On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical $L^{p}$-space

1.   Introduction

In this paper, we mainly study the following Kirchhoff-type system

under the condition of $(C.1), (C.2)$ where $(C.1)$: $a_{1}, a_{2}\geq0, \; b_{1}, b_{2} > 0, \; \alpha, \beta > 1, \; \alpha+\beta = 2^* = 2 \times3 /(3-2) = 6, \; \varepsilon > 0$; $(C.2)$: $f(x), g(x)\geq0, \; f(x), g(x)\not\equiv 0, \; f(x), g(x)\in L^{\frac{6}{5}}(\mathbb{R}^{3})$. $2^* = \frac{2 N}{N-2}$ is the critical Sobolev exponent where $N = 3$.

It is well known that the critical problem

does not have a minimizing energy solution when $\Omega$ is a domain different from $\mathbb{R}^N$. This result is derived from the sharp Sobolev inequality on $\mathbb{R}^N$ ^[1,2,3]. However, the situation is different if there is a nonhomogeneous term $f(x)$:

Tarantello ^[4] showed that problem (1.3) has at least two solutions in bounded domains. The main idea is to use the Ekeland variational principle to get one solution $u_0$ which is a local minimum solution and use the mountain pass theorem to obtain the second solution $u_1$ with the energy

where $S$ is the best constant for the Sobolev embedding $D^{1, 2}(\mathbb{R}^{N})\hookrightarrow L^{\frac{2N}{N-2}}(\mathbb{R}^{N}),$ namely

Let $U$ be a solution for the following problem:

Then, we know

which are all positive solutions of (1.4) (see ^[5,6]) for any $\varepsilon > 0$ and $y\in \mathbb{R}^{N}$. Moreover, we know that U satisfies

where $\|U\| = (\int_{\mathbb{R}^N}|\nabla u|^2 d x)^{\frac{1}{2}}$ and $|U|_s = (\int_{\mathbb{R}^N}|u|^s d x)^{\frac{1}{s}}$ are the norms of the Sobolev space $D^{1, 2}\left(\mathbb{R}^N\right)$ and Lebesgue space $L^s\left(\mathbb{R}^N\right), s \in\left[2, 2^*\right]$, respectively. Han ^[7] extended (1.3) to the following elliptic system

in bounded domains. The author used upper and lower solution methods and variational methods to prove that the problem (1.6) has at least two solutions for both subcritical and critical cases. The first solution $(\tilde{u}_0, \tilde{v}_0)$ of (1.6) is also a local minimizer of the associated functional $I$ and they obtained the second solution $(\tilde{u}_1, \bar{v}_1)$ with

where

Indeed, according to ^[7] we know that

is a solution of the following system

where $U$ is a solution of (1.4),

and

We are certain that the existence of solutions will be affected by the nonhomogeneous term $f(x)$ or $g(x)$ and the existence of a second solution for (1.3) or (1.6) will also be affected by Eq (1.2) or system (1.7).

When $a_{1} = a_{2},$ $b_{1} = b_{2},$ $f = g$ and $u = v$, system (1.1) transforms into the following individual equation.

which is related to the stationary analogue of the equation

presented by Kirchhoff in ^[8]. Here, the parameters in (1.8) carry the following interpretations: $\rho$ represents the mass density, $P_0$ the initial tension, $h$ the cross-sectional area, $E$ the Young's modulus of the material and $L$ the length of the string. It was underscored in ^[9] that the Kirchhoff-type problem models a variety of physical and biological systems where $u$ characterizes a process that relies on its own average (e.g., population density). Early investigations into the Kirchhoff-type problem can be traced back to the work of Bernstein ^[10] and Pohoẑaev ^[11]. Nevertheless, Eq (1.8) only attracted significant attention after Lions ^[12] introduced an abstract framework for such problems.

Liu et al. ^[5] discovered that problem (1.8) has at least two positive solutions when $N = 3, 4$ under certain assumptions for $f(x)$. Specifically, to consider the existence of a second solution for problem (1.8), they needed to establish the existence of a unique positive solution for (1.8) when $\varepsilon = 0$. In fact, they found that the positive solutions of

can be expressed as

with

The solvability or multiplicity of the Kirchhoff type equation with critical exponent has been extensively studied in recent years; see, for instance, ^{[5,13,14,15,16,17,18,19,20,21,22,23,24,25]} and references therein.

Inspired by the ideas presented in ^[5] and ^[4], we discuss system (1.1) with $\varepsilon > 0$ small enough. First, we establish the existence of solutions for problem (1.1) when $\varepsilon = 0$:

Theorem 1.1. Assume that $\varepsilon = 0$ and $(u_0, v_0)$ is a positive solution of (1.7) and

Then, we have

(i) If $a_1 = a_2 = 0$, problem (1.1) has a unique positive solution $z = (u', v')$ where

(ii) If $a_{1}$ and $a_{2}$ are not equal to $0$ at the same time, problem (1.1) has a unique positive solution

Indeed, if $a_1 = 0, a_2\not = 0$,

(iii) If $a_{1}\not = 0(>0), a_{2}\not = 0(>0)$, problem (1.1) has at least a positive solution $z = (u', v') = (\lambda_{1}u_{0}, \lambda_{2}v_{0})$ where

and

Remark 1.1. In fact, from the following equations

and

we find that the unique or explicit form of $(\lambda_1, \lambda_2)$ is affected by $a_1, a_2, b_1, b_2$. Thus, we can only obtain the uniqueness of $(\lambda_1, \lambda_2)$ in cases $(i)-(ii)$ and the explicit form of $(\lambda_1, \lambda_2)$ in case $(i)$. Furthermore, we believe that if $a_1, a_2, b_1, b_2$ satisfy certain assumptions, $(\lambda_1, \lambda_2)$ will be unique, and we can also obtain the explicit form of $(\lambda_1, \lambda_2)$.

We define $\varepsilon^{\ast} = \frac{g(\sqrt{\frac{9bS_{\alpha, \beta}^{3}}{20}})\sqrt{S}}{|f|{\frac{6}{5}}+|g|{\frac{6}{5}}}$ where $g(\sqrt{\frac{9bS_{\alpha, \beta}^{3}}{20}}) = \frac{b}{4}(\sqrt{\frac{9bS_{\alpha, \beta}^{3}}{20}})^{3}-\frac{1}{3S^{3}}(\sqrt{\frac{9bS_{\alpha, \beta}^{3}}{20}})^{5},$ $b = \frac{1}{2}\min(b_{1}, b_{2})$. Next, we consider the existence of a local minimum solution for problem (1.1) by applying the Ekeland variational principle.

Theorem 1.2. Assuming conditions $(C.1)$ and $(C.2)$ hold, system (1.1) has a local minimum solution for any $\varepsilon\in(0, \varepsilon^{\ast})$.

Remark 1.2. First, we demonstrate that the minimal value of the set minimization problem can be attained by $(u, v)$ and then we prove that $(u, v)$ is a solution of (1.1). Unlike with single equations, due to the mutual interaction of $(u, v)$, it is challenging to obtain $\|u\| \geq\left\|u_n\right\|+o(1), \|v\| \geq\left\|v_n\right\|+o(1)$ where $(u_{n}, v_{n})$ represents the minimizing sequence of $c_{0}$. The definitions of $u_n, v_n, u, v, c_{0}$ can be found in the proof of Theorem 1.2 in Section 4. By drawing upon the proof method from Theorem 1.3 in ^[5] and employing meticulous estimates, we can overcome this difficulty.

Finally, we investigate the existence of a second solution for problem (1.1) by applying the mountain pass lemma and the concentration compactness principle. To obtain the energy estimation of the associated functional $\Phi_{\varepsilon}$ for problem (1.1), we will need the explicit form of $(\lambda_1, \lambda_2)$. Therefore, when $a_1 = a_2 = 0$ we have:

Theorem 1.3. Assume $\alpha = \beta = 3$ and $b_{1} = b_{2}$, there exists $\varepsilon^{\ast\ast}\in(0, \varepsilon^{\ast}]$ such that for any $\varepsilon\in(0, \varepsilon^{\ast\ast})$, problem (1.1) has another solution. The value of $\varepsilon^*$ is defined in Theorem 1.2.

Remark 1.3. First, we prove that the associated functional $\Phi_{\varepsilon}$ for problem (1.1) satisfies the mountain pass structure, from which we obtain a $(PS)$ sequence. Then, we establish the $(PS)$ condition by using the concentration compactness principle. From this, we obtain another solution for (1.1). Owing to the lack of compactness ($\alpha + \beta = 2^*$), the mutual action of $(u, v)$ and the influence of the nonlocal term, there arises a new challenge in employing the concentration compactness principle. Moreover, it is difficult to derive an explicit expression when the values of $a_1$ and $a_2$ are non-zero. The reason for only considering the case where $\alpha = \beta = 3$ is that after extensive estimation, it is only when $\alpha = \beta = 3$ that $\Lambda$ reaches its minimum value $\Lambda_{min}$. Therefore, the estimate satisfied by $m$ contradicts Lemma 4.4(i). The definitions of $\Lambda$, $\Lambda_{min}$, and $m$ as well as the details of their related proofs can be found in Remark 4.2 and the Proof of Theorem 1.3.

Remark 1.4. The innovation of this paper lies in overcoming the lack of compactness ($\alpha + \beta = 2^*$) and the mutual interaction of $(u, v)$ to demonstrate the existence of at least two solutions for systems (1.1). This extends the results from single equations in ^[5] to a system of equations. We accomplished this by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle as well as through some precise estimates.

The structure of this paper is as follows: Section 2 provides some preliminary background knowledge. Section 3 is dedicated to the proof of Theorem 1.1. Finally, we present the proofs for Theorem 1.2 and Theorem 1.3.

2.   Notations and preliminary results

First, we introduce the following notations, which will be useful for proving the upcoming theorems in this section.

$\bullet$ The function space corresponding to problem (1.1) is $E = D^{1, 2}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3})$ with the space norm defined as $\|(u, v)\| = (\|u\|^{2}+\|v\|^{2})^{\frac{1}{2}}$. $E^*$ is the dual space of $E$.

$\bullet (u, v)\in E, \; B_{\rho} = \{u\in E:\|(u, v)\| < \rho\}$.

$\bullet \overline{B_{\rho}} = \{(u, v)\in E :\|(u, v)\|\leq\rho\}, \; \partial B_{\rho} = \{(u, v)\in E:\|(u, v)\| = \rho\}.$

$\bullet$ The following elliptic system

has a positive radial vector solution $z_{0} = (u_{0}, v_{0})$ under the condition (C.1) (see^[7]).

$\bullet\; \; u^{+} = \max\{0, u\}, \; u^{-} = \max\{0, -u\}.$

Let us denote the energy functional $\Phi_{\varepsilon}:E\rightarrow \mathbb{R}$ corresponding to (1.1) by

Obviously, $\Phi_{\varepsilon}$ is of $C^{1}$ and has the derivative given by

Next, we present the following lemma which can be utilized in the proof of Theorem 1.3.

Lemma 2.1. Assume $\alpha, \beta > 1, \alpha+\beta = 6$ and define

Then,

where $a, \ b$ is any real number.

Proof. Refer to ^[26] for the proof of $S_{\alpha, \beta} = [(\frac{\alpha}{\beta})^{\frac{\beta}{6}}+(\frac{\alpha}{\beta})^{-\frac{\alpha}{6}}]S$; we will provide the proof of $\tilde{S}_{\alpha, \beta}$ later. Assume that $\omega_{n}$ is a minimizing sequence for S, let $s, t > 0$ to be chosen. Taking $u_{n} = \frac{s}{\sqrt{a}}\omega_{n}, \, \, \, v_{n} = \frac{t}{\sqrt{b}}\omega_{n}$ in (2.2), we have that

Noting that

we can proceed to define the function as follows.

When $x = \sqrt{\frac{\alpha}{\beta}},$ there exists the minimum value

Considering (2.3)–(2.5), we get

Then, we need to prove that

Let $(u_{n}, v_{n})$ be a minimizing sequence for $\tilde{S}_{\alpha, \beta}$. Define $z_{n} = s_{n}v_{n}$, for some $s_{n} > 0$ such that

By Young's inequality

By (2.8), we have

Using (2.10), we have

where $h(s_{n}) = as_{n}^{\frac{2\beta}{\alpha+\beta}}+bs_{n}^{\frac{-2\alpha}{\alpha+\beta}}$. Then, we get

Therefore, (2.7) is proved. Combining (2.6), we get

□ Ekeland's variational principle is a tool used to obtain a local minimum solution. We include it here for the convenience of the readers.

Theorem 2.1. (^[27], Theorem 4.1) Let $M$ be a complete metric space with metric $d$ and let $I$ : $M \mapsto(-\infty, +\infty$] be a lower semicontinuous function, bounded from below and not identical to $+\infty$. Let $\epsilon > 0$ be given and $u \in M$ be such that

Then, there exists $v \in M$ such that

For each $w \in M$, one has

3.   Proof of Theorem 1.1

In this section, we provide the proof of Theorem 1.1. The key idea is to observe that the right-hand side of problem (1.1), namely $a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx$ and $a_{2}+b_{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx$, can be regarded as two constants. This insight guides us to construct the solution for problem (1.1) by utilizing the solution for problem (2.1) and the method of undetermined coefficients.

Proof of Theorem 1.1.The proof of Theorem 1.1 is inspired by the idea presented in ^[5]. For any $C_{1}, C_{2} > 0$, let $(u', v') = (\lambda_{1}u_{0}, \lambda_{2}v_{0})$ where $(u', v')$ is a vector solution of (3.1), $\lambda_{1} > 0$, $\lambda_{2} > 0$.

By (3.1) and the fact that $(u_{0}, v_{0})$ satisfies (2.1), we can obtain

which implies

Now, we consider the equations

Let $\int_{\mathbb{R}^{3}}|\nabla u_{0}|^{2}dx = S_{1}, \; \int_{\mathbb{R}^{3}}|\nabla v_{0}|^{2}dx = S_{2}$. Thus, $C_{1}, C_{2}$ satisfy

Next, we consider the existence and uniqueness of the positive solution from (1.1) when $\varepsilon = 0$.

(i) If $a_{1} = 0, \; a_{2} = 0$, we deduce from (3.4) that

Combining with (3.3), we have

Hence, we have

(ii) If $a_{1}$ and $a_{2}$ are not equal to 0 at the same time, we can assume $a_{1} = 0$ and $a_{2}\neq0$ which implies that

According to the above, we define

$A = (b_{2}S_{2})(b_{1}S_{1})^{\frac{\alpha}{2+\beta}} > 0, \; \; \; \; -\frac{1}{3} < k = \frac{\beta-2}{2+\beta} < \frac{3}{7}, B = a_{2} > 0$

where $x > B.$ We want to determine the number of $C_{2}$; we only need to find the solution of

$(1')$ When $\beta\in(1, 2)$, $k\in(-\frac{1}{3}, 0)$, we can get

Considering $f(x) > 0$ as $x\rightarrow B$, $f(x)\rightarrow -\infty$ as $x\rightarrow +\infty$. Thus, there exists a unique $C_{2} > B$ such that $f(C_{2}) = 0$.

$(2')$ When $\beta = 2, \; k = 0$, from (3.6)

Considering (3.2), (3.3), we have

$(3')$ When $\beta\in(2, 5), \; k\in(0, \frac{3}{7})$, we know $x_{0} = (Ak)^{-\frac{1}{k-1}}$ is the only maximum from

Considering $f(x) > 0$ as $x\rightarrow B$, $f(x)\rightarrow -\infty$ as $x\rightarrow +\infty$. Thus, there exists a unique $C_{2} > B$ such that $f(C_{2}) = 0$. So, we can prove that there exists a unique $(u', v')$ as a result of the only $\lambda_{1}, \lambda_{2}$. If $a_{1}\neq0$ and $a_{2} = 0$, we can get a unique $(u', v')$ in the same way.

(iii) If $a_{1}\not = 0(>0), a_{2}\not = 0(>0)$, we deduce from (3.4) that

Let $C_{1} = \lambda C_{2}.$ By the above equality, we have

Because $\lambda = \frac{C_{1}}{C_{2}} > 0,$ we deduce from (3.7) that

Then, we have

Let $A^{'} = (\frac{1}{2})^{\frac{\alpha}{4}}(b_{2}S_{2})^{\frac{4-\alpha}{4}}, \; \; \; \; B^{'} = a_{1}b_{2}S_{2}, \; \; \; \; C^{'} = a_{2}, \; \; \; \; D^{'} = 4b_{1}b_{2}S_{1}S_{2}$ and

As a result of

Then, there exists a $C_{2} > 0$ such that $h(C_{2}) = 0$. Because the uniqueness is not clear, we have some difficulties in considering the existence of the second solution of problem (1.1). Hence, the Theorem 1.1 is proved.

4.   Proofs of Theorem 1.2 and Theorem 1.3

In this section, we establish the existence of two solutions for (1.1) by using some variational methods. First, we will present the proofs of Theorem 1.2 and Theorem 1.3 utilizing various Lemmas for each proof, respectively. We consider the existence of a local minimum solution for problem (1.1) by applying the Ekeland variational principle.

Lemma 4.1. Assume that $(C.1), (C.2)$ hold. Then, there exists $\rho > 0$ such that for any $\varepsilon\in(0, \varepsilon^{\ast}),$ one has $\Phi_{\varepsilon}|_{\partial B\rho}\geq \widetilde{\alpha}$ for some $\widetilde{\alpha} > 0$. For the definition of $\varepsilon^{\ast}$, please refer to Page 4, line 26.

Proof. By the Hölder inequalities, $b_{1} > 0, \ b_{2} > 0$, the complete square formula and Sobolev inequalities, one has

where $b = \frac{1}{2}\min(b_{1}, b_{2})$. Let $g(t) = \frac{b}{4}t^{3}-\frac{1}{3S^{3}}t^{5},$ then $\max\limits_{t\geq0}g(t) = g(\rho) > 0$ with $\rho = \sqrt{\frac{9bS_{\alpha, \beta}^{3}}{20}}.$ Owing to $\varepsilon^{\ast} = \frac{g(\rho)\sqrt{S}}{|f|_{\frac{6}{5}}+|g|_{\frac{6}{5}}}$, for any $\varepsilon\in(0, \varepsilon^{\ast}),$ we have

Hence, we conclude the proof. □

Lemma 4.2. Suppose that $(C.1), (C.2)$ hold. Then, for any $\varepsilon\in(0, \varepsilon^{\ast})$ one has $c_{0} = \inf\limits_{u\in\overline{B_{\rho}}}\Phi_{\varepsilon}(u, v)\in(-\infty, 0)$ where $\rho, \ \varepsilon^{\ast}$ is given by Lemma 4.1.

Proof. Firstly, we choose a $(u, v)\in E$ such that $\int_{\mathbb{R}^{3}}(fu+gv)dx > 0$. Then, for any $t > 0$ we have

Hence, there exists a sufficiently small $t > 0$ such that $\|t(u, v)\| \leq \rho$ and $\Phi_{\varepsilon}(tu, tv) < 0$ which leads to

As $\|(u, v)\| \leq \rho$, we obtain

Consequently, we can establish that $c_{0} \in (-\infty, 0)$, thereby completing the proof. □

Proof of Theorem 1.2. By applying Lemma 4.2, we find that $c_{0} = \inf\limits_{u\in\overline{B_{\rho}}}\Phi_{\varepsilon}(u, v)\in(-\infty, 0)$. Moreover, we know that $\Phi_{\varepsilon}|_{\partial B\rho} > 0$ from Lemma 4.1. Therefore, we deduce that the minimum cannot be attained on $\partial B{\rho}$. According to Lemma 4.1, Lemma 4.2 and Theorem 2.1 (Ekeland variational principle), there exists $(u_{n}, v_{n})\in B_{\rho}$ such that $\Phi_{\varepsilon}(u_{n}, v_{n})\rightarrow c_{0}$ and $\Phi_{\varepsilon}'(u_{n}, v_{n})\rightarrow0$ in $E^{\ast}$. The above proof can be referred to [^[28], pp. 534-535]. Consequently, there exists $(u, v)\in E$ satisfying:

First, we prove that $(u, v)$ is a minimizer for $c_{0}$. Noting that $\overline{B_{\rho}}$ is closed and convex,

Therefore, what we need to prove next is that $\Phi_{\varepsilon}(u, v)\leq c_{0}$. The key idea here is that since $\Phi_{\varepsilon}(u_{n}, v_{n})$ converges to $c_{0}$, we need to establish the inequality relationship between $\Phi_{\varepsilon}(u, v)$ and $\Phi_{\varepsilon}(u_{n}, v_{n})$.

In order to eliminate $\frac{1}{4}(b_{1}\|u\|^{4}+b_{2}\|v\|^{4}),$ we have the following estimates. From $\Phi'_{\varepsilon}(u_{n}, v_{n})\rightarrow 0$ in $E^{\ast}$, it holds that

Combining above equality and weakly lower semi-continuity of norm, we have

Hence, we get $(u, v)$ is a minimizer for $c_{0}$.

Now, we need to prove $(u, v)$ is a solution of (1.1) with $\Phi_{\varepsilon}(u, v) = c_{0}$. On one hand, we have

On the other hand, we have

Hence, we have

So, we obtain

Following the above steps, we have

Thus, we obtain

Using again the weakly lower semi-continuity of norm, we get $\|(u, v)\| = \left\|\left(u_n, v_n\right)\right\|+o(1)$. Combining $u_{n}\rightharpoonup u$, $v_{n}\rightharpoonup v$ in $E$, we have $u_{n}\rightarrow u, v_{n}\rightarrow v$ in $E$ and then $\Phi_{\varepsilon}\left(u_n, v_n\right) \rightarrow$ $c_0 = \Phi_{\varepsilon}(u, v), \Phi_{\varepsilon}^{\prime}\left(u_n, v_n\right) \rightarrow \Phi_{\varepsilon}^{\prime}(u, v) = 0$ in $E^*$. We complete the proof.

Now, we give proof of the second positive solution by the mountain pass lemma and the concentration compactness principle.

Lemma 4.3. Suppose that $(C.1), (C.2)$ hold. Then, there exists $(u^{\ast}, v^{\ast})\in E$ such that $\|(u^{\ast}, v^{\ast})\| > \rho$ and $\Phi_{\varepsilon}(u^{\ast}, v^{\ast}) < 0$ where $\rho$ is given by Lemma 2.

Proof. Let $u_{0} = kU, \; v_{0} = lU,$ we can obtain $\frac{k}{l} = \sqrt{\frac{\alpha}{\beta}}$ and

where U is a solution of (1.4). Then, for any $t > 0$ it holds that

Hence, there exists a sufficiently large $t_{0} > 0$ such that

Let $(u^{^{\ast}}, v^{\ast}) = (t_{0}u_{0}, t_{0}v_{0}).$ This completes the proof. □

According to lemma 4.1, lemma 4.3, we can find $(u, v)$ such that

Then, we define

where $A$ is the set of all passes which connect 0 and $e = (u^{^{\ast}}, v^{\ast}), i, e.,$

Remark 4.1. For any $\varepsilon\in(0, \varepsilon^{\ast})$, we can obtain a nonnegative bounded $(PS)$ sequence.

Proof. By the mountain pass theorem ^[6], there exists $(u_n, v_n)\in E$ such that $I(u_{n}, v_{n})\rightarrow m$ and $I'(u_{n}, v_{n})\rightarrow 0$ in $E^{\ast}$. Thus, we can get

where $Q_1$ is a positive constant with a sufficiently small value.

By the Hölder inequality and the Sobolev inequality, we get

where $C(\varepsilon)$ is a a sufficiently small constant that depends only on $\varepsilon$. Combining $(4.1)$ with $(4.2)$, we conclude that $(u_n, v_n)$ is bounded in $E$.

Since $u = u^{+}-u^{-}, v = v^{+}-v^{-},$ we have

which implies $\|u_{n}^{-}\| = 0, \; \|v_{n}^{-}\| = 0, n\rightarrow \infty.$ According to Hölder and Sobolev inequality, we have

Therefore,

Next, we need to verify that

Given that $\|u_{n}^{-}\| = 0, \; \|v_{n}^{-}\| = 0, n\rightarrow \infty$ and $\varepsilon\int_{\mathbb{R}^{3}}(fu_{n}^{-}+gv_{n}^{-})dx = 0, n\rightarrow \infty$, we have

Given that $\|u_{n}^{-}\| = 0, \; \|v_{n}^{-}\| = 0, n\rightarrow \infty$ and $\varepsilon\int_{\mathbb{R}^{3}}(fu_{n}^{-}+gv_{n}^{-})dx = 0, n\rightarrow \infty$, we have

Then, we can obtain a nonnegative bounded sequence for $\Phi_{\varepsilon}$. We complete the proof.

Lemma 4.4. Suppose that $(C.1), (C.2)$ hold. Then, there exists $\varepsilon^{\ast\ast}\in (0, \varepsilon^{\ast})$ such that for any $\varepsilon\in(0, \varepsilon^{\ast\ast})$ where $\Lambda$ is the Maximum value for p(t), the following statements hold:

$Proof.$ Let

and

Then, there exists $t_{1} > 0$ such that $p'(t_{1}) = 0$. In this case, we have

On the other hand, we know that $(u', v')$ satisfies

Combining (4.3) and (4.4), we obtain $t_{1} = 1$ and

Let $\varepsilon_{1}\in(0, \varepsilon^{{\ast}}]$. Then, for $t_{2}\in(0, t_{1})$ and $\varepsilon\in(0, \varepsilon_{1})$, we have:

(i) when $a_{1} = a_{2} = 0,$

(ii) when $a_{1} = 0, a_{2}\neq0,$

(iii) when $a_{1}\neq0, a_{2}\neq0,$

Choosing $\varepsilon^{\ast\ast}\in (0, \varepsilon_{1}]$ for any $\varepsilon \in (0, \varepsilon^{\ast\ast})$, we can deduce that for all $t\geq t_{2}$,

Furthermore, we obtain the following inequalities:

(i) when $a_{1} = a_{2} = 0,$

(ii) when $a_{1} = 0, a_{2}\neq0,$

(iii) when $a_{1}\neq0, a_{2}\neq0,$

Therefore, we complete the proof. □

Proof of Theorem 1.3. According to Remark 4.1, we can get that $\{(u_{n}, v_{n})\}$ is bounded and nonnegative. Up to a subsequence, there exists $(u, v)\subset E$ such that $u_{n}\rightharpoonup u$, $v_{n}\rightharpoonup v$ in E, $u_{n}\rightarrow u$, $v_{n}\rightarrow v$ in $L_{loc}^{s}(\mathbb{R}^{3})\times L_{loc}^{s}(\mathbb{R}^{3})(1\leq s < 2^{\ast})$ and $u_{n}\rightarrow u, v_{n}\rightarrow v \; a.e\; in\; \mathbb{R}^{3}$. By applying the concentration compactness principle (see Proposition 2.2 in ^[29]), we can find non-negative measures $\mu$ and $\nu$ on $\mathbb{R}^{3}$, a vector function $(u, v)$ and an at most countable set $\Gamma$ such that as $n\rightarrow \infty$,

in the sense of measure and

Here, $\delta_{x_{i}}$ is the Dirac delta measure concentrated $x_{i}$. We claim that $\Gamma = \varnothing$. Suppose by contradiction that $\Gamma \neq \varnothing$. To obtain a contradiction, we estimate $m = \lim_{n\rightarrow \infty}\Phi_{\varepsilon}(u_{n}, v_{n})$ by utilizing the assumption $\Gamma \neq \varnothing$ and the concentration compactness principle. By comparing this estimation of $m$ with the one provided in Lemma 4.1, we deduce a contradiction. To do this, we first present the following relevant estimates. Fix $k \in \Gamma$. For $\rho > 0$, assume that $\varphi_\rho^k \in C_0^{\infty}\left(\mathbb{R}^3\right)$ satisfies $\varphi_\rho^k \in[0, 1]$,

and $|\nabla \varphi_{\rho}^{k}|\leq\frac{\rho}{2}.$ It follows from $\langle(\Phi_{\rho}^{'}(u_{n}, v_{n}), (\varphi_{\rho}^{k}u_{n}, 0)\rangle\rightarrow0$ that

In the same way, it follows from $\langle(\Phi_{\rho}^{'}(u_{n}, v_{n}), (0, \varphi_{\rho}^{k}v_{n})\rangle\rightarrow0$ that

First, we need to solve the lack of compactness problem from the critical Sobolev exponent which causes the invariance of dilation. Combining (4.7), (4.8) and Hölder inequality, we have

and

where $B_{\rho}(a_{k}) = \{x\in R^{3}:|x-a_{k}| < \rho\}.$ By (4.5) and (4.6), we have

We can deduce from (4.7)–(4.13) that

So, we have

For $R > 0,$ assume that $\varphi_{R}\in C_{0}^{\infty}(\mathbb{R}^{3})$ satisfies $\varphi_{R}\in[0, 1],$

and $|\nabla \varphi_{R}| < \frac{2}{R}.$ By applying the concentration compactness principle, we obtain

Hence, we can infer that

(i) If $a_{1} = a_{2} = 0,$ by (4.14), we need to demonstrate that

By Lemma 2.1 and the fact that $(u_{0}, v_{0})$ satisfies (2.1), we can obtain:

and

Combing with (4.16), (4.17) and (3.5), $(u', v')$ satisfies

Consequently, we have

Based on $p'(t) = 0$, we can determine that $t = 1$. Therefore, there exists

On one hand, considering

we have

Hence,

On the other hand, we can derive

Therefore, it is only when $\alpha = \beta = 3$ and $b_{1} = b_{2}$ that

which contradicts Lemma 4.4 (i).

Remark 4.2. The reason for only considering the case where $\alpha = \beta = 3$ is that after extensive estimation, it is only when $\alpha = \beta = 3$ that $\Lambda$ reaches its minimum value $\Lambda_{min}$. Therefore, the estimate satisfied by $m$ contradicts Lemma 4.4 (i). For the cases in Lemma 4.4 (ii) and (iii), obtaining the result is challenging due to the mutual interaction of $(u, v)$ adding complexity to our computations. This will be our main task in the following work.

Moving forward, we will only consider the case where $a_{1} = a_{2} = 0,$ $\alpha = \beta = 3$ and $b_{1} = b_{2}$. We need to solve the lack of compactness problem from the region $\mathbb{R}^3$ which causes the invariance of translation.

For $R > 0$, define

By concentration compactness principle, we obtain

and $S_{\alpha, \beta}\mu_{\infty}^{\frac{1}{3}}\leq\nu_{\infty}$. Next, we estimate $\nu_{\infty}$ and $\mu_{\infty}.$ Assume that $\chi_{R}\in C_{0}^{\infty}(\mathbb{R}^{3})$ satisfy $\chi_{R}\in[0, 1]$, we have

where $|\nabla \chi_{R}| < \frac{3}{R}$. It follows from $\langle(\Phi_{\varepsilon}^{'}(u_{n}, v_{n}), (\chi_{R}u_{n}, 0)\rangle\rightarrow0$ that

In this way, we can also have from $\langle(\Phi_{\varepsilon}^{'}(u_{n}, v_{n}), (0, \chi_{R}v_{n})\rangle\rightarrow0$ that

By Hölder inequality, we have

and

Combining (4.21), we have

and

Otherwise, we get

Combining (4.23)–(4.29), we have

We obtain one of the following two cases holds:

(1) $\nu_{\infty} = 0;\; \; \mu_{\infty} = 0.$

(2)

Suppose that case (2) holds. We deduce that

Considering as the same as (4.15)–(4.20), we get

which is a contradiction. Thus, case (1) holds.

Combining (4.5), (4.22) with $\Gamma = \varnothing$, we have:

Applying Fatou's lemma, we obtain:

Thus, we have

Set $\left\|u_n\right\| \rightarrow d$. Then, by $\lim_{n\rightarrow \infty}\int_{\mathbb{R}^{3}}u_{n}^{\alpha}v_{n}^{\beta}dx = \int_{\mathbb{R}^{3}}u^{\alpha}v^{\beta}dx,$ we have

and

which deduces $d = \|u\|$. Combining $u_n \rightharpoonup u$ in $D^{1, 2}(\mathbb{R}^3)$, we obtain $u_n \rightarrow u$ in $D^{1, 2}(\mathbb{R}^3)$. Following the same approach and steps, we can also establish that $v_{n} \rightarrow v$ in $D^{1, 2}(\mathbb{R}^3)$. According to Remark 4.1 and Lemma 4.4, there exists a non-negative bounded sequence ${(u_{n}, v_{n})} \subset E$ that satisfies

Consequently, by $u_n \rightarrow u$ in $D^{1, 2}(\mathbb{R}^3)$ and $v_{n} \rightarrow v$ in $D^{1, 2}(\mathbb{R}^3)$ we have

This completes the proof of the existence of the second solution.

5.   Conclusions

In this paper, we first consider the existence of a local minimum solution for problem (1.1) by applying the Ekeland variational principle. Next, we investigate the existence of a second solution for problem (1.1) by applying the mountain pass lemma and the concentration compactness principle. To obtain the energy estimation of the associated functional $\Phi_{\varepsilon}$ for problem (1.1), we will need the explicit form of $(\lambda_1, \lambda_2)$. Therefore, when $a_1 = a_2 = 0$ we have: Assume $\alpha = \beta = 3$ and $b_{1} = b_{2}$, there exists $\varepsilon^{\ast\ast}\in(0, \varepsilon^{\ast}]$ such that for any $\varepsilon\in(0, \varepsilon^{\ast\ast})$, problem (1.1) has another solution. The value of $\varepsilon^*$ is defined in Theorem 1.2. The reason for only considering the case where $\alpha = \beta = 3$ is that after extensive estimation, it is only when $\alpha = \beta = 3$ that $\Lambda$ reaches its minimum value $\Lambda_{min}$. Therefore, the estimate satisfied by $m$ contradicts Lemma 4.4 (i). For the cases in Lemma 4.4 (ii) and (iii), obtaining the result is challenging due to the mutual interaction of $(u, v)$ adding complexity to our computations. This will be our main task in the following work.

Use of AI tools declaration

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

Acknowledgments

The authors would like to express their sincere gratitude to Prof. HaiYang He for her invaluable guidance throughout this research. Her expertise and insightful comments have greatly contributed to the quality of this work.

Conflict of interest

The authors declare there is no conflicts of interest.