

$ϵ\mathrm{\Delta }u+V\left(x\right)u-ϵk\left(\mathrm{\Delta }\left(|u{|}^2\right)\right)u=\mu |u{|}^{p-1}u,u>0x\in {\mathbb{R}}^N,N>2,(1.3)$

where V(x) = W(x) – F is the new potential function. The problem (1.3) has been studied by many academics. In [9], the existence results of multiple solutions were studied via dual approach techniques and variational methods when k > 0 was small enough. The existence of soliton solutions was established by a minimization argument; see [10]. The Mountain Pass Theorem is combined with the principle of symmetric criticality to establish the multiplicity of solutions in [11]. In [12], the author proved the existence of soliton solutions via making a change in variables and creating a suitable Orlicz space. The minimax principles for lower semicontinuous functionals were used to find solutions in [13].

In [14], the authors used the method developed by [1, 15] to divide the energy functional into two parts and established the existence of ground-state solutions for a type of quasilinear Schrödinger equation like 1.3. Inspired by [14], we try to find the existence of ground-state solutions for system 1.1. This achievement can enrich the relatively few existing results about this system.

The main result of this paper is the following:

Theorem 1.1. When k < 0, α > 1, β > 1, and α + β < 2*, then (1.1) has a ground-state solution.

This paper is organized as follows. In Section 2, preparation work is completed. In Section 3, we reformulate this problem and prove Theorem 1.1. In this paper, C is defined as different constants.

2 Reformulation of the problem and preliminaries

First, we explain that $L^q({\mathbb{R}}^N)$ denotes the Lebesgue space with the norm

${‖u‖}_p={\left({\int }_{{\mathbb{R}}^N}|u{|}^{p}dx\right)}^{\frac{1}{p}},$

where 1 ≤ p < ∞. $L^q=L^q({\mathbb{R}}^N)\times L^q({\mathbb{R}}^N)$ with the norm

${‖\left(u,v\right)‖}_p={\left({\int }_{{\mathbb{R}}^N}|u{|}^{p}dx\right)}^{\frac{1}{p}}+{\left({\int }_{{\mathbb{R}}^N}|v{|}^{p}dx\right)}^{\frac{1}{p}},$

where 1 ≤ p < ∞.

$H^1=\left\{\left(u,v\right):u,v\in L^2\left({\mathbb{R}}^N\right),\nabla u,\nabla v\in L^2\left({\mathbb{R}}^N\right)\right\}$

with norms

$‖\left(u,v\right)‖=‖u‖+‖v‖={\left({\int }_{{\mathbb{R}}^N}\left(|\nabla u{|}^2+u^2\right)dx\right)}^{\frac{1}{2}}+{\left({\int }_{{\mathbb{R}}^N}\left(|\nabla v{|}^2+v^2\right)dx\right)}^{\frac{1}{2}}$

and

${‖\left(u,v\right)‖}^2={‖u‖}^2+{‖v‖}^2.$

The embedding H¹↪L^q is continuous and compact for q ∈ (2, 2*).

In (1.1), the Euler–Lagrange functional associated with Equation 1.1 is given by

$\begin{array}{cc}\hfill I\left(u,v\right)& =\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(1-k{u}^2\right)|\nabla u{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}|u{|}^{2}dx\hfill \\ \hfill & +\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(1-k{v}^2\right)|\nabla v{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}|v{|}^{2}dx-\frac{2}{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|u{|}^{\alpha }|v{|}^{\beta }dx.\hfill \end{array}$

For (u, v), constructing the variable like [16, 17], we have

$dz=\sqrt{-k}\sqrt{1-k{u}^2}du,z=h\left(u\right)=\frac{1}{2}\sqrt{-k}u\sqrt{1-k{u}^2}+\frac{1}{2}ln\left(\sqrt{-k}u+\sqrt{1-k{u}^2}\right),$

$dw=\sqrt{-k}\sqrt{1-k{v}^2}dv,w=h\left(v\right)=\frac{1}{2}\sqrt{-k}v\sqrt{1-k{v}^2}+\frac{1}{2}ln\left(\sqrt{-k}v+\sqrt{1-k{v}^2}\right).$

Since h is strictly monotone, it has a well-defined inverse function f and u = f(z), v = f(w). Note that

$h\left(u\right)\sim \left\{\begin{array}{c}\sqrt{-k}u,|u|\ll \sqrt{\frac{1}{-k}}\hfill \\ \frac{-k}{2}u|u|,|u|\gg \sqrt{\frac{1}{-k}}\hfill \end{array},\right.h^{\prime }\left(u\right)=\sqrt{-k}\sqrt{1-k{u}^2}$

and

$f\left(z\right)\sim \left\{\begin{array}{c}\frac{1}{\sqrt{-k}}z,|z|\ll \sqrt{\frac{1}{-k}}\hfill \\ \sqrt{\frac{2}{-k|z|}}z,|z|\gg \sqrt{\frac{1}{-k}}\hfill \end{array},\right.$

$f^{\prime }\left(z\right)=\frac{1}{h^{\prime }\left(u\right)}=\frac{1}{\sqrt{-k}\sqrt{1-k{v}^2}}=\frac{1}{\sqrt{-k}\sqrt{1-k{f\left(z\right)}^2}}.$

Similarly, the same operation holds true for v = f(w).

Using the variable, (1.1) will become

$\left\{\begin{array}{cc}-\frac{1}{k}\mathrm{\Delta }z+f\left(z\right)f^{\prime }\left(z\right)=\frac{2\alpha }{\alpha +\beta }|f\left(z\right){|}^{\alpha -2}f\left(z\right)|f\left(w\right){|}^{\beta },\hfill & x\in {\mathbb{R}}^N,\hfill \\ -\frac{1}{k}\mathrm{\Delta }w+f\left(w\right)f^{\prime }\left(w\right)=\frac{2\beta }{\alpha +\beta }|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta -2}f\left(w\right),\hfill & x\in {\mathbb{R}}^N,\hfill \end{array}\right.(2.1)$

where $f:[0,\infty )\to \mathbb{R}$ and

$f^{\prime }=\frac{1}{\sqrt{-k}\sqrt{1-k{f}^2}}$

on [0, ∞), f(0) = 0, and f(−t) = f(t) on [0, ∞). From the above facts, if (z, w) is a weak solution for (2.1), then $(u,v)=\left(f(z),f(w)\right)$ is a weak solution for (1.1). The energy functional I(u, v) reduces to the following functional:

$\begin{array}{cc}\hfill \varphi \left(z,w\right)& =\frac{1}{2}{\int }_{{\mathbb{R}}^N}\frac{1}{\sqrt{-k}}|\nabla z{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}{f}^2\left(z\right)dx\hfill \\ \hfill & +\frac{1}{2}{\int }_{{\mathbb{R}}^N}\frac{1}{\sqrt{-k}}|\nabla w{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}{f}^2\left(w\right)dx-\frac{2}{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta }dx.\hfill \end{array}(2.2)$

There are some properties of $f:\mathbb{R}\to \mathbb{R}$ as follows, which are proved in [16, 17].

Lemma 2.1. The function f(t) and its derivative satisfy the following properties:

(i) $\frac{f(t)}{t}\to 1$ as t → 0;

(ii) f(t) ≤ |t| for any $t\in \mathbb{R}$;

(iii) $f(t)\le 2^{\frac{1}{4}}\sqrt{|t|}$ for all $t\in \mathbb{R}$;

(iv) $\frac{f^2(t)}{2}\le tf(t)f^{\prime }(t)\le f^2(t)$ for all $t\in \mathbb{R}$;

(v) there exists a positive constant C such that

$|f\left(t\right)|\ge \left\{\begin{array}{c}C|t|,\mathit{\text{if}}t\le 1,\hfill \\ C|t{|}^{\frac{1}{2}},\mathit{\text{if}}t>1;\hfill \end{array}\right.$

(vi) $|f(t)f^{\prime }(t)|\le \frac{1}{\sqrt{2}}$ for all $t\in \mathbb{R}$.

3 Proof of theorem 1.1

In this section, we will complete the proof of Theorem 1.1. First, we will recall the critical point theorem in [1], which is crucial for proving Theorem 1.1.

Theorem 3.1. Let $\left(X,‖(\cdot ,\cdot )‖\right)$ be a Banach space and $L\subset {\mathbb{R}}_{+}$ an interval. Consider the following family of C¹-functionals on X:

${\mathrm{\Phi }}_{\lambda }\left(z,w\right)=A\left(z,w\right)+\lambda B\left(z,w\right),\lambda \in L,$

with B being non-negative and either A(z, w) → +∞ or B(z, w) → +∞ as $‖(z,w)‖\to \infty$. Assume that there are two points (z₁, w₁), (z₂, w₂) ⊂ X such that

$c_{\lambda }=\underset{\gamma \in {\mathrm{\Gamma }}_{\lambda }}{\inf }\underset{\left(t_1,t_2\right)\in \left[\mathrm{0,1}\right]\times \left[\mathrm{0,1}\right]}{\max }{\mathrm{\Phi }}_{\lambda }\left(\gamma \left(t_1,t_2\right)\right)>\max \left\{{\mathrm{\Phi }}_{\lambda }\left(z_1,w_1\right),{\mathrm{\Phi }}_{\lambda }\left(z_2,w_2\right)\right\}\mathit{\text{for all}}\lambda \in L,$

where Γ_λ = {γ ∈ C([0, 1] × [0, 1], X): γ(0, 0) = (z₁, w₁), γ(1, 1) = (z₂, w₂)}. Then, for almost every λ ∈ L, there is a sequence {(z_n, w_n)} ⊂ X such that

(i) (z_n, w_n) is bounded;

(ii) Φ_λ(z, w) → c_λ;

(iii) ${\mathrm{\Phi }}_{\lambda }^{\prime }(z_n,w_n)\to 0$ in the dual X⁻¹ of X.

Moreover, the map λ → c_λ is non-increasing and continuous from the left.

Let λ ∈ L be an arbitrary but fixed value where $c_{\lambda }^{\prime }$ exists, where $c_{\lambda }^{\prime }$ is the derivative of c_λ with respect to λ. Let {λ_n} ⊂ L be a strictly increasing sequence such that λ_n → λ. To prove Theorem 3.1, we will show the following lemmas:

Lemma 3.1. There exists a sequence of path {γ_n} ⊂ Γ and $K=K(c_{\lambda }^{\prime })>0$ such that

(i) $‖{\gamma }_n(t_1,t_2)‖\le K$ if γ_n(t₁, t₂) satisfies

$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\lambda }\left({\gamma }_n\left(t_1,t_2\right)\right)\ge c_{\lambda }-\left(\lambda -{\lambda }_n\right);\end{array}(3.1)$

(ii) ${\max }_{(t_1,t_2)\in [\mathrm{0,1}]}{\mathrm{\Phi }}_{\lambda }({\gamma }_n(t_1,t_2))\le c_{\lambda }+(-c_{\lambda }^{\prime }+2)(\lambda -{\lambda }_n)$.

Proof. The proof is standard; see [1].

Lemma 3.1. means that there exists a sequence of paths {γ_n} ⊂ Γ such that

$\underset{\left(t_1,t_2\right)\in \left[\mathrm{0,1}\right]\times \left[\mathrm{0,1}\right]}{\max }{\mathrm{\Phi }}_{\lambda }\left({\gamma }_n\left(t_1,t_2\right)\right)\to c_{\lambda },$

for all $n\in \mathbb{N}$ sufficiently large; starting from a level strictly below c_λ, all the “top” of the path is contained in the ball centered at the origin of fixed radius $K=K(c_{\lambda }^{\prime })>0$. Now, for α > 0, we define

$F_{\alpha }=\left\{\left(z,w\right)\in X:‖\left(z,w\right)‖\le K+1\text{\hspace{0.17em}and\hspace{0.17em}}|{\mathrm{\Phi }}_{\lambda }\left(z,w\right)-c_{\lambda }|\le \alpha \right\},$

where K is given in lemma 3.1.

Lemma 3.2. For all α > 0,

$\begin{array}{c}\hfill \inf \left\{‖{\mathrm{\Phi }}_{\lambda }^{\prime }\left(z,w\right)‖:\left(z,w\right)\in F_{\alpha }\right\}=0.\end{array}(3.2)$

Proof. We assume that (3.2) does not hold. Then, there exists α > 0 such that for any (z, w) ∈ F_α, we obtain

$\begin{array}{c}\hfill ‖{\mathrm{\Phi }}_{\lambda }^{\prime }\left(z,w\right)‖\ge \alpha .\end{array}(3.3)$

Without loss of generality, we can assume that

$0<\alpha <\frac{1}{2}\left[c_{\lambda }-\max \left\{{\mathrm{\Phi }}_{\lambda }\left(z_1,w_1\right),{\mathrm{\Phi }}_{\lambda }\left(z_2,w_2\right)\right\}\right].$

A classical deformation argument then says that there exists ϵ ∈ [0, α] and a homeomorphism η: X → X such that

$\begin{array}{c}\hfill \eta \left(u\right)=u,\text{if}|{\mathrm{\Phi }}_{\lambda }\left(z,w\right)-c_{\lambda }|\ge \alpha ,\end{array}(3.4)$

$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\lambda }\left(\eta \left(z,w\right)\right)\le {\mathrm{\Phi }}_{\lambda }\left(z,w\right),\forall \left(z,w\right)\in X,\end{array}(3.5)$

$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\lambda }\left(\eta \left(z,w\right)\right)\le c_{\lambda }-ϵ,\forall \left(z,w\right)\in X,\text{\hspace{0.17em}satisfying\hspace{0.17em}}‖\left(z,w\right)‖\le K\text{\hspace{0.17em}and\hspace{0.17em}}{\mathrm{\Phi }}_{\lambda }\left(z,w\right)\le c_{\lambda }+ϵ.\end{array}(3.6)$

Let {γ_n} ⊂ Γ be the sequence obtained in lemma 3.1. We choose and fix $m\in \mathbb{N}$ sufficiently large in order that

$\begin{array}{c}\hfill \left(-c_{\lambda }^{\prime }+2\right)\left(\lambda -{\lambda }_m\right)\le ϵ.\end{array}(3.7)$

By lemma 3.1 and (3.4), η(γ_m) ∈ Γ. Now if (z, w) = γ_m(t₁, t₂) satisfies

${\mathrm{\Phi }}_{\lambda }\left(z,w\right)\le c_{\lambda }-\left(\lambda -{\lambda }_m\right),$

then (3.5) implies that

$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\lambda }\left(\eta \left(z,w\right)\right)\le c_{\lambda }-\left(\lambda -{\lambda }_m\right).\end{array}(3.8)$

If (z, w) = γ_m(t₁, t₂) satisfies

${\mathrm{\Phi }}_{\lambda }\left(z,w\right)>c_{\lambda }-\left(\lambda -{\lambda }_m\right),$

by lemma 3.1 and (3.7), we obtain (z, w) such that $‖(z,w)‖\le K$ with Φ_λ(z, w) ≤ c_λ + ϵ. From (3.6), we obtain

$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\lambda }\left(\eta \left(z,w\right)\right)\le c_{\lambda }-ϵ\le c_{\lambda }-\left(\lambda -{\lambda }_m\right).\end{array}(3.9)$

Combining (3.8) with (3.9), we obtain

$\underset{\left(t_1,t_2\right)\in \left[\mathrm{0,1}\right]\times \left[\mathrm{0,1}\right]}{\max }{\mathrm{\Phi }}_{\lambda }\left(\eta \left({\gamma }_m\left(t_1,t_2\right)\right)\right)\le c_{\lambda }-\left(\lambda -{\lambda }_m\right),$

which contradicts the variational characterization of c_λ.

Next, we prove theorem 3.1.

Proof. Since lemma 3.2 is true, there exists a PS sequence for Φ_λ at the level $c_{\lambda }\in \mathbb{R}$, which is contained in the ball of radius K + 1 centered at the origin. Hence, this is proved.

Let $L=\left[\frac{1}{2},1\right]$, we define the following energy functional:

$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{\lambda }\left(z,w\right)& =\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla z{|}^2+z^2+\frac{1}{\sqrt{-k}}|\nabla w{|}^2+w^2\right)dx\hfill \\ \hfill & -\lambda {\int }_{{\mathbb{R}}^N}\left(\frac{1}{2}\left(z^2-f^2\left(z\right)+w^2-f^2\left(w\right)\right)+\frac{2}{\alpha +\beta }|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta }\right)dx,\hfill \end{array}(3.10)$

where λ ∈ L. Moreover, let

$A\left(z,w\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla z{|}^2+z^2+\frac{1}{\sqrt{-k}}|\nabla w{|}^2+w^2\right)dx$

and

$B\left(z,w\right)=\lambda {\int }_{{\mathbb{R}}^N}\left(\frac{1}{2}\left(z^2-f^2\left(z\right)+w^2-f^2\left(w\right)\right)+\frac{2}{\alpha +\beta }|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta }\right)dx.$

Letting $‖(z,w)‖\to +\infty$, then A(z, w) → +∞ and B(z, w) ≥ 0.

By a standard argument in [18, 19], we have the following Pohožaev-type identity:

Lemma 3.3. If (z, w) ∈ H¹ is a critical point of (3.10), then (z,w) satisfies P_λ(z, w) = 0, where

$\begin{array}{cc}\hfill P_{\lambda }\left(z,w\right)≔& \frac{N-2}{2}{\int }_{{\mathbb{R}}^N}\frac{1}{\sqrt{-k}}\left(|\nabla z{|}^2+|\nabla w{|}^2\right)dx+\frac{N}{2}{\int }_{{\mathbb{R}}^N}\left(f^2\left(z\right)+f^2\left(w\right)\right)dx\hfill \\ \hfill & -\frac{2N\lambda }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta }dx.\hfill \end{array}(3.11)$

Similar to [9], we obtain the following lemma:

Lemma 3.4. Φ_λ(z, w) meet the conditions as follows:

(i) there exists (z, w) ∈ H¹ \{(0, 0)} such that Φ_λ(z, w) < 0 for all λ ∈ L;

(ii) for c_λ, we obtain

$c_{\lambda }=\underset{\gamma \in \mathrm{\Gamma }}{\inf }\underset{\left(t_1,t_2\right)\in \left[\mathrm{0,1}\right]\times \left[\mathrm{0,1}\right]}{\max }{\mathrm{\Phi }}_{\lambda }\left(\gamma \left(t_1,t_2\right)\right)>\max \left\{{\mathrm{\Phi }}_{\lambda }\left(\mathrm{0,0}\right),{\mathrm{\Phi }}_{\lambda }\left(z,w\right)\right\},$

for all λ ∈ L, where

$\mathrm{\Gamma }=\left\{\gamma \in C\left(\left[\mathrm{0,1}\right]\times \left[\mathrm{0,1}\right],H^1\right):\gamma \left(\mathrm{0,0}\right)=\left(\mathrm{0,0}\right),\gamma \left(\mathrm{1,1}\right)=\left(z,w\right)\right\}.$

Proof. (i) Let (z, w) ∈ H¹ \{(0, 0)} be fixed. For any $\lambda \in L=\left[\frac{1}{2},1\right]$, we obtain

$\begin{array}{cc}\hfill & {\mathrm{\Phi }}_{\lambda }\left(z,w\right)\le {\mathrm{\Phi }}_{\frac{1}{2}}\left(z,w\right)\hfill \\ \hfill =& \frac{1}{2}{\int }_{{\mathbb{R}}^N}\frac{1}{\sqrt{-k}}\left(|\nabla z{|}^2+|\nabla w{|}^2\right)dx\hfill \\ \hfill & +\frac{1}{4}{\int }_{{\mathbb{R}}^N}\left(z^2+f^2\left(z\right)+w^2+f^2\left(w\right)\right)dx-\frac{1}{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta }dx.\hfill \end{array}$

As [20, 21], we consider $\varphi ,\phi \in C_0^{\infty }(\mathbb{R})$ such that 0 ≤ ϕ(x) ≤ 1, 0 ≤ φ(x) ≤ 1 and

$\varphi \left(x\right)=\left\{\begin{array}{c}1,\text{if}|x|\le 1,\hfill \\ 0,\text{if}|x|\ge 1,\hfill \end{array}\right.\phi \left(x\right)=\left\{\begin{array}{c}1,\text{if}|x|\le 1,\hfill \\ 0,\text{if}|x|\ge 1.\hfill \end{array}\right.$

By Lemma 2.1 (ii) and (v), we obtain

$|f\left(t\varphi \right)|\ge C|t\varphi |\ge Cf\left(t\right)\varphi .$

By Lemma 2.1 (ii),

$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{\lambda }\left(t_1\varphi ,t_2\phi \right)\le & \frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla t_1\varphi {|}^2+t_1^2{\varphi }^2\right)dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla t_2\phi {|}^2+t_2^2{\phi }^2\right)dx\hfill \\ \hfill & -\frac{1}{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(t_1\varphi \right){|}^{\alpha }|f\left(t_2\phi \right){|}^{\beta }dx\hfill \\ \hfill \le & \frac{t_1^2}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla \varphi {|}^2+{\varphi }^2\right)dx+\frac{t_2^2}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla \phi {|}^2+{\phi }^2\right)dx\hfill \\ \hfill & -\frac{C\left(|f\left(t_1\right){|}^{\alpha }+|f\left(t_2\right){|}^{\beta }\right)}{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|\varphi {|}^{\alpha }|\phi {|}^{\beta }dx.\hfill \end{array}$

It follows that Φ_λ(t₁ϕ, t₂φ) → −∞ as (t₁, t₂) → (+∞, + ∞). Thus, there exists (t₃, t₄) > 0 such that Φ_λ(t₃ϕ, t₄φ) < 0. Thus, taking (z, w) = (t₃ϕ, t₄φ), we obtain Φ_λ(z, w) < 0 for all λ ∈ L.

(ii) As [20, 22], there exists C > 0 and ρ₁ > 0 small enough such that

${\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla z{|}^2+f^2\left(z\right)+\frac{1}{\sqrt{-k}}|\nabla w{|}^2+f^2\left(w\right)\right)dx\ge C‖\left(z,w\right)‖,$

for $‖(z,w)‖\le {\rho }_1$. From Lemma 2.1 (iii) and Hölder inequality, we obtain

$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{\lambda }\left(z,w\right)& \ge \frac{1}{2}{\int }_{{\mathbb{R}}^N}\frac{1}{\sqrt{-k}}|\nabla z{|}^2+f^2\left(z\right)dx+\frac{1}{2}{\int }_{{\mathbb{R}}^N}\frac{1}{\sqrt{-k}}|\nabla w{|}^2+f^2\left(w\right)dx\hfill \\ \hfill & -\frac{1}{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta }dx\hfill \\ \hfill & \ge C‖\left(z,w\right)‖-\frac{1}{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z\right){|}^{\alpha }|f\left(w\right){|}^{\beta }dx\hfill \\ \hfill & \ge C‖\left(z,w\right)‖-C{‖z^{{\alpha }_1}‖}_p{‖w^{{\beta }_1}‖}_{p^{\prime }}\text{for\hspace{0.17em}all}‖\left(z,w\right)‖\le {\rho }_1,\hfill \end{array}$

where α₁ = α or $\frac{\alpha }{2}$, β₁ = β or $\frac{\beta }{2}$, and $(\frac{1}{p}+\frac{1}{p^{\prime }})=1$. It can conclude that Φ_λ has a strict local minimum at 0, and hence, c_λ > 0.

By Theorem 3.1, it is easy to know that for every $\lambda \in \left[\frac{1}{2},1\right]$, there exists a bounded sequence (z_n, w_n) ⊂ H¹ such that Φ_λ(z_n, w_n) → c_λ and ${\mathrm{\Phi }}_{\lambda }^{\prime }(z_n,w_n)\to 0$.

Lemma 3.5. If (z_n, w_n) ⊂ H¹ is the sequence obtained above, then for almost every $\lambda \in L=[\frac{1}{2},1]$, there exists (z_λ, w_λ) ∈ H¹ \{(0, 0)} such that Φ_λ(z_λ, w_λ) → c_λ and ${\mathrm{\Phi }}_{\lambda }^{\prime }(z_{\lambda },w_{\lambda })\to 0$.

Proof. Since (z_n, w_n) is bounded in H¹, up to a subsequence, there exists (z_λ, w_λ) ∈ H¹ such that

$\left(z_n,w_n\right)\rightharpoonup \left(z_{\lambda },w_{\lambda }\right)\mathit{\text{in}}{H}^1,$

$\left(z_n,w_n\right)\to \left(z_{\lambda },w_{\lambda }\right)\text{\hspace{0.17em}in\hspace{0.17em}}{L}^s\text{\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}}2<s<2^{*},$

$\left(z_n\left(x\right),w_n\left(x\right)\right)\to \left(z_{\lambda }\left(x\right),w_{\lambda }\left(x\right)\right)\text{\hspace{0.17em}a.\hspace{0.17em}e.\hspace{0.17em}in\hspace{0.17em}}{\mathbb{R}}^N.$

Since ${\mathrm{\Phi }}_{\lambda }^{\prime }(z_n,w_n)\to 0$, by the Lebesgue dominated convergence theorem, it is easy to get ${\mathrm{\Phi }}_{\lambda }^{\prime }(z_n,w_n)\to {\mathrm{\Phi }}_{\lambda }^{\prime }(z_{\lambda },w_{\lambda })$, that is, ${\mathrm{\Phi }}_{\lambda }^{\prime }(z_{\lambda },w_{\lambda })=0$, as shown in [23]. Similar to [22, 24, 25], there exists C > 0 such that

$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla \left(z_n-z_{\lambda }\right){|}^2+\left(f\left(z_n\right)f^{\prime }\left(z_n\right)-f\left(z_{\lambda }\right)f^{\prime }\left(z_{\lambda }\right)\right)\left(z_n-z_{\lambda }\right)\right)dx\hfill \\ \hfill \ge & C{‖z_n-z_{\lambda }‖}^2,\hfill \end{array}(3.12)$

$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla \left(w_n-w_{\lambda }\right){|}^2+\left(f\left(w_n\right)f^{\prime }\left(w_n\right)-f\left(w_{\lambda }\right)f^{\prime }\left(w_{\lambda }\right)\right)\left(w_n-w_{\lambda }\right)\right)dx\hfill \\ \hfill \ge & C{‖w_n-w_{\lambda }‖}^2.\hfill \end{array}(3.13)$

By Hölder inequality and Lemma 2.1(ii) and (iv), we deduce that

$\begin{array}{cc}\hfill & \frac{2\alpha }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z_n\right){|}^{\alpha -2}f\left(z_n\right)f^{\prime }\left(z_n\right)|f\left(w_n\right){|}^{\beta }\left(z_n-z_{\lambda }\right)dx\hfill \\ \hfill & +\frac{2\beta }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z_n\right){|}^{\alpha }|f\left(w_n\right){|}^{\beta -2}f\left(w_n\right)f^{\prime }\left(w_n\right)\left(w_n-w_{\lambda }\right)dx\hfill \\ \hfill \le & \frac{2\alpha }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|z_n{|}^{\alpha -1}|w_n{|}^{\beta }\left(z_n-z_{\lambda }\right)dx+\frac{2\beta }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|z_n{|}^{\alpha }|w_n{|}^{\beta -1}\left(w_n-w_{\lambda }\right)dx\le \hfill & \hfill \frac{2\alpha }{\alpha +\beta }{\left({\int }_{{\mathbb{R}}^N}|z_n{|}^{\frac{1}{\beta }}|w_n{|}^{\frac{1}{\alpha -1}}dx\right)}^{\left(\alpha -1\right)\beta }{‖z_n-z_{\lambda }‖}_{p_1}\\ \hfill & +\frac{2\alpha }{\alpha +\beta }{\left({\int }_{{\mathbb{R}}^N}|z_n{|}^{\frac{1}{\beta -1}}|w_n{|}^{\frac{1}{\alpha }}dx\right)}^{\left(\beta -1\right)\alpha }{‖w_n-w_{\lambda }‖}_{p_2}\to 0,\hfill \end{array}(3.14)$

where $p_1=\frac{1}{(\alpha -1)\beta }$ and $p_2=\frac{1}{(\beta -1)\alpha }$. Similarly, we obtain

$\begin{array}{cc}\hfill & \frac{2\alpha }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z_{\lambda }\right){|}^{\alpha -2}f\left(z_{\lambda }\right)f^{\prime }\left(z_{\lambda }\right)|f\left(w_{\lambda }\right){|}^{\beta }\left(z_n-z_{\lambda }\right)dx\hfill \\ \hfill & +\frac{2\beta }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}|f\left(z_{\lambda }\right){|}^{\alpha }|f\left(w_{\lambda }\right){|}^{\beta -2}f\left(w_{\lambda }\right)f^{\prime }\left(w_{\lambda }\right)\left(w_n-w_{\lambda }\right)dx\to 0.\hfill \end{array}(3.15)$

Following (3.12), 3.13, 3.14, and .3.15, we obtain

$\begin{array}{cc}\hfill 0\leftarrow & ⟨{\mathrm{\Phi }}_{\lambda }^{\prime }\left(z_n,w_n\right)-{\mathrm{\Phi }}_{\lambda }^{\prime }\left(z_{\lambda },w_{\lambda }\right),\left(z_n-z_{\lambda },w_n-w_{\lambda }\right)⟩\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla \left(z_n-z_{\lambda }\right){|}^2+f\left(z_n-z_{\lambda }\right)f^{\prime }\left(z_n-z_{\lambda }\right)\left(z_n-z_{\lambda }\right)\right)dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}\left(\frac{1}{\sqrt{-k}}|\nabla \left(w_n-w_{\lambda }\right){|}^2+f\left(w_n-w_{\lambda }\right)f^{\prime }\left(w_n-w_{\lambda }\right)\left(w_n-w_{\lambda }\right)\right)dx\hfill \\ \hfill & -\frac{2\alpha }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}\left[|f\left(z_n\right){|}^{\alpha -2}f\left(z_n\right)f^{\prime }\left(z_n\right)|f\left(w_n\right){|}^{\beta }\right.\hfill \\ \hfill & \left.-|f\left(z_{\lambda }\right){|}^{\alpha -2}f\left(z_{\lambda }\right)f^{\prime }\left(z_{\lambda }\right)|f\left(w_{\lambda }\right){|}^{\beta }\right]\left(z_n-z_{\lambda }\right)dx\hfill \\ \hfill & -\frac{2\beta }{\alpha +\beta }{\int }_{{\mathbb{R}}^N}\left[|f\left(z_n\right){|}^{\alpha }|f\left(w_n\right){|}^{\beta -2}f\left(w_n\right)f^{\prime }\left(w_n\right)\right.\hfill \\ \hfill & \left.-|f\left(z_{\lambda }\right){|}^{\alpha }|f\left(w_{\lambda }\right){|}^{\beta -2}f\left(w_{\lambda }\right)f^{\prime }\left(w_{\lambda }\right)\right]\left(w_n-w_{\lambda }\right)dx\hfill \\ \hfill \ge & C{‖z_n-z_{\lambda }‖}^2+C{‖w_n-w_{\lambda }‖}^2+o_n\left(1\right),\hfill \end{array}(3.16)$

which implies that (z_n, w_n) → (z_λ, w_λ) in H¹. Thus, (z_λ, w_λ) is a non-trivial critical point of Φ_λ(z, w) with Φ_λ(z_λ, w_λ) = c_λ.

Next, we prove Theorem 1.1.

Proof. At first, using Theorem 3.1, for arbitrary $\lambda \in L=\left[\frac{1}{2},1\right]$, there is a (z_λ, w_λ) ∈ H¹ such that

$\left(z_n,w_n\right)\rightharpoonup \left(z_{\lambda },w_{\lambda }\right)\ne \left(\mathrm{0,0}\right)\text{\hspace{0.17em}in\hspace{0.17em}}{H}^1,$

${\mathrm{\Phi }}_{\lambda }\left(z_n,w_n\right)\to c_{\lambda }\text{\hspace{0.17em}and\hspace{0.17em}}{\mathrm{\Phi }}_{\lambda }^{\prime }\left(z_n,w_n\right)\to 0.$

By Lemma 3.5, we obtain

${\mathrm{\Phi }}_{\lambda }\left(z_{\lambda },w_{\lambda }\right)\to c_{\lambda }\text{\hspace{0.17em}and\hspace{0.17em}}{\mathrm{\Phi }}_{\lambda }^{\prime }\left(z_{\lambda },w_{\lambda }\right)=0.$

Thus, there exists ${\lambda }_n\subset \left[\frac{1}{2},1\right]$ such that

${\lambda }_n\to 1,\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)\in H^1,$

${\mathrm{\Phi }}_{{\lambda }_n}^{\prime }\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)=0\text{\hspace{0.17em}and\hspace{0.17em}}{\mathrm{\Phi }}_{{\lambda }_n}\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)=c_{{\lambda }_n}.$

Next, we prove that $\{(z_{{\lambda }_n},w_{{\lambda }_n})\}$ is bounded in H¹. From Lemma 3.4

${\mathrm{\Phi }}_{{\lambda }_n}\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)=c_{\frac{1}{2}},{\mathrm{\Phi }}_{{\lambda }_n}^{\prime }\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)=0,$

it follows that

$\begin{array}{cc}\hfill c_{\frac{1}{2}}& \ge {\mathrm{\Phi }}_{{\lambda }_n}\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)\hfill \\ ={\mathrm{\Phi }}_{{\lambda }_n}\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)-\frac{1}{N}{P}_{{\lambda }_n}\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)\hfill \\ \hfill & =\frac{N-2}{2N}{\int }_{{\mathbb{R}}^N}\left(\frac{2}{N-2}\frac{1}{\sqrt{-k}}\left(|\nabla z_{{\lambda }_n}{|}^2+|\nabla w_{{\lambda }_n}{|}^2\right)+f^2\left(z_{{\lambda }_n}\right)+f^2\left(w_{{\lambda }_n}\right)\right)dx.\hfill \end{array}(3.17)$

By Lemma 2.1 (v) and Sobolev inequality, it follows that

${\int }_{|z_{{\lambda }_n}|\le 1}{z}_{{\lambda }_n}^{2}dx\le C{\int }_{{\mathbb{R}}^N}{f}^2\left(z_{{\lambda }_n}\right)dx,{\int }_{|w_{{\lambda }_n}|\le 1}{w}_{{\lambda }_n}^{2}dx\le C{\int }_{{\mathbb{R}}^N}{f}^2\left(w_{{\lambda }_w}\right)dx$

and

${\int }_{|z_{{\lambda }_n}|>1}{z}_{{\lambda }_n}^{2}dx\le {\int }_{|z_{{\lambda }_n}|>1}{z}_{{\lambda }_n}^{2^{*}}dx\le C{\left({\int }_{{\mathbb{R}}^N}|\nabla z_{{\lambda }_n}{|}^{2}dx\right)}^{\frac{2^{*}}{2}},$

${\int }_{|w_{{\lambda }_n}|>1}{w}_{{\lambda }_n}^{2}dx\le {\int }_{|w_{{\lambda }_n}|>1}{w}_{{\lambda }_n}^{2^{*}}dx\le C{\left({\int }_{{\mathbb{R}}^N}|\nabla w_{{\lambda }_n}{|}^{2}dx\right)}^{\frac{2^{*}}{2}}.$

Therefore,

$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}\left(z_{{\lambda }_n}^2+w_{{\lambda }_n}^2\right)dx\hfill \\ \hfill =& {\int }_{|z_{{\lambda }_n}|\le 1}{z}_{{\lambda }_n}^{2}dx+{\int }_{|z_{{\lambda }_n}|>1}{z}_{{\lambda }_n}^{2}dx+{\int }_{|w_{{\lambda }_n}|\le 1}{w}_{{\lambda }_n}^{2}dx+{\int }_{|w_{{\lambda }_n}|>1}{w}_{{\lambda }_n}^{2}dx\hfill \\ \hfill \le & C{\int }_{{\mathbb{R}}^N}{f}^2\left(z_{{\lambda }_n}\right)dx+C{\int }_{{\mathbb{R}}^N}{f}^2\left(w_{{\lambda }_w}\right)dx\hfill \\ \hfill & +C{\left({\int }_{{\mathbb{R}}^N}|\nabla z_{{\lambda }_n}{|}^{2}dx\right)}^{\frac{2^{*}}{2}}+C{\left({\int }_{{\mathbb{R}}^N}|\nabla w_{{\lambda }_n}{|}^{2}dx\right)}^{\frac{2^{*}}{2}}.\hfill \end{array}(3.18)$

Combining (3.17) and (3.18), we infer that there exists C > 0 such that

${\int }_{{\mathbb{R}}^N}\left(z_{{\lambda }_n}^2+w_{{\lambda }_n}^2\right)dx\le C.$

Thus, there exists C > 0 independent of n such that

${‖\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)‖}^2={\int }_{{\mathbb{R}}^N}\left(|\nabla z_{{\lambda }_n}{|}^2+z_{{\lambda }_n}^2\right)dx+{\int }_{{\mathbb{R}}^N}\left(|\nabla w_{{\lambda }_n}{|}^2+w_{{\lambda }_n}^2\right)dx\le C.$

Next, we can assume that the limit of ${\mathrm{\Phi }}_{{\lambda }_n}(z_{{\lambda }_n},w_{{\lambda }_n})$ exists. By Theorem 3.1, we know that λ → c_λ is continuous from the left. Thus, we obtain

$0\le \underset{n\to \infty }{\lim }{\mathrm{\Phi }}_{{\lambda }_n}\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)\le c_{\frac{1}{2}}.$

Then, by using the fact that

$\mathrm{\Phi }\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)={\mathrm{\Phi }}_{{\lambda }_n}\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)+\frac{\left({\lambda }_n-1\right)}{\alpha \beta }{\int }_{{\mathbb{R}}^N}\frac{2}{\alpha +\beta }|f\left(z_{{\lambda }_n}\right){|}^{\alpha }|f\left(w_{{\lambda }_n}\right){|}^{\beta }dx$

and

$\begin{array}{cc}\hfill \langle {\mathrm{\Phi }}^{\prime }\left(z_{{\lambda }_n},w_{{\lambda }_n}\right),\left(\varphi ,\psi \right)\rangle & =\langle {\mathrm{\Phi }}_{{\lambda }_n}^{\prime }\left(z_{{\lambda }_n},w_{{\lambda }_n}\right),\left(\varphi ,\psi \right)\rangle \hfill \\ \hfill & +\frac{\left({\lambda }_n-1\right)}{\beta }{\int }_{{\mathbb{R}}^N}\frac{2}{\alpha +\beta }|f\left(z_{{\lambda }_n}\right){|}^{\alpha -1}{f}^{\prime }\left(z_{{\lambda }_n}\right)\varphi |f\left(w_{{\lambda }_n}\right){|}^{\beta }dx\hfill \\ \hfill & +\frac{\left({\lambda }_n-1\right)}{\alpha }{\int }_{{\mathbb{R}}^N}\frac{2}{\alpha +\beta }|f\left(z_{{\lambda }_n}\right){|}^{\alpha }|f\left(w_{{\lambda }_n}\right){|}^{\beta -1}{f}^{\prime }\left(w_{{\lambda }_n}\right)\psi dx,\hfill \end{array}$

for any $\varphi ,\psi \in C_0^{\infty }({\mathbb{R}}^N)$ and $‖(z_{{\lambda }_n},w_{{\lambda }_n})‖\le C$, it follows that

$\underset{n\to \infty }{\lim }\mathrm{\Phi }\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)=c_1,\underset{n\to \infty }{\lim }{\mathrm{\Phi }}^{\prime }\left(z_{{\lambda }_n},w_{{\lambda }_n}\right)=0.$

Up to a subsequence, there exists a subsequence $(z_{{\lambda }_n},w_{{\lambda }_n})$ denoted by (z_n, w_n) and (z₀, w₀) ∈ H¹ such that (z_n, w_n) ⇀ (z₀, w₀) in H¹. Using the same method as Lemma 3.5, we will obtain the existence of a non-trivial solution (z₀, w₀) for Φ and Φ′(z₀, w₀) = 0 and Φ(z₀, w₀) = c₁.

To find ground-state solutions, we need to define that

$m≔\inf \left\{\mathrm{\Phi }\left(z,w\right):\left(z,w\right)\ne \left(\mathrm{0,0}\right),{\mathrm{\Phi }}^{\prime }\left(z,w\right)=0\right\}.$

By Lemma 3.3, it follows that

$P\left(z,w\right)=P_1\left(z,w\right)=0.$

According to (3.17), we have m ≥ 0. Let (z_n, w_n) be a sequence such that

${\mathrm{\Phi }}^{\prime }\left(z_n,w_n\right)=0\text{\hspace{0.17em}and\hspace{0.17em}}\mathrm{\Phi }\left(z_n,w_n\right)\to m.$

Similar to Lemma 3.5, we can prove that there exists (z′, w′) ∈ H¹ such that

${\mathrm{\Phi }}^{\prime }\left(z^{\prime },w^{\prime }\right)=0\text{\hspace{0.17em}and\hspace{0.17em}}\mathrm{\Phi }\left(z^{\prime },w^{\prime }\right)=m,$

which implies that $(u^{\prime },v^{\prime })=\left(f(z^{\prime }),f(w^{\prime })\right)$ is a ground-state solution of (1.1). The proof is complete.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding authors.

Author contributions

XZ: writing–original draft and writing–review and editing. JZ: writing–original draft and writing–review and editing.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. JZ was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (nos 2022MS01001), the Key Laboratory of Infinite-dimensional Hamiltonian System and Its Algorithm Application (Inner Mongolia Normal University), the Ministry of Education (No. 2023KFZD01), the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (No. NJYT23100), the Fundamental Research Funds for the Inner Mongolia Normal University (No. 2022JBQN072), and the Mathematics First-class Disciplines Cultivation Fund of Inner Mongolia Normal University (No. 2024YLKY14). XZ was supported by the Fundamental Research Funds for the Inner Mongolia Normal University (2022JBXC03) and the Graduate Students Research Innovation Fund of Inner Mongolia Normal University (CXJJS22100).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

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