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Multiple positive solutions for a Schrödinger–Poisson system with critical and supercritical growths J. Leia, H. Suob a School of Mathematics and Statistics, Guizhou University, Guiyang, China b School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, China
Англоязычная версия:
Izvestiya: Mathematics, 2023, Volume 87, Issue 1, Pages DOI: https://doi.org/10.4213/im9244e 
Реферативные базы данных:
    § 1. Introduction and main resultWe consider the solvability of the Schrödinger–Poisson system
$$
\begin{equation}
\begin{cases} -\Delta u+u+\lambda\phi u= Q(x)|u|^{4}u+\mu h(x)|u|^{q-2}u &\text{in }\mathbb{R}^3, \\ -\Delta \phi=u^{2} &\text{in }\mathbb{R}^3, \end{cases}
\end{equation}
\tag{1.1}
$$
where $h(x)=|x|^\beta/(1+|x|^\beta)$, $0< \beta<3$, $6<q<6+2\beta$, $\lambda,\mu> 0$ are real parameters. We assume that the coefficient function $Q(x)\in C(\mathbb{R}^3)$, $Q(x)> 0$ a. e. $x\in \mathbb{R}^3$ and satisfies the following assumption: $\mathrm{(Q_1)}$ There exist points $a_1,a_2,\dots,a_k\in \mathbb{R}^3$ such that $Q(a_i)$ are strict local maxima satisfying and
$$
\begin{equation*}
Q(x)=Q(a_i)+O(|x-a_i|^2)\quad\text{as}\quad x\to a_i,\qquad 1\leqslant i \leqslant k.
\end{equation*}
\notag
$$
The Schrödinger–Poisson system
$$
\begin{equation}
\begin{cases} -\dfrac{\hbar^2}{2m}\, \Delta u+e\omega u+\phi u=f(x,u) &\text{in }\mathbb{R}^3, \\ -\Delta\phi=4\pi eu^2 &\text{in }\mathbb{R}^3, \end{cases}
\end{equation}
\tag{1.2}
$$
was first introduced in [1] describing the behavior of a quantum particle in three- dimensional space interacting with the electromagnetic field. Here $m$ and $e$ are the mass and the charge of the particle, respectively, while $\omega$ denotes the phase, $\hbar$ is Planck’s constant. The unknowns of the system are the field $u$ associated with the particles and the electric potential $\phi$. The presence of the nonlinear term $f(x,u)$ simulates the interaction between many particles or external nonlinear perturbations. The class of nonlinear terms $f(x,u)=\lambda |u|^{p-2}u$, $\lambda> 0$, $p>2$, contains the Slater correction term $C_\mathrm{s}|u|^{2/3}u$, where $C_\mathrm{s}$ is the Slater constant depending on the particles. It is important in various physical frameworks: gravitation, plasma physics, semiconductor theory, quantum chemistry (see [2], [3]). The class of nonlinear terms $f(x,u)=a|u|^{p-2}u+b|u|^{q-2}u$ with $a,b\in \mathbb{R}$ and $2<p<q< \infty$ arises in various regions of mathematical physics. For example, when $a> 0$, $b< 0$, $p=4$ and $q=6$, it models the evolution of a monochromatic wave complex envelope in a medium with weakly saturating nonlinearity. This Schrödinger equation with double power nonlinearities appears in boson gas interaction, nonlinear optics, nuclear hydrodynamics and so on (see [4], [5]). For more physical background of this system, we refer to [1], [6]. In recent years, many papers have been devoted to studying the existence and multiplicity of positive solutions of the system (1.2) under various assumptions on the nonlinear term. We refer the reader to [7]–[11]. The existence of nontrivial solutions for the system (1.2) was obtained in [12]–[14]. The existence of ground state solutions for the system (1.2) has been well studied by various authors, see [15]–[20], [3]. It is worth pointing out that the system (1.2) with critical growth has been studied extensively, for example, in [8], [10], [16], [18]–[22]. In addition, many researchers proved by the variational method that the elliptic equations with critical exponent have at least $k$ positive solutions. We refer the reader to [23]–[28] and references therein. Specially, Huang [26] studied the following nonhomogeneous equations involving the $p$-Laplacian:
$$
\begin{equation}
\begin{cases} -\Delta_p u=Q(x)|u|^{p^*-2}u+\theta h(x),\quad x\in{\mathbb{R}^N}, \\ u\in D^{1,p}(\mathbb{R}^N), \end{cases}
\end{equation}
\tag{1.3}
$$
where $1< p< N$, $p^*=Np/(N-p)$ is the critical Sobolev exponent, $h(x)> 0$ a. e. in $\mathbb{R}^N$, $\theta> 0$ is a real parameter. $Q(x)$ satisfies appropriate assumptions. By the variational method, the author obtained the existence of $k$ positive solutions of (1.3). Recently, the authors established the multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system
$$
\begin{equation}
\begin{cases} -\Delta u+\lambda u+K(x)\phi u= Q(x)|u|^{p-2}u &\text{in }\mathbb{R}^3, \\ -\Delta \phi=K(x)u^{2} &\text{in }\mathbb{R}^3, \end{cases}
\end{equation}
\tag{1.4}
$$
where $\lambda$ is a positive parameter. In [7], Chen, Kuo, and Wu proved that the system (1.4) has $k$ positive solutions with $4\leqslant p<6$, $Q(x)\in C(\mathbb{R}^3)$ and $K(x)\equiv 1$. When $2<p<6$, Sun, Wu, and Feng [11] obtained the number of positive solutions of (1.4) under certain conditions on $Q(x)$ and $K(x)$. In [14], Li and Gu studied the following Schrödinger–Poisson system with critical and supercritical nonlinearities:
$$
\begin{equation}
\begin{cases} -\Delta u+u+\phi u= |u|^{p-2}u+u^5+b(x)|u|^{q-2}u &\text{in } \mathbb{R}^3, \\ -\Delta \phi=u^{2} &\text{in } \mathbb{R}^3, \end{cases}
\end{equation}
\tag{1.5}
$$
where $p\in(4,6)$, $q\in(6,+\infty)$. They proved the existence of a nontrivial solution of the system (1.5) by using the variational method. Compare our work with [14], where the authors considered the Schrödinger–Poisson system with critical and supercritical nonlinear terms and obtained that the system has a nontrivial solution. Our result is different since we suppose that $Q(x)$ has $k$ strict local maxima and obtain the number of positive solutions. Since the system (1.1) is of critical growth, the embedding $H_{r}^1(\mathbb{R}^3)\hookrightarrow L^{6}(\mathbb{R}^3)$ need not be compact. We overcome this difficulty by using the Brézis–Lieb lemma. Moreover, we obtain a compactness result for the supercritical term by using the potential function and thus overcome another difficulty. Throughout this paper, we use the following notation. Let $H^1(\mathbb{R}^3)$, $D^{1,2}(\mathbb{R}^3)$ be the usual Sobolev spaces defined by $H^1(\mathbb{R}^3)=\{u\in L^2(\mathbb{R}^3) \colon |\nabla u|\in L^2(\mathbb{R}^3)\}$, $D^{1,2}(\mathbb{R}^3)=\{u\in L^6(\mathbb{R}^3) \colon |\nabla u|\in L^2(\mathbb{R}^3)\}$, and let $H_r^1(\mathbb{R}^3)$, $D_r^{1,2}(\mathbb{R}^3)$ be the corresponding subspaces of radial functions. The corresponding norms and inner products are defined by
$$
\begin{equation*}
\begin{gathered} \, \langle u,v\rangle=\int_{\mathbb{R}^3}(\nabla u\,\nabla v+uv)\, dx, \\ \|u\|_{H^1(\mathbb{R}^3)}=\biggl(\int_{\mathbb{R}^3}(|\nabla u|^2+u^2)\, dx\biggr)^{1/2},\qquad u,v\in H^1(\mathbb{R}^3), \\ \langle u,v\rangle=\int_{\mathbb{R}^3}\nabla u\,\nabla v\, dx,\quad \|u\|_{D^{1,2}(\mathbb{R}^3)}=\biggl(\int_{\mathbb{R}^3}\nabla u\,\nabla v\, dx\biggr)^{1/2},\qquad u,v\in D^{1,2}(\mathbb{R}^3). \end{gathered}
\end{equation*}
\notag
$$
In this paper, we consider the system (1.1) in the subspace $H_r^1(\mathbb{R}^3)$ and write $\|\,{\cdot}\,\|$ for the norm of $H_r^1(\mathbb{R}^3)$. The norm in $L^p(\mathbb{R}^3)$ is denoted by $|\,{\cdot}\,|_p$ for $p\in [1,+\infty)$. We write $C, C_1, C_2,\dots$ for various positive constants, which may vary from line to line. Let $S$ be the best constant for the Sobolev embedding $D^{1,2}(\mathbb{R}^3)\hookrightarrow L^{6}(\mathbb{R}^3)$, namely
$$
\begin{equation*}
S=\inf_{u\in D^{1,2}(\mathbb{R}^3)\setminus \{0\}}\frac{\int_{\mathbb{R}^3}|\nabla u|^2\, dx}{\bigl(\int_{\mathbb{R}^3}|u|^{6}\, dx\bigr)^{1/3}}.
\end{equation*}
\notag
$$
We now state our main result. Theorem 1.1. Assume that $(Q_1)$ holds. Then there exist $\lambda_0,\mu_0> 0$ small enough such that for all $\lambda\in(0,\lambda_0)$ and $\mu\in(0,\mu_0)$, the system (1.1) has at least $k$ positive solutions. Remark 1.2. A simple example satisfying $(Q_1)$ is where $a_i=\pi/2+i\pi$ for all $i$, $1\leqslant i\leqslant k$. § 2. PreliminariesWith the help of the Lax–Milgram theorem, we see that for every $u\in H_r^1(\mathbb{R}^3)$ there exists a unique $\phi_u\in D_r^{1,2}(\mathbb{R}^3)$ such that $-\Delta\phi_u=u^2$. Hence the system (1.1) can be transformed into the following semilinear nonlocal elliptic equation:
$$
\begin{equation}
-\Delta u+u+\lambda\phi_u u=Q(x)|u|^{4}u+\mu h(x)|u|^{q-2}u \qquad\text{in } \mathbb{R}^3.
\end{equation}
\tag{2.1}
$$
The variational functional associated with (2.1) is of the form
$$
\begin{equation*}
I_{\lambda,\mu}(u)=\frac{1}{2}\|u\|^2+\frac{\lambda}{4}\int_{\mathbb{R}^3}\phi_u u^2\, dx -\frac{1}{6}\int_{\mathbb{R}^3}Q(x)|u|^6 \, dx-\frac{\mu}{q}\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx.
\end{equation*}
\notag
$$
We say that $u\in H_{r}^1(\mathbb{R}^3)$ is a weak solution of (2.1) if $u$ satisfies
$$
\begin{equation*}
\int_{\mathbb{R}^3}|\nabla u|^2\, dx +\int_{\mathbb{R}^3}u^2\, dx+\lambda\int_{\mathbb{R}^3}\phi_u u^2\, dx-\int_{\mathbb{R}^3}Q(x)|u|^{6}\, dx-\mu\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx=0.
\end{equation*}
\notag
$$
So if a nonzero solution exists, then it must lie in the Nehari manifold $\mathcal{N}_{\lambda,\mu}$, which is defined by
$$
\begin{equation*}
\begin{aligned} \, \mathcal{N}_{\lambda,\mu} &=\biggl\{u\in H_{r}^1(\mathbb{R}^3)\setminus\{0\}\colon \int_{\mathbb{R}^3}|\nabla u|^2 \, dx+\int_{\mathbb{R}^3}u^2\, dx+\lambda\int_{\mathbb{R}^3}\phi_u u^2\, dx \\ &\qquad-\int_{\mathbb{R}^3}Q(x)|u|^{6}\, dx-\displaystyle\mu\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx=0\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
By [18], [3], the function $\phi_u$ has the following properties. Lemma 2.1. For every $u\in H_r^1(\mathbb{R}^3)$ there is a unique solution $\phi_u\in D_r^{1,2}(\mathbb{R}^3)$ of Moreover, the following assertions hold. (1) $\phi_u\geqslant 0$ for $x\in \mathbb{R}^3$ and $\phi_{tu}=t^{2}\phi_u$ for all $t> 0$. (2) If $u$ is a radial function, then $\phi_u$ is radial. (3)
$$
\begin{equation*}
\|\phi_u\|_{D^{1,2}(\mathbb{R}^3)}^{2}=\int_{\mathbb{R}^3}\phi_uu^{2}\, dx\leqslant S^{-1}|u|_{12/5}^{4}\leqslant C\|u\|^4.
\end{equation*}
\notag
$$
(4) Assume that $u_n\rightharpoonup u$ in $H_r^1(\mathbb{R}^3)$. Then $\phi_{u_n}\to \phi_u$ in $D_r^{1,2}(\mathbb{R}^3)$ and Lemma 2.2. Assume that $0< \beta<3$ and $6<q<6+2\beta$. If $\{u_n\}$ is bounded and $u_n\rightharpoonup u$ in $H_{r}^1(\mathbb{R}^3)$, then Proof. By Lemma 2.2 in [14], we have
$$
\begin{equation*}
|u_n(x)|\leqslant C\|u_n\|\frac{1}{|x|^{1/2}},\quad\text{for a. e. }\ x\in \mathbb{R}^3.
\end{equation*}
\notag
$$
By the boundedness of $\{u_n\}$, for $6<q<6+2\beta$, one has
$$
\begin{equation*}
\begin{aligned} \, \int_{|x|\leqslant1}\frac{|x|^\beta}{1+|x|^\beta}|u_n|^q\, dx &\leqslant C\int_{|x|\leqslant1}\frac{|x|^\beta}{1+|x|^\beta}\, \frac{dx}{|x|^{q/2}} \\ &\leqslant C\int_{|x|\leqslant1} \frac{dx}{|x|^{q/2-\beta}}= C\int_0^1 \frac{dt}{t^{q/2-\beta-2}} < +\infty, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \int_{|x|\geqslant1}\frac{|x|^\beta}{1+|x|^\beta}|u_n|^q\, dx &\leqslant C\int_{|x|\geqslant1}\frac{|x|^\beta}{1+|x|^\beta}\, \frac{dx}{|x|^{q/2}} \\ &\leqslant C\int_{|x|\geqslant1}\, \frac{dx}{|x|^{q/2}}= C\int_1^{+\infty} \frac{dt}{t^{q/2-2}}< +\infty. \end{aligned}
\end{equation*}
\notag
$$
Therefore, by the Lebesgue dominated convergence theorem, we get as $n\to \infty$. The proof is complete. Lemma 2.3. Suppose that $0< \beta<3$ and $6<q<6+2\beta$, for all $\lambda, \mu> 0$. Then the functional $I_{\lambda, \mu}$ satisfies the $(\mathrm{PS})_{c}$ condition for every $c\in(-\infty, S^{3/2}/(3Q_M^{1/2}))$. Proof. Let $\{u_n\}\subset H_{r}^1(\mathbb{R}^3)$ be a $(\mathrm{PS})_{c}$ sequence, that is,
$$
\begin{equation}
I_{\lambda,\mu}(u_n)\to c\quad\text{and}\quad I'_{\lambda,\mu}(u_n)\to 0\quad\text{as}\quad n\to\infty.
\end{equation}
\tag{2.2}
$$
It follows from (2.2) that
$$
\begin{equation*}
\begin{aligned} \, |c|+1+o{(\|u_n\|)} &\geqslant I_{\lambda,\mu}(u_n)-\frac{1}{4}\langle I'_{\lambda,\mu}(u_n),u_n\rangle \\ &=\biggl(\frac{1}{2}-\frac{1}{4}\biggr)\|u_n\|^2 +\biggl(\frac{1}{4}-\frac{1}{6}\biggr)\int_{\mathbb{R}^3}Q(x)|u_n|^6 \, dx \\ &\qquad+\mu\biggl(\frac{1}{4}-\frac{1}{q}\biggr) \int_{\mathbb{R}^3}h(x)|u_n|^{q}\, dx \geqslant\frac{1}{4}\|u_n\|^2. \end{aligned}
\end{equation*}
\notag
$$
Therefore, $\{u_n\}$ is bounded in $H_{r}^1(\mathbb{R}^3)$. Thus, we may assume up to a subsequence, still denoted by $\{u_n\}$, that there exists $u\in H_{r}^1(\mathbb{R}^3)$ such that
$$
\begin{equation}
\begin{cases} u_n\rightharpoonup u &\text{weakly in }H_{r}^1(\mathbb{R}^3), \\ u_n\to u &\text{strongly in }L^{p}(\mathbb{R}^3),\ 2\leqslant p<6, \\ u_n(x)\to u(x) &\text{a. e. in }\mathbb{R}^3 \end{cases}
\end{equation}
\tag{2.3}
$$
as $n\to \infty$. Next, we prove that $u_n\to u$ strongly in $H_{r}^1(\mathbb{R}^3)$. Set $w_n=u_n-u$ and $\lim_{n\to\infty}\|w_n\|^2=l$. By (2.2), Brézis–Lieb’s lemma, Lemma 2.1, and Lemma 2.2, we have
$$
\begin{equation}
\begin{aligned} \, &\|w_n\|^2+\|u\|^2+\lambda\int_{\mathbb{R}^3}\phi_u u^2\, dx-\int_{\mathbb{R}^3}Q(x)|w_n|^6\, dx-\int_{\mathbb{R}^3}Q(x)|u|^6\, dx \nonumber \\ &\qquad-\mu\int_{\mathbb{R}^3}h(x)|u|^q\, dx=o(1), \end{aligned}
\end{equation}
\tag{2.4}
$$
and
$$
\begin{equation}
\|u\|^2+\lambda\int_{\mathbb{R}^3}\phi_u u^2\, dx-\int_{\mathbb{R}^3}Q(x)|u|^6\, dx-\mu\int_{\mathbb{R}^3}h(x)|u|^q\, dx=0.
\end{equation}
\tag{2.5}
$$
It follows from (2.4) and (2.5) that Since $\|w_n\|^2\to l$, by (2.6), one has
$$
\begin{equation*}
\int_{\mathbb{R}^3}Q(x)|w_n|^6\, dx\to l\quad\text{as}\quad n\to\infty.
\end{equation*}
\notag
$$
Applying the Sobolev inequality, by (2.6), we have
$$
\begin{equation}
\|w_n\|^2\leqslant Q_M \int_{\mathbb{R}^3}|w_n|^6\, dx+o(1)\leqslant Q_M S^{-3}\|w_n\|^6+o(1).
\end{equation}
\tag{2.7}
$$
Thus, by (2.7), we can deduce that $l^3\geqslant S^3l/Q_M$ as $n\to \infty$, which implies that either $l= 0$ or $l\geqslant S^{3/2}/Q_M^{1/2}$ as $n\to \infty$. If $l\geqslant S^{3/2}/Q_M^{1/2}$, then, by (2.2),
$$
\begin{equation*}
c=\biggl(\frac{1}{2}-\frac{1}{6}\biggr)l+I_{\lambda,\mu}(u)\geqslant \frac{S^{3/2}}{3Q_M^{1/2}}+I_{\lambda,\mu}(u)
\end{equation*}
\notag
$$
as $n\to \infty$. On one hand, by the definition of $c$, we get
$$
\begin{equation*}
I_{\lambda,\mu}(u)\leqslant c-\frac{S^{3/2}}{3Q_M^{1/2}}<0.
\end{equation*}
\notag
$$
On the other hand, by (2.5), we have This is a contradiction. Hence, $l= 0$ and $u_n\to u$ strongly in $H_{r}^1(\mathbb{R}^3)$. The proof is complete. Lemma 2.4. Assume that $0< \beta<3$, $6<q<6+2\beta$ and $\mu,\lambda> 0$. Then $\mathcal{N}_{\lambda,\mu}\neq\varnothing$. Proof. Suppose that $u\in H_{r}^1(\mathbb{R}^3)\setminus\{0\}$. We define a fibering map $J_u\colon t\to I_{\lambda,\mu}(tu)$ by
$$
\begin{equation*}
J_u(t)=\frac{t^2}{2}\|u\|^2+\lambda\frac{t^4}{4}\int_{\mathbb{R}^3}\phi_{u}u^2\, dx-\frac{t^6}{6}\int_{\mathbb{R}^3}Q(x)|u|^{6}\, dx -\mu\frac{t^q}{q}\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx
\end{equation*}
\notag
$$
for all $t\geqslant 0$. Then $J_u(0)= 0$ and $J_u(t)\to-\infty$ as $t\to + \infty$. Moreover, we get We can see that there exists a unique $t_u> 0$ such that $J'_u(t_u)= 0$, that is, Therefore, $t_u u\in\mathcal{N}_{\lambda,\mu}$ and $\mathcal{N}_{\lambda,\mu}\neq\varnothing$. The proof is complete. Lemma 2.5. Suppose that $0< \beta<3$, $6<q<6+2\beta$, and $\lambda,\mu> 0$. Then the functional $I_{\lambda,\mu}(u)$ is coercive and bounded from below on $\mathcal{N_{\lambda,\mu}}$. Proof. For all $u\in\mathcal{N_{\lambda,\mu}}$, we have
$$
\begin{equation*}
\begin{aligned} \, I_{\lambda,\mu}(u)&=\frac{1}{2}\|u\|^2+\frac{\lambda}{4}\int_{\mathbb{R}^3}\phi_u u^2\, dx -\frac{1}{6}\int_{\mathbb{R}^3}Q(x)|u|^6 \, dx -\frac{\mu}{q}\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx \\ &=\biggl(\frac{1}{2}-\frac{1}{4}\biggr)\|u\|^2+\biggl(\frac{1}{4}-\frac{1}{6}\biggr) \int_{\mathbb{R}^3}Q(x)|u|^6 \, dx \\ &\qquad+\mu\biggl(\frac{1}{4}-\frac{1}{q}\biggr) \int_{\mathbb{R}^3}h(x)|u|^{q}\, dx \geqslant\frac{1}{4}\|u\|^2, \end{aligned}
\end{equation*}
\notag
$$
which implies that $I_{\lambda,\mu}(u)$ is coercive and bounded from below on $u\in\mathcal{N_{\lambda,\mu}}$. The proof is complete. Lemma 2.6. Assume that $u_0$ is a local minimizer of $I_{\lambda,\mu}$ on $\mathcal{N_{\lambda,\mu}}$. Then $I_{\lambda,\mu}'(u_0)= 0$ in $H_r^{-1}(\mathbb{R}^3)$. Proof. If $u_0$ is a local minimizer for $I_{\lambda,\mu}$ on $\mathcal{N_{\lambda,\mu}}$, then, by the definition of $J_u(t)$ in Lemma 2.4, there exists a neighborhood $U$ of $u_0$ in $H_r^1(\mathbb{R}^3)$ such that
$$
\begin{equation*}
I_{\lambda,\mu}(u_0)=\min_{u\in\mathcal{N_{\lambda,\mu}}}I_{\lambda,\mu}(u)=\min_{u\in U\setminus\{0\},\,J'_u(1)=0}I_{\lambda,\mu}(u)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
J'_u(1)=\|u\|^2+\lambda\int_{\mathbb{R}^3}\phi_u u^2\, dx-\int_{\mathbb{R}^3}Q(x)|u|^{6}\, dx -\mu\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx.
\end{equation*}
\notag
$$
Combining this with Lemma 2.4 and the equality $J'_u(1)= 0$, we get
$$
\begin{equation*}
\begin{aligned} \, J''_u(1)&=\|u\|^2+3\lambda\int_{\mathbb{R}^3}\phi_u u^2\, dx-5\int_{\mathbb{R}^3}Q(x)|u|^{6}\, dx -(q-1)\mu\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx \\ &=(1-3)\|u\|^2-(5-3)\int_{\mathbb{R}^3}Q(x)|u|^{6}\, dx -(q-1-3)\mu\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx <0. \end{aligned}
\end{equation*}
\notag
$$
It follows from the theory of Lagrange multipliers that there exists a $\theta\in\mathbb{R}$ such that $I'_{\lambda,\mu}(u_0)=\theta J''_{u_0}$. Moreover, since $u_0\in \mathcal{N_{\lambda,\mu}}$, one has which implies that $\theta= 0$. Therefore, we obtain $I'_{\lambda,\mu}(u_0)= 0$ in $H_r^{-1}(\mathbb{R}^3)$. The proof is complete. Choose $r> 0$ small enough such that $\bigcup_{i=1}^k\overline{B_{r}(a_i)}\subset \mathbb{R}^3$ and $\overline{B_{r}(a_i)}\cap\overline{B_{r}(a_j)}=\varnothing$ for $i\neq j$, $i,j=1,2,\dots,k$, where $\overline{B_{r}(a_i)}=\{x\in \mathbb{R}^3\colon |x-a_i|\leqslant r\}$. We minimize the functional $I_{\lambda,\mu}$ on some submanifolds of $\mathcal{N_{\lambda,\mu}}$. For this we define a barycenter map $g\colon H_r^{1}(\mathbb{R}^3)\setminus\{0\}\to \mathbb{R}^3$ by
$$
\begin{equation*}
g(u)=\frac{\int_{\mathbb{R}^3}x|u|^{6}\, dx}{\int_{\mathbb{R}^3}|u|^{6}\, dx},
\end{equation*}
\notag
$$
see [23] or [28]. Set
$$
\begin{equation*}
\begin{alignedat}{2} \mathcal{N}_{\lambda,\mu}^{i} &=\{u\in\mathcal{N_{\lambda,\mu}}\colon |g(u)-a_i|<r\}, &\qquad &1\leqslant i\leqslant k, \\ \mathcal{O}_{\lambda,\mu}^i &=\{u\in\mathcal{N_{\lambda,\mu}}\colon |g(u)-a_i|=r\}, &\qquad &1\leqslant i\leqslant k. \end{alignedat}
\end{equation*}
\notag
$$
According to Lemma 2.4, we know that $\mathcal{N}_{\lambda,\mu}^{i} \neq \varnothing $ for $1\leqslant i\leqslant k$. By Lemma 2.5, we can define
$$
\begin{equation*}
m_{\lambda,\mu}^i=\inf_{u\in \mathcal{N}_{\lambda,\mu}^{i}}I_{\lambda,\mu}(u)\quad\text{and}\quad \overline{m}_{\lambda,\mu}^i=\inf_{u\in \mathcal{O}_{\lambda,\mu}^i}I_{\lambda,\mu}(u), \qquad 1\leqslant i\leqslant k.
\end{equation*}
\notag
$$
As is well known, the function
$$
\begin{equation*}
U(x)=\frac{3^{1/4}}{(1+|x|^2)^{1/2}},\qquad x\in \mathbb{R}^3,
\end{equation*}
\notag
$$
satisfies Let $\rho> 0$ be small enough such that $0\notin B_\rho(a_i)\subset \mathbb{R}^3$ for $i=1,2,\dots,k$. Choose a cut-off function $\varphi_i(x)\in C_0^\infty(\mathbb{R}^3)$ such that $0\leqslant \varphi_i(x)\leqslant 1$ for $x\in \mathbb{R}^3$ and
$$
\begin{equation*}
\varphi_i(x)= \begin{cases} 1, &|x-a_i|\leqslant\dfrac{\rho}{2}, \\ 0, &|x-a_i|\geqslant\rho. \end{cases}
\end{equation*}
\notag
$$
Let
$$
\begin{equation*}
u_\varepsilon^i(x)=\varepsilon^{-1/2}\varphi_i(x)U\biggl(\frac{x-a_i}{\varepsilon}\biggr) =\frac{\varphi_i(x)3^{1/4} \varepsilon^{1/2}}{[\varepsilon^2+|x-a_i|^2]^{1/2}},\qquad\varepsilon>0.
\end{equation*}
\notag
$$
Using the estimates obtained by Brézis and Nirenberg [29], we have the following results:
$$
\begin{equation}
\begin{cases} |\nabla u_\varepsilon^i|_2^2= S^{3/2}+O(\varepsilon), \\ |u_\varepsilon^i|_6^6= S^{3/2}+O(\varepsilon^{3}) \end{cases}
\end{equation}
\tag{2.8}
$$
and
$$
\begin{equation}
|u_\varepsilon^i|_t^t= \begin{cases} O(\varepsilon^{t/2}), &t\in[2,3), \\ O(\varepsilon^{t/2}|{\ln\varepsilon}|), &t=3, \\ O(\varepsilon^{(6-t)/2}), &t\in(3,6). \end{cases}
\end{equation}
\tag{2.9}
$$
Then we have the following lemma. Proof. By Lemma 2.4, one can find $t_\varepsilon^i> 0$ and $t_\varepsilon^iu_\varepsilon^i\in\mathcal{N}_{\lambda, \mu}$ in such a way that $\sup_{t\geqslant0}I_{\lambda,\mu}(tu_\varepsilon^i)=I_{\lambda,\mu}(t_\varepsilon^i u_\varepsilon^i)$. By Lemma 3.2 in [14], we can assume that there exist positive constants $t_1$, $t_2$ such that $0< t_1<t_\varepsilon^i<t_2< + \infty$. Moreover, by the definition of $g(u)$, we have
$$
\begin{equation*}
\begin{aligned} \, g(t_\varepsilon^iu_\varepsilon^i)&=\frac{\int_{\mathbb{R}^3}x|\varepsilon^{-1/2}t_\varepsilon^i \varphi_i(x)U((x-a_i)/\varepsilon)|^6\, dx} {\int_{\mathbb{R}^3}|\varepsilon^{-1/2}t_\varepsilon^i\varphi_i(x)U((x-a_i)/\varepsilon)|^6\, dx} \\ &= \frac{\int_{\mathbb{R}^3}(a_i+\varepsilon y)|t_\varepsilon^i\varphi_i(a_i+\varepsilon y)U(y)|^6\, dy}{\int_{\mathbb{R}^3}|t_\varepsilon^i\varphi_i(a_i+\varepsilon y)U(y)|^6\, dy} \to a_i\quad\text{as}\quad\varepsilon\to0. \end{aligned}
\end{equation*}
\notag
$$
Therefore, there exists an $\varepsilon_0> 0$ such that $g(t_\varepsilon^iu_\varepsilon^i)\in B_{r}(a_i)$ for every $\varepsilon$, $0< \varepsilon<\varepsilon_{0}$. It follows that $t_\varepsilon^i u_\varepsilon^i\in \mathcal{N}_{\lambda,\mu}^{i}$, $1\leqslant i\leqslant k$. Then we have
$$
\begin{equation*}
m_{\lambda,\mu}^i\leqslant I_{\lambda,\mu}(t_\varepsilon^i u_\varepsilon^i)=\sup_{t\geqslant0}I_{\lambda,\mu}(tu_\varepsilon^i).
\end{equation*}
\notag
$$
In addition, by $(Q_1)$, for every $\eta> 0$ there exists a $\delta> 0$ such that $|Q(x)-Q(a_i)|<\eta|x-a_i|^2$ for $0< |x-a_i|<\delta$. When $\varepsilon> 0$ is small enough, we have
$$
\begin{equation*}
\begin{aligned} \, &\biggl|\int_{\mathbb{R}^3}Q(x)|u_\varepsilon^i|^6\, dx -\int_{\mathbb{R}^3}Q(a_i)|u_\varepsilon^i|^6\, dx\biggr| \leqslant\int_{\mathbb{R}^3}|Q(x)-Q(a_i)||u_\varepsilon^i|^6\, dx \\ &\qquad< \int_{B_{\delta}(a_i)}\eta|x-a_i|^2\frac{(3\varepsilon^2)^{3/2}}{(\varepsilon^2+|x-a_i|^2)^3}\, dx+ C\int_{\mathbb{R}^3\setminus{B_{\delta}(a_i)}} \frac{(3\varepsilon^2)^{3/2}}{(\varepsilon^2+|x-a_i|^2)^3}\, dx \\ &\qquad\leqslant C \eta\varepsilon^3\int_{0}^{\delta}\frac{y^{4}}{(\varepsilon^2+y^2)^{3}}\, dy+C\varepsilon^3\int_{\delta}^{+\infty}\frac{y^2}{(\varepsilon^2+y^2)^{3}}\, dy \\ &\qquad\leqslant C \eta\varepsilon^2\int_{0}^{\delta/\varepsilon}\frac{t^{4}}{(1+t^2)^3}\, dt+C\int_{\delta/\varepsilon}^{+\infty}\frac{t^2}{(1+t^2)^3}\, dt \\ &\qquad\leqslant C \eta\varepsilon^2\int_{0}^{+\infty}\frac{t^{4}}{(1+t^2)^3}\, dt+C\int_{\delta/\varepsilon}^{+\infty}\frac{1}{t^4}\, dt \\ &\qquad\leqslant C_1 \eta\varepsilon^2+C_2\varepsilon^3, \end{aligned}
\end{equation*}
\notag
$$
where $C_1,C_2> 0$ (independent of $\eta$, $\varepsilon$). From this we derive that
$$
\begin{equation}
\limsup_{\varepsilon\to 0}\frac{\bigl|\int_{\mathbb{R}^3}Q(x)|u_\varepsilon^i|^6\, dx-\int_{\mathbb{R}^3}Q(a_i)|u_\varepsilon^i|^6\, dx\bigr|}{\varepsilon^2}\leqslant C_1\eta.
\end{equation}
\tag{2.10}
$$
Then from the arbitrariness of $\eta> 0$, by (2.8) and (2.10), one has
$$
\begin{equation}
\int_{\mathbb{R}^3}Q(x)|u_\varepsilon^i|^6\, dx=Q(a_i)\int_{\mathbb{R}^3}|u_\varepsilon^i|^6\, dx+O(\varepsilon^2)=Q_MS^{3/2}+O(\varepsilon^2).
\end{equation}
\tag{2.11}
$$
By Lemma 2.2 and (2.9), we have the following estimate:
$$
\begin{equation}
\int_{\mathbb{R}^3}\phi_{u_\varepsilon^i}|u_\varepsilon^i|^2\, dx\leqslant C|u_\varepsilon^i|_{12/5}^4\leqslant O(\varepsilon^2).
\end{equation}
\tag{2.12}
$$
To estimate the supercritical term, we use the following inequality:
$$
\begin{equation*}
|a+b|^p\leqslant 2^p(|a|+|b|),\qquad a,b\in \mathbb{R},\quad p>0.
\end{equation*}
\notag
$$
According to the definition of $u_\varepsilon^i$, by (2.8) and $0< \beta<3$, there exists $C_3> 0$ such that
$$
\begin{equation}
\begin{aligned} \, &\int_{\mathbb{R}^3}\frac{|x|^\beta}{1+|x|^\beta}|u_\varepsilon^i|^q\, dx \geqslant C\int_{|x-a_i|\leqslant \rho/2}\frac{|x|^\beta}{1+|x|^\beta} \frac{\varepsilon^{q/2}}{(\varepsilon^2+|x-a_i|^2)^{q/2}}\, dx \nonumber \\ &\geqslant C\varepsilon^{q/2}\int_{|x-a_i|\leqslant \rho/2}\frac{|x|^\beta}{1+(|a_i|+\rho/2)^\beta}\, \frac{1}{(\varepsilon^2+|x-a_i|^2)^{q/2}}\, dx \nonumber \\ &\geqslant C\varepsilon^{q/2}\int_{|x-a_i|\leqslant \rho/2}\frac{|x|^\beta +|x-a_i|^\beta-|x-a_i|^\beta}{(\varepsilon^2+|x-a_i|^2)^{q/2}}\, dx \nonumber \\ &\geqslant C\varepsilon^{q/2}\int_{|x-a_i|\leqslant \rho/2}\frac{2^{-\beta}|a_i|^\beta}{(\varepsilon^2\,{+}\,|x-a_i|^2)^{q/2}}\, dx- C\varepsilon^{q/2}\int_{|x-a_i|\leqslant \rho/2} \frac{|x-a_i|^\beta}{(\varepsilon^2{+}\,|x-a_i|^2)^{q/2}}\, dx \nonumber \\ &\geqslant C\varepsilon^{q/2}\int_0^{\rho/2}\frac{r^2}{(\varepsilon^2+r^2)^{q/2}}\, dr -C\varepsilon^{q/2}\int_0^{\rho/2}\frac{r^{2+\beta}}{(\varepsilon^2+r^2)^{q/2}}\, dr \nonumber \\ &= C\varepsilon^{3-q/2}\int_0^{\rho/2\varepsilon}\frac{t^2}{(1+t^2)^{q/2}}\, dt -C\varepsilon^{3+\beta-q/2}\int_0^{\rho/2\varepsilon}\frac{t^{2+\beta}}{(1+t^2)^{q/2}}\, dt \nonumber \\ &\geqslant C\varepsilon^{3-q/2}\int_0^1t^2\, dt -C\varepsilon^{3+\beta-q/2}\biggl(\int_0^1t^{2+\beta}\, dt+\int_1^{+\infty}t^{2+\beta-q}\, dt\biggr) \nonumber \\ &\geqslant C\varepsilon^{3-q/2} -C\varepsilon^{3+\beta-q/2}\biggl(\frac{1}{3+\beta} +\frac{1}{3+\beta-q}t^{3+\beta-q}\bigg|_1^{+\infty}\biggr) \geqslant C_3\varepsilon^{3-q/2}. \end{aligned}
\end{equation}
\tag{2.13}
$$
Combining this with (2.11), (2.12) and (2.13), we have
$$
\begin{equation*}
\begin{aligned} \, m_{\lambda,\mu}^i&\leqslant I_{\lambda,\mu}(t_\varepsilon^i u_\varepsilon^i)=\sup_{t\geqslant0}I_{\lambda,\mu}(tu_\varepsilon^i) \leqslant\sup_{t\geqslant0}\biggl\{\frac{t^2}{2}\|u_\varepsilon^i\|^2 +\lambda\frac{t^4}{4}\int_{\mathbb{R}^3}\phi_{u_\varepsilon^i}|u_\varepsilon^i|^2\, dx \\ &\qquad\qquad\qquad\qquad-\frac{t^6}{6}\int_{\mathbb{R}^3}Q(x)|u_\varepsilon^i|^{6}\, dx-\mu\frac{t^q}{q}\int_{\mathbb{R}^3}\frac{|x|^\beta}{1+|x|^\beta}|u_\varepsilon^i|^{q}\, dx\biggr\} \\ &\leqslant \sup_{t\geqslant0}\biggl\{\frac{t^2}{2}\int_{\mathbb{R}^3}|\nabla u_\varepsilon^i|^2\, dx-\frac{t^6}{6}\int_{\mathbb{R}^3}Q(x)|u_\varepsilon^i|^{6}\, dx\biggr\}+O(\varepsilon)+\lambda C_4\varepsilon^2-C_3\mu\varepsilon^{3-q/2} \\ &\leqslant\frac{1}{3}\frac{\bigl(\int_{\mathbb{R}^3}|\nabla u_\varepsilon^i|^2\, dx\bigr)^{3/2}}{\bigl(\int_{\mathbb{R}^3}Q(x)|u_\varepsilon^i|^{6}\, dx\bigr)^{1/2}}+O(\varepsilon)+\lambda C_4\varepsilon^2-C_3\mu\varepsilon^{3-q/2} \\ &\leqslant \frac{S^{3/2}}{3Q_M^{1/2}}+O(\varepsilon)+\lambda C_4\varepsilon^2-C_3\mu\varepsilon^{3-q/2}< \frac{S^{3/2}}{3Q_M^{1/2}}, \end{aligned}
\end{equation*}
\notag
$$
where $C_4> 0$. The proof is complete. Lemma 2.8. There exist $\lambda_0, \mu_0> 0$ small enough such that the following inequalities hold whenever $0< \lambda<\lambda_0$ and $0< \mu<\mu_0$: Proof. Indeed, arguing by contradiction, assume that there are positive sequences $\{\lambda_n\}$ and $\{\mu_n\}$ such that $\lambda_n\to 0$ and $\mu_n\to 0$ as $n\to \infty$ and
$$
\begin{equation*}
\overline{m}_{\lambda_n,\mu_n}^{\,i}\to m \leqslant\frac{S^{3/2}}{3Q_M^{1/2}}.
\end{equation*}
\notag
$$
Hence there is a sequence $\{u_n\}\subset \mathcal{O}_{\lambda_n,\mu_n}^i$ such that $I_{\lambda_n,\mu_n}(u_n)\to m$ as $n\to \infty$ and
$$
\begin{equation}
\|u_n\|^2+\lambda_n\int_{\mathbb{R}^3}\phi_{u_n} u_n^2\, dx =\int_{\mathbb{R}^3}Q(x)|u_n|^{6}\, dx+\mu_n\int_{\mathbb{R}^3}h(x)|u_n|^{q}\, dx.
\end{equation}
\tag{2.14}
$$
Since $\{u_n\}$ is bounded in $H_r^{1}(\mathbb{R}^3)$, it follows from Lemma 2.1 and Lemma 2.2 that
$$
\begin{equation*}
\lim_{n\to\infty}\lambda_n\int_{\mathbb{R}^3}\phi_{u_n} u_n^2\, dx\leqslant\lim_{n\to \infty}\lambda_nC_1\|u_n\|^4=0
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\lim_{n\to\infty}\mu_n\int_{\mathbb{R}^3}h(x)|u_n|^{q}\, dx\leqslant\lim_{n\to \infty}\mu_nC_2\|u_n\|^q=0,
\end{equation*}
\notag
$$
where $C_1,C_2> 0$ are constants. By (2.14) and the Sobolev inequality, one has
$$
\begin{equation*}
\|u_n\|^2\leqslant \|u_n\|^2+\lambda_n\int_{\mathbb{R}^3}\phi_u u^2\, dx\leqslant Q_MS^{-3}\|u_n\|^6+\mu_n C_2\|u_n\|^q,
\end{equation*}
\notag
$$
which implies that there exists a positive constant $C_3> 0$ such that $\|u_n\|\geqslant C_3$ for every $\lambda_n\in(0,\lambda_0)$ and $\mu_n\in(0,\mu_0)$. Thus, we can fix a constant $A> 0$ such that
$$
\begin{equation*}
\|u_n\|^2 \geqslant A\quad\text{and}\quad\int_{\mathbb{R}^3}Q(x)|u_n|^{6}\, dx\geqslant A.
\end{equation*}
\notag
$$
Then, up to a subsequence, there exists an $a> 0$ such that
$$
\begin{equation}
\lim_{n\to\infty} \|u_n\|^2 =\lim_{n\to\infty}\int_{\mathbb{R}^3}Q(x)|u_n|^6\, dx=a.
\end{equation}
\tag{2.15}
$$
It follows from (2.15) that
$$
\begin{equation}
a\leqslant Q_M\lim_{n\to\infty}\int_{\mathbb{R}^3}|u_n|^6\, dx\leqslant Q_MS^{-3}\lim_{n\to\infty}\biggl(\int_{\mathbb{R}^3}|\nabla u_n|^2\, dx\biggr)^3\leqslant Q_MS^{-3}a^3.
\end{equation}
\tag{2.16}
$$
We deduce that $a\geqslant S^{3/2}/Q_M^{1/2}$. On the other hand, we have
$$
\begin{equation}
\begin{aligned} \, \frac{1}{3}a &=\frac{1}{2} \|u_n\|^2 +\frac{\lambda_n}{4}\int_{\mathbb{R}^3}\phi_{u_n} u_n^2\, dx -\frac{1}{6}\int_{\mathbb{R}^3}Q(x)|u_n|^{6}\, dx \nonumber \\ &\qquad-\frac{\mu_n}{q}\int_{\mathbb{R}^3}h(x)|u_n|^{q}\, dx+o(1) = I_{\lambda_n,\mu_n}(u_n)+o(1)\leqslant\frac{S^{3/2}}{3Q_M^{1/2}}. \end{aligned}
\end{equation}
\tag{2.17}
$$
Hence, it follows from (2.16) and (2.17) that $a=S^{3/2}/Q_M^{1/2}$ and
$$
\begin{equation*}
\lim_{n\to\infty}\int_{\mathbb{R}^3}Q_M|u_n|^6\, dx=\frac{S^{3/2}}{Q_M^{1/2}}.
\end{equation*}
\notag
$$
Then, we have
$$
\begin{equation}
\lim_{n\to\infty}\int_{\mathbb{R}^3}(Q_M-Q(x))|u_n|^6\, dx=0.
\end{equation}
\tag{2.18}
$$
Setting $w_n=u_n/|u_n|_{6}$, we have $|w_n|_{6}= 1$ and
$$
\begin{equation*}
\lim_{n\to\infty}\int_{\mathbb{R}^3}|\nabla w_n|^2\, dx=\lim_{n\to\infty}\frac{|\nabla u_n|_2^2}{|u_n|_{6}^2}=S.
\end{equation*}
\notag
$$
Therefore, $\{w_n\}$ is a minimizing sequence for $S$. By [30], there exists an $x_0\in\mathbb{R}^3$ such that
$$
\begin{equation*}
|\nabla w_n|^2\rightharpoonup d\tilde{\mu}=S\delta_{x_0},\qquad |w_n|^6\rightharpoonup d\tilde{\nu}=\delta_{x_0}
\end{equation*}
\notag
$$
weakly in the sense of measure, where $\tilde{\mu},\tilde{\nu}$ are finite measures and $\delta_{x_0}$ is the Dirac mass at $x_0\in\mathbb{R}^3$. Since $w_n\in \mathcal{O}_{\lambda_n,\mu_n}^i$, we have
$$
\begin{equation*}
g(w_n)=\frac{\int_{\mathbb{R}^3}x|w_n|^{6}\, dx}{\int_{\mathbb{R}^3}|w_n|^{6}\, dx}\to x_0,
\end{equation*}
\notag
$$
as $n\to \infty$, which implies that $x_0 \notin \{a_i\colon i=1,2,\dots,k\}$. Consequently, by (2.18), we have This is a contradiction because $Q(x)$ is not a constant function. The proof is complete. Lemma 2.9. For every $u\in \mathcal{N}_{\lambda,\mu}^{i}$, $1\leqslant i\leqslant k$, one can find a number $\delta> 0$ and a differentiable function $f\colon B_{\delta}(0)\subset H_r^{1}(\mathbb{R}^3)\to \mathbb{R}$ such that $f(0)= 1$, $f(v)(u-v)\in\mathcal{N}_{\lambda,\mu}^{i}$ for every $v\in B_{\delta}(0)$ and Proof. It is similar to the proof of Lemma 4.4 in [25]. Given any $u\in\mathcal{N}_{\lambda,\mu}^{i}$, we define a function $G\colon \mathbb{R}^+\times H_r^{1}(\mathbb{R}^3)\to \mathbb{R}$ by
$$
\begin{equation*}
\begin{aligned} \, G(t,v)&= t\int_{\mathbb{R}^3}(|\nabla(u-v)|^2+|u-v|^2)\, dx+\lambda t^3\int_{\mathbb{R}^3}\phi_{u-v} |u-v|^2\, dx \\ &\qquad-t^5\int_{\mathbb{R}^3}Q(x)|u-v|^{6}\, dx-\mu t^{q-1}\int_{\mathbb{R}^3}h(x)|u-v|^{q}\, dx. \end{aligned}
\end{equation*}
\notag
$$
Therefore, $G(1,0)=\langle I'_{\lambda,\mu}(u),u\rangle= 0$ and
$$
\begin{equation*}
\begin{aligned} \, G_t(1,0)&= G_t(1,0)-3\langle I'_{\lambda,\mu}(u),u\rangle \\ &= (1-3)\int_{\mathbb{R}^3}(|\nabla u|^2+|u|^2)\, dx-(5-3)\int_{\mathbb{R}^3}Q(x)|u|^{6}\, dx \\ &\qquad-\mu(q-1-3)\int_{\mathbb{R}^3}h(x)|u|^{q}\, dx <0. \end{aligned}
\end{equation*}
\notag
$$
Then, according to the implicit function theorem, there exists $\delta> 0$ small enough and a differentiable function $f\colon B_{\delta}(0)\subset H_r^{1}(\mathbb{R}^3)\to \mathbb{R}$ such that $f(0)= 1$ and $G(f(v),v)= 0$ for every $v\in B_{\delta}(0)$. Since $G(f(v),v)= 0$, it is easy to check that $f(v)(u-v)\in\mathcal{N}_{\lambda,\mu}^{i}$ and
$$
\begin{equation*}
\begin{aligned} \, &\langle f'(0),\varphi\rangle= -\frac{\langle G_v(1,0),\varphi\rangle}{G_t(1,0)} \\ &{=}\,\frac{2\!\int_{\mathbb{R}^3}(\nabla u\, \nabla\varphi\,{+}\,u\varphi) \, dx\,{+}\,4\lambda\!\int_{\mathbb{R}^3}\!\phi_uu\varphi \, dx\,{-}\,6\!\int_{\mathbb{R}^3}\!Q(x)|u|^4u\varphi \, dx\,{-}\,\mu q\!\int_{\mathbb{R}^3}\!h(x)|u|^{q{-}2}u\varphi \, dx}{\int_{\mathbb{R}^3}(|\nabla u|^2\,{+}\,u^2) \, dx\,{+}\,3\lambda\!\int_{\mathbb{R}^3}\!\phi_u|u|^2\, dx\,{-}\,5\!\int_{\mathbb{R}^3}\!Q(x)|u|^{6}\, dx \,{-}\,\mu(q\,{-}\,1)\!\int_{\mathbb{R}^3}\!h(x)|u|^{q}\, dx}. \end{aligned}
\end{equation*}
\notag
$$
The proof is complete. Proof of Theorem 1.1. Combining Lemma 2.7 and Lemma 2.8, we infer that
$$
\begin{equation*}
m_{\lambda,\mu}^i<\frac{S^{3/2}}{3Q_M^{1/2}}<\overline{m}_{\lambda,\mu}^{\,i},
\end{equation*}
\notag
$$
for all $\lambda\in(0,\lambda_0)$ and $\mu\in(0,\mu_0)$. Consequently, it follows that
$$
\begin{equation*}
m_{\lambda,\mu}^i=\inf\{I_{\lambda,\mu}(u)\colon u\in\mathcal{N}_{\lambda,\mu}^{i}\cup\mathcal{O}_{\lambda,\mu}^i\}.
\end{equation*}
\notag
$$
Let $\{u_n^i\}\subset \mathcal{N}_{\lambda,\mu}^{i}\cup\mathcal{O}_{\lambda,\mu}^i$ be a minimizing sequence for $m_{\lambda,\mu}^i$. Then, by the Ekeland variational principle [31], there exists a subsequence, still denoted by $\{u_n^i\}$ such that
$$
\begin{equation}
I_{\lambda,\mu}(u_n^i)\leqslant m_{\lambda,\mu}^i+\frac{1}{n},\qquad I_{\lambda,\mu}(w)\geqslant I_{\lambda,\mu}(u_n^i)-\frac{1}{n}\|w-u_n^i\|,\quad w\in{\mathcal{N}_{\lambda,\mu}^{i}\cup\mathcal{O}_{\lambda,\mu}^i}.
\end{equation}
\tag{2.19}
$$
We now prove that $I'_{\lambda,\mu}(u_n^i)\to 0$ as $n\to \infty$. Indeed, by Lemma 2.9, there exists a $\delta_n^i> 0$ and a differentiable function $f_n^i\colon B_{\delta_n^i}(0)\subset H_r^{1}(\mathbb{R}^3)\to \mathbb{R}$ such that $f_n^i(0)= 1$, $f_n^i(v)(u_n^i-v)\in \mathcal{N}_{\lambda,\mu}^{i}$ for all $v\in B_{\delta_n^i}(0)$. Set $\varphi\in H_r^{1}(\mathbb{R}^3)$ with $\|\varphi\|= 1$ and $0< \delta<\delta_n^i$. Choosing $v=\delta\varphi$, one has $v=\delta\varphi\in B_{\delta_n^i}(0)$ and $f_n^i(\delta \varphi)(u_n^i-\delta \varphi)\in \mathcal{N}_{\lambda,\mu}^{i}$, it follows from (2.19) and the Taylor expansion that
$$
\begin{equation*}
\begin{aligned} \, &\frac{\|f_n^i(\delta\varphi)(u_n^i-\delta \varphi)-u_n^i\|}{n} \geqslant I_{\lambda,\mu}(u_n^i)-I_{\lambda,\mu}[f_n^i(\delta\varphi)(u_n^i-\delta\varphi)] \\ &=\langle I'_{\lambda,\mu}(u_n^i),u_n^i-f_n^i(\delta\varphi)(u_n^i-\delta\varphi)\rangle +o\bigl(\|u_n^i-f_n^i(\delta\varphi)(u_n^i-\delta\varphi)\|\bigr) \\ &=\delta f_n^i(\delta\varphi)\langle I'_{\lambda,\mu}(u_n^i),\varphi\rangle+(1-f_n^i(\delta\varphi))\langle I'_{\lambda,\mu}(u_n^i),{u_n^i} \rangle +o\bigl(\|u_n^i-f_n^i(\delta\varphi)(u_n^i-\delta\varphi)\|\bigr) \\ &= \delta f_n^i(\delta\varphi)\langle I'_{\lambda,\mu}(u_n^i),\varphi\rangle +o\bigl(\|u_n^i-f_n^i(\delta\varphi)(u_n^i-\delta\varphi)\|\bigr). \end{aligned}
\end{equation*}
\notag
$$
Hence, letting $\delta\to 0^+$, we get
$$
\begin{equation*}
\begin{aligned} \, \langle I'_{\lambda,\mu}(u_n^i),\varphi\rangle &\leqslant\frac{\|u_n^i-f_n^i(\delta \varphi)(u_n^i-\delta\varphi)\|(1/n+|o(1)|)}{\delta |f_n^i(\delta \varphi)|} \\ &\leqslant \frac{\|u_n^i(f_n^i(\delta\varphi)-f_n^i(0))-\delta f_n^i(\delta\varphi)\varphi\|(1/n+|o(1)|)}{\delta |f_n^i(\delta\varphi)|} \\ &\leqslant \frac{\bigl(\|u_n^i\||f_n^i(\delta\varphi)-f_n^i(0)|+\delta |f_n^i(\delta\varphi)|\|\varphi\|\bigr)(1/n+|o(1)|)}{\delta |f_n^i(\delta\varphi)|} \\ &\leqslant\biggl(1+\|u_n^i\|\frac{|f_n^i(\delta\varphi)-f_n^i(0)|}{\delta|f_n^i(\delta\varphi)|}\biggr) \biggl(\frac{1}{n}+|o(1)|\biggr) \\ &\leqslant \bigl(1+\|u_n^i\|\|(f_n^i)'(0)\|\bigr)\biggl(\frac{1}{n}+|o(1)|\biggr). \end{aligned}
\end{equation*}
\notag
$$
Since $\{u_n^i\}$ and $\{(f_n^i)'(0)\}$ are bounded, we deduce that $I'_{\lambda,\mu}(u_n^i)\to 0$ as $n\to \infty$. Therefore, $\{u_n^i\}$ is a $(\mathrm{PS})_{m_{\lambda,\mu}^i}$ sequence for $I_{\lambda,\mu}$ at the level $m_{\lambda,\mu}^i$. Since $I_{\lambda,\mu}(|u_n^i|)=I_{\lambda,\mu}(u_n^i)$, it follows that $u_n^i\geqslant 0$. By Lemma 2.3 and the inequality $m_{\lambda,\mu}^i<S^{3/2}/(3Q_M^{1/2})$, there exists a subsequence of $\{u_n^i\}$, still denoted by $\{u_n^i\}$, and a function $u^i\in H_r^{1}(\mathbb{R}^3)$ with $u^i\geqslant 0$ such that $u_n^i\to u^i$ strongly in $H_r^{1}(\mathbb{R}^3)$ for all $i$, $1\leqslant i\leqslant k$, as $n\to \infty$. It follows from Lemma 2.5 and Lemma 2.8 that $\|u_n^i\|\geqslant A> 0$ and
$$
\begin{equation*}
\lim_{n\to\infty}I_{\lambda,\mu}(u_n^i)=I_{\lambda,\mu}(u^i)=m_{\lambda,\mu}^i>0,
\end{equation*}
\notag
$$
whence $u^i\not\equiv 0$. Consequently, applying the strong maximum principle, we see that the system (1.1) has at least $k$ positive solutions $u^i$, $1\leqslant i\leqslant k$, for all $\lambda\in(0,\lambda_0)$ and $\mu\in(0,\mu_0)$. The proof is complete. 
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