1. Introduction and statement of main result
In this article, we are devoted to the following biharmonic Choquard type equation
\begin{equation}
\Delta^2u-\beta\Delta u=\lambda u+(I_\mu*F(u))f(u),
\quad\mbox{in}\ \ \mathbb{R}^4,
\end{equation}with prescribed mass
\begin{equation}
\displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^4),
\end{equation} where Δ2 denotes the biharmonic operator, 
$\beta\geq0$, c > 0, 
$\lambda\in \mathbb{R}$, 
$I_\mu=\frac{1}{|x|^\mu}$ with 
$\mu\in (0,4)$, F(u) is the primitive function of f(u), which satisfies some suitable conditions that will be specified later.
The study of (1.1) originates from seeking the standing wave solutions of the following time-dependent nonlinear Schrödinger equation with the mixed dispersion (see [Reference Fibich, Ilan and Papanicolaou17, Reference Karpman22]) and a general Choquard nonlinear term
\begin{equation}
i\psi_t-\Delta^2\psi+\beta\Delta \psi+(I_\mu*F(\psi))f(\psi)=0, \quad\psi(0,x)=\psi_0(x), \quad \psi(t,x):\mathbb{R}\times \mathbb{R}^4\rightarrow \mathbb{C},
\end{equation}where i denotes the imaginary unit and f satisfies:
$(H_1)$ 
$ f(t)\in \mathbb{R}$ for 
$t\in \mathbb{R}$ and 
$f(e^{i\theta }z)=e^{i\theta }f(z)$ for any 
$\theta \in \mathbb{R}$, 
$z\in \mathbb{C}$;
$(H_2)$ 
$\displaystyle F(z)=\int_{0}^{|z|}f(t)dt$ for any 
$z\in \mathbb{C}$.
To look for standing wave solutions 
$\psi(t,x)=e^{-i\lambda t}u(x)$ for (1.3), where 
$\lambda\in \mathbb{R}$ and 
$u\in H^2(\mathbb{R}^4)$ is a time-independent real valued function, one leads to problem (1.1). In particular, we are interested in looking for ground state solutions, i.e., solutions minimizing the associated energy functional among all non-trivial solutions. If 
$\lambda\in \mathbb{R}$ is a fixed parameter, there are many results for equation (1.1) by variational methods, we refer to [Reference Battaglia and Van Schaftingen6, Reference Cingolani, Gallo and Tanaka14, Reference Ghimenti and Van Schaftingen18, Reference Moroz and Van Schaftingen35–Reference Moroz and Van Schaftingen37, Reference Ruiz and Van Schaftingen44] and references therein.
Another alternative choice is to find solutions of (1.1) with prescribed mass, and 
$\lambda\in \mathbb{R}$ arises as a Lagrange multiplier. This type of solution is called normalized solution, and this approach is particularly meaningful from the physical point of view, since, in addition to there is a conservation of mass by 
$(H_1)$ and 
$(H_2)$, the mass has often an important physical meaning, e.g., it shows the power supply in nonlinear optics, or the total number of atoms in Bose-Einstein condensation.
For the nonlinear Schrödinger equation with a normalization constraint
\begin{align}
\begin{split}
\left\{
\begin{array}{ll}
-\Delta u=\lambda u+f(u),
\quad\mbox{in}\ \ \mathbb{R}^N, \\
\displaystyle\int_{\mathbb{R}^N}|u|^2dx=c^2,\quad {u\in H^1(\mathbb{R}^N)}.
\end{array}
\right.
\end{split}
\end{align} If 
$f(u)=|u|^{p-2}u$, by the Gagliardo-Nirenberg inequality [Reference Nirenberg38], 
$\bar{p}:=2+\frac{4}{N}$ is called the L 2-critical exponent. In this case, the associated energy functional of (1.4) is defined by
\begin{equation*}
\mathcal{J}_1(u)=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\frac{1}{p}\int_{\mathbb{R}^N}|u|^pdx.
\end{equation*} If the problem is L 2-subcritical, i.e., 
$p\in(2,2+\frac{4}{N})$, then 
$\mathcal{J}_1$ is bounded from below on 
$\widehat{S}(c):=\{u\in H^1(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^2dx=c^2\}$, so the ground state solution of (1.4) can be found as a global minimizer of 
$\mathcal{J}_1$ on 
$\widehat{S}(c)$. Stuart [Reference Stuart48, Reference Stuart49] first obtained the existence of normalized solutions for (1.4) by the bifurcation theory. Other results for the L 2-subcritical problems can be found, we refer the readers to [Reference Cazenave and Lions11, Reference Lions28, Reference Shibata45] and references therein.
However, if the problem is L 2-critical or supercritical, i.e., 
$p\in[2+\frac{4}{N},2^*)$, where 
$2^*=\infty$ if 
$N\leq2$ and 
$2^*=\frac{2N}{N-2}$ if 
$N\geq3$, then 
$\mathcal{J}_1$ is unbounded from below on 
$\widehat{S}(c)$, so it seems impossible to search for a global minimizer to obtain a solution of (1.4). Furthermore, since 
$\lambda\in \mathbb{R}$ is unknown and 
$H^1(\mathbb{R}^N)\hookrightarrow L^2(\mathbb{R}^N)$ is not compact, some classical methods cannot be directly used to prove the boundedness and compactness of any (PS) sequence. This case was first studied by Jeanjean in [Reference Jeanjean20]. For quite a long time, the work of Jeanjean [Reference Jeanjean20] is the only one in this aspect, and the idea of [Reference Jeanjean20] has been exploited and further developed in the search for normalized solutions when the energy functional is unbounded from below on the L 2 constraint, see [Reference Alves, Ji and Miyagaki1, Reference Bartsch and Soave3–Reference Bartsch and Soave5, Reference Bieganowski and Mederski7, Reference Ding and Zhong15, Reference Jeanjean and Lu21, Reference Li25, Reference Soave46, Reference Soave47, Reference Wei and Wu51, Reference Zhang, Zhang and Zhong56] and so on for normalized solutions in 
$\mathbb{R}^N$, [Reference Noris, Tavares and Verzini39–Reference Pierotti, Verzini and Yu41] for normalized solutions in bounded domains, and [Reference Bartsch, Liu and Liu2, Reference Li and Ye24, Reference Li26, Reference Yao, Chen, Rădulescu and Sun54, Reference Yuan, Chen and Tang55] for normalized solutions of Choquard equations. It is worth pointing out that in [Reference Bartsch and Soave4, Reference Bartsch and Soave5], Bartsch and Soave presented a new approach that is based on a natural constraint associated to the problem and proved the existence of normalized solutions for (1.4) by using a minimax principle based on the homotopy stable family. Borrowing some arguments from Bartsch and Soave [Reference Bartsch and Soave4, Reference Bartsch and Soave5], Jeanjean and Lu [Reference Jeanjean and Lu21] also complemented and generalized the results of [Reference Jeanjean20].
Let us mention the following mixed dispersion biharmonic nonlinear Schrödinger equation with a prescribed L 2-norm constraint
\begin{align}
\begin{split}
\left\{
\begin{array}{ll}
\Delta^2u-\beta\Delta u=\lambda u+f(u),
\quad\mbox{in}\ \ \mathbb{R}^N, \\
\displaystyle\int_{\mathbb{R}^N}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^N).
\end{array}
\right.
\end{split}
\end{align} In this case, this kind of problem gives a new L 2-critical exponent 
$\bar{q}:=2+\frac{8}{N}$. If β > 0, 
$f(u)=|u|^{q-2}u$, and 
$q\in (2,2+\frac{8}{N})$, Bonheure et al. [Reference Bonheure, Casteras, dos Santos and Nascimento8] studied (1.5) and established the existence, qualitative properties of minimizers. By using the minimax principle, Bonheure et al. [Reference Bonheure, Casteras, Gou and Jeanjean9] obtained the existence of ground state solutions, radial positive solutions, and the multiplicity of radial solutions for (1.5) when 
$q\in (2+\frac{8}{N},4^*)$, where 
$4^*=\infty$ if 
$N\leq4$ and 
$4^*=\frac{2N}{N-4}$ if 
$N\geq5$. If β < 0, the problem is more involved, see [Reference Boussaïd, Fernández and Jeanjean10, Reference Luo, Zheng and Zhu31] for 
$q\in (2,2+\frac{8}{N})$ and [Reference Luo and Yang32] for 
$q\in (2+\frac{8}{N},4^*)$. Recently, Fernandez et al. have improved some results of [Reference Bonheure, Casteras, dos Santos and Nascimento8] and [Reference Bonheure, Casteras, Gou and Jeanjean9], we refer to [Reference Fernández, Jeanjean, Mandel and Mariş16] for more details. Moreover, Luo and Zhang [Reference Luo and Zhang30] studied normalized solutions of (1.5) with a general nonlinear term f, where 
$\beta\in \mathbb{R}$ and f satisfies the suitable L 2-subcritical assumptions.
More recently, Chen and Chen [Reference Chen and Chen12] studied normalized solutions for the following biharmonic Choquard equation
\begin{align}
\begin{split}
\left\{
\begin{array}{ll}
\Delta^2 u=\lambda u+\omega|u|^{q-2}u+(I_\mu*|u|^{4_\mu^*})|u|^{4_\mu^*-2}u,
\quad\mbox{in}\ \ \mathbb{R}^N, \\
\displaystyle\int_{\mathbb{R}^N}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^N),
\end{array}
\right.
\end{split}
\end{align} where 
$N\geq5$, ω > 0, 
$2 \lt q \lt 2+\frac{8}{N}$, and 
$4_\mu^*=\frac{2N-\mu}{N-4}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality (see [Reference Lieb and Loss27]). By using the mountain-pass theorem, the authors obtained two solutions, one is the local minimizer, the other one is the critical point of mountain-pass type. It’s easy to find that 
$4_\mu^*=\infty$ if 
$N\leq4$. Moreover, it is well known that in 
$\mathbb{R}^4$, the natural growth is given by the Adams inequality [Reference Ruf and Sani43]. Since the embedding 
$H^2(\mathbb{R}^4)\hookrightarrow L^\infty(\mathbb{R}^4)$ is not compact, by using Jeanjean’s trick [Reference Jeanjean20], the authors in [Reference Chen and Wang13] have studied the existence of normalized solutions for (1.5) with a Choquard nonlinear term involving exponential critical growth in 
$\mathbb{R}^4$. Inspired by [Reference Bartsch and Soave4, Reference Bartsch and Soave5, Reference Jeanjean and Lu21], we are going to give an another view to study the normalized solution of equations (1.1) and (1.2). Compared with the previous work [Reference Chen and Wang13], there are some differences and features of this paper as follows:
(i) Different from [Reference Chen and Wang13], by using the minimax principle based on the homotopy stable family, we work directly in 
$H^2(\mathbb{R}^4)$ to obtain a 
$(PS)_{E(c)}$ sequence, where E(c) is the ground state energy defined in (1.8). Then by showing that E(c) is non-increasing with respect to c and strictly decreasing for some c, we overcome the compactness.
(ii) Based on the Adams functions [Reference Lu and Yang29], we give a more natural condition, see 
$(f_5)$, to estimate the upper bound of E(c), which is crucial to rule out the trivial case for the weak limit of a 
$(PS)_{E(c)}$ sequence. However, this Adams functions cannot be directly applied to study normalized solutions, because the functions are unknown in some ranges and the L 2-norms of u, 
$\nabla u$, 
$\Delta u$ are given as 
$O(1/\log n)$, 
$n\in \mathbb{N}^+$, after a normalization, it is not conducive for us to obtain a refined estimation. With regard to this point, we make some appropriate modifications to the Adams functions, then using the following condition 
$(f_5)$, through a careful calculation, we complete the estimation.
Now, we introduce the precise assumptions on f. Assume that f satisfies:
$(f_1)$ 
$f\in C(\mathbb{R},\mathbb{R})$ is odd or even, 
$\displaystyle\lim\limits_{t\rightarrow0} \frac{|f(t)|}{|t|^\nu}=0$ for some 
$\nu \gt 2-\frac{\mu}{4}$;
$(f_2)$ f has exponential critical growth at infinity, i.e.,
\begin{align*}
\begin{split}
\lim\limits_{|t|\rightarrow+\infty}\frac{|f(t)|}{e^{\alpha t^2}}=\left\{
\begin{array}{ll}
0,&\quad \text{for} \,\,\,\alpha \gt 32\pi^2,\\
+\infty,&\quad \text{for} \,\,\,0 \lt \alpha \lt 32\pi^2;
\end{array}
\right.
\end{split}
\end{align*} 
$(f_3)$ there exists a constant 
$\theta \gt 3-\frac{\mu}{4}$ such that
\begin{equation*}
0 \lt \theta F(t)\leq tf(t), \,\,\,\text{for all}\,\, t\in \mathbb{R}\backslash \{0\};
\end{equation*} 
$(f_4)$ there exist 
$M_0 \gt 0$ and 
$R_0 \gt 0$ such that 
$F(t)\leq M_0|f(t)|$, for all 
$|t|\geq R_0$;
$(f_5)$ there exists ϱ > 0 such that 
$\liminf\limits_{t\rightarrow+\infty}\frac{f(t)}{e^{32\pi^2 t^2}}\geq\varrho$;
$(f_6)$ for any 
$t\in \mathbb{R}\backslash \{0\}$, let 
$\overline{F}(t):=f(t)t-(2-\frac{\mu}{4})F(t)$, then
\begin{equation*}
\frac{\overline{F}(t)}{|t|^{3-\frac{\mu}{4}}}\,\,\text{is non-increasing in}\,\, (-\infty,0)\,\, \text{and non-decreasing in}\,\, (0,+\infty).
\end{equation*}Our main result can be stated as follows:
Theorem 1.1 Assume that f satisfies 
$(f_1)$-
$(f_6)$, then there exists 
$\beta_* \gt 0$ such that for all 
$\beta\in [0,\beta_*)$, problem (1.1) and (1.2) admit a ground state solution.
Remark 1.1. A typical example satisfying 
$(f_1)$-
$(f_6)$ is 
$
f(t)=|t|^{p-2}te^{32\pi t^2}$ for any 
$p \gt \max\{3,\nu+1,\theta\}$.
Remark 1.2. Unlike [Reference Chen and Wang13], which imposes a restriction on f, the main result of this paper is formulated in terms of β.
We define the energy functional 
$\mathcal J:H^2(\mathbb{R}^4)\rightarrow\mathbb{R}$ by
\begin{align*}
\mathcal J(u)=\frac{1}{2}\int_{\mathbb{R}^4}|\Delta u|^2dx+\frac{\beta}{2}\int_{\mathbb{R}^4}|\nabla u|^2dx-\frac{1}{2}\int_{\mathbb{R}^4}(I_\mu*F(u))F(u)dx,
\end{align*} where 
$ H^2(\mathbb{R}^4)=\big\{u\in L^2(\mathbb{R}^4):\nabla u\in L^2(\mathbb{R}^4),\ \Delta u\in L^2(\mathbb{R}^4)\big\}$ endowed with the norm
\begin{equation*}
\|u\|=(\|\Delta u\|_2^2+2\|\nabla u\|_2^2+\|u\|_2^2)^{\frac{1}{2}}.
\end{equation*}Recalling the following interpolation inequality
\begin{equation*}
\int_{\mathbb{R}^4}|\nabla u|^2dx\leq \Big(\int_{\mathbb{R}^4}|\Delta u|^2dx\Big)^{\frac{1}{2}}\Big(\int_{\mathbb{R}^4}| u|^2dx\Big)^{\frac{1}{2}},
\end{equation*} we can see that 
$\|\cdot\|$ is equivalent to the norm 
$ \|u\|_{H^2(\mathbb{R}^4)}:=(\|\Delta u\|_2^2+\|u\|_2^2)^{\frac{1}{2}}$. By 
$(f_1)$ and 
$(f_2)$, fix q > 2, for any ξ > 0 and 
$\alpha \gt 32\pi^2$, there exists a constant 
$C_\xi \gt 0$, which depends on 
$\xi,q,\alpha,\nu$, such that
\begin{equation*}
|f(t)|\leq \xi|t|^\nu+C_\xi|t|^q(e^{\alpha t^2}-1),\quad\text{for all}\,\, t\in \mathbb{R},
\end{equation*} and using 
$(f_3)$, we have
\begin{equation}
|F(t)|\leq \xi|t|^{\nu+1}+C_\xi|t|^{q+1}(e^{\alpha t^2}-1),\quad\text{for all}\,\, t\in \mathbb{R}.
\end{equation} By (1.7), using the Hardy-Littlewood-Sobolev inequality and Adams inequality [Reference Yang53], we obtain 
$\mathcal{J}$ is well defined in 
$H^2(\mathbb{R}^4)$ and 
$\mathcal{J}\in C^1(H^2(\mathbb{R}^4),\mathbb{R})$ with
\begin{align*}
\langle\mathcal J'(u),v\rangle=&\int_{\mathbb{R}^4}\Delta u \Delta vdx+\beta\int_{\mathbb{R}^4}\nabla u \cdot \nabla vdx-\int_{\mathbb{R}^4}(I_\mu*F(u))f(u)vdx,
\end{align*} for any 
$u, v\in H^2(\mathbb{R}^4)$. For any c > 0, set
\begin{equation*}
S(c):=\Big\{u\in H^2(\mathbb{R}^4):\int_{\mathbb{R}^4}|u|^2dx=c^2\Big\}.
\end{equation*}Define
\begin{equation}
E(c):=\inf\limits_{u\in \mathcal{P}(c)}\mathcal J(u),
\end{equation} where 
$\mathcal{P}(c)$ is the Pohozaev manifold defined by
\begin{equation*}\mathcal{P}(c)=\Big\{u\in S(c):P(u)=0\Big\}
\end{equation*}with
\begin{align*}
P(u)&=2\int_{\mathbb{R}^4}|\Delta u |^2\ dx+\beta\int_{\mathbb{R}^4}|\nabla u|^2 dx+\frac{8-\mu}{2} \int_{\mathbb{R}^4} (I_\mu*F(u))F(u)dx\\
\notag& \quad -2\int_{\mathbb{R}^4} (I_\mu*F(u))f(u)u dx.
\end{align*} By Lemma 3.2, we know 
$\mathcal{P}(c)$ is non-empty, and from Lemma 2.8, we can see that any critical point of 
$\mathcal{J}$ on S(c) stays in 
$\mathcal{P}(c)$, thus any critical point u of 
$\mathcal{J}$ on S(c) with 
$\mathcal{J}(u)=E(c)$ is a ground state solution of (1.1) and (1.2).
For any 
$s\in \mathbb{R}$ and 
$u\in H^2(\mathbb{R}^4)$, we define
\begin{equation*}
\mathcal{H}(u,s)(x):=e^{2s}u(e^sx),\quad\text{for a.e.}\,\, x\in \mathbb{R}^4.
\end{equation*} For simplicity, we always write 
$\mathcal{H}(u,s)$. One can easily check that 
$\| \mathcal{H}(u,s)\|_2=\|u\|_2$ for any 
$s\in \mathbb{R}$, and 
$\mathcal{H}(u,s_1+s_2)=\mathcal{H}(\mathcal{H}(u,s_1),s_2)=\mathcal{H}(\mathcal{H}(u,s_2),s_1)$ for any 
$s_1,s_2\in \mathbb{R}$.
Remark 1.3. When f satisfies 
$(f_1)-(f_3)$ and 
$(f_6)$, it follows from Lemma 3.4 that E(c) is well defined and strictly positive. As established in Lemma 5.1, by invoking 
$(f_3)$, 
$(f_4)$, and the minimax principle for homotopy-stable families, we construct a bounded 
$(PS)_{E(c)}$ sequence 
$\{u_n\} \subset H^2(\mathbb{R}^4)$. Up to a subsequence and translations in 
$\mathbb{R}^4$, its weak limit 
$u_c \in H^2(\mathbb{R}^4)$ is a non-trivial weak solution to (1.1). Moreover, define 
$c_1:= \|u_c\|_{L^2} \in (0, c]$, satisfying 
$P(u_c) = 0$. By 
$(f_3)$ and Fatou’s lemma, we deduce that 
$E(c_1) \leq \mathcal{J}(u_c) \leq E(c)$. Thus, if E(c) can be shown to be strictly decreasing for some c, it follows that 
$\mathcal{J}(u_c) = E(c)$ and 
$\|u_c\|_{L^2} = c$, meaning uc is a ground state solution to (1.1) and (1.2). In light of this, studying the monotonicity of the function 
$c \mapsto E(c)$ becomes a natural next step.
Our paper is arranged as follows. The forthcoming section contains some preliminary results. In 
$\S$ 3, we introduce the modified Adams functions, and estimate the upper bound of E(c). The monotonicity of the function 
$c\mapsto E(c)$ is studied in 
$\S$ 4. In 
$\S$ 5, we use the minimax principle based on the homotopy stable family to construct a bounded 
$(PS)_{E(c)}$ sequence. Section 6 is devoted to the proof of theorem 1.1.
In this paper, we use C to denote any positive constants possibly different from line to line. 
$B_r(x)$ denotes the open ball centered at 
$x\in \mathbb{R}^4$ with radius r > 0. 
$(E^*, \|\cdot\|_{*})$ is the dual space of Banach space 
$(E, \|\cdot\|)$. 
$L^p(\mathbb{R}^4)$ is the usual Lebesgue space endowed with the norm 
$\|u\|_{p}=(\int_{\mathbb{R}^4}|u|^pdx)^{\frac{1}{p}}$ when 
$1 \lt p \lt \infty$, 
$\|u\|_\infty=\inf\{C \gt 0, |u(x)|\leq C \,\,\text{a.e. in} \,\,\mathbb{R}^4\}$.
2. Preliminaries
In this section, we present some preliminaries. For nonlocal problems involving Riesz potentials, the Hardy-Littlewood-Sobolev (HLS) inequality, a fundamental tool in the theory, will be employed throughout the sequel.
Proposition 2.1. [Reference Lieb and Loss27, theorem 4.3]
Assume that 
$1 \lt r$, 
$t \lt \infty$, 
$0 \lt \mu \lt 4$ and 
$
\frac{1}{r}+\frac{\mu}{4}+\frac{1}{t}=2.
$ Then there exists 
$C(\mu,r,t) \gt 0$ such that
\begin{align}
\Big|\int_{\mathbb R^{4}}(I_{\mu}\ast g(x))h(x)dx\Big|\leq
C(\mu,r,t)\|g\|_r
\|h\|_t
\end{align} for all 
$g\in L^r(\mathbb{R}^4)$ and 
$h\in L^t(\mathbb{R}^4)$.
Lemma 2.1. (Cauchy-Schwarz type inequality) [Reference Mattner33] For 
$g,h\in L_{loc}^1(\mathbb{R}^4)$, there holds
\begin{equation}
\int_{\mathbb R^{4}}(I_{\mu}\ast |g(x)|)|h(x)|dx\leq \Big(\int_{\mathbb R^{4}}(I_{\mu}\ast |g(x)|)|g(x)|dx\Big)^{\frac{1}{2}}\Big(\int_{\mathbb R^{4}}(I_{\mu}\ast |h(x)|)|h(x)|dx\Big)^{\frac{1}{2}}.
\end{equation}Lemma 2.2. (Gagliardo-Nirenberg inequality) [Reference Nirenberg38] For any 
$u\in H^2(\mathbb{R}^4)$ and 
$p\geq2$, then it holds
\begin{equation}
\|u\|_p\leq B_{p}\|\Delta u\|_2^{\frac{p-2}{p}}\| u\|_2^{\frac{2}{p}},
\end{equation}where Bp is a constant depending on p.
Lemma 2.3. (i) [Reference Yang53, theorem 1.2] If α > 0 and 
$u\in H^2(\mathbb{R}^4)$, then
\begin{equation*}
\int_{\mathbb{R}^4}(e^{\alpha u^2}-1)dx \lt +\infty;
\end{equation*}(ii) [Reference Miyagaki, Santana and Vieira34, proposition 7] There exists a constant C > 0 such that
\begin{equation*}
\sup\limits_{u\in H^2(\mathbb{R}^4), \|\Delta u\|_2\leq1}\int_{\mathbb{R}^4}(e^{\alpha u^2}-1)dx\leq C
\end{equation*} for all 
$0 \lt \alpha\leq 32\pi^2$.
Lemma 2.4. [Reference Kavian23, lemma 4.8] Let 
$\Omega\subseteq\mathbb R^{4}$ be any open set. For 
$1 \lt s \lt \infty$, let 
$\{u_n\}$ be bounded in 
$L^s(\Omega)$ and 
$u_n(x)\rightarrow u(x)$ a.e. in Ω. Then 
$u_n(x)\rightharpoonup u(x) \,\,in \,\,L^s(\Omega)$.
Definition 2.1. [Reference Ghoussoub19, definition 3.1] Let B be a closed subset of X. A class of 
$\mathcal{F}$ of compact subsets of X is a homotopy stable family with boundary B provided
(i) Every set in 
$\mathcal{F}$ contains B;
(ii) For any set 
$A\in \mathcal{F}$ and any 
$\eta\in C([0,1]\times X,X)$ satisfying 
$\eta(t,v)=v$ for all 
$(t,v)\in (\{0\}\times X)\cup([0,1]\times B)$, one has 
$\eta(\{1\}\times A)\in\mathcal{F}$.
Lemma 2.5. [Reference Ghoussoub19, theorem 3.2]
Let ψ be a C 1-functional on a complete connected C 1-Finsler manifold X (without boundary), and consider a homotopy stable family 
$\mathcal{F}$ with a closed boundary B. Set 
$\widetilde{c} =\widetilde{c}(\psi,\mathcal{F})=\inf\limits_{A\in \mathcal{F}}\max\limits_{v\in A}\psi(v)$ and suppose that
\begin{equation*}
\sup\psi(B) \lt \widetilde{c}.
\end{equation*} Then, for any sequence of sets 
$\{A_n\} \subset\mathcal{F}$ satisfying 
$\lim\limits_{n\rightarrow\infty}\sup\limits_{v\in A_n}\psi(v)=\widetilde{c}$, there exists a sequence 
$\{v_n\}\subset X$ such that
(i) 
$\lim\limits_{n\rightarrow\infty}\psi(v_n)=\widetilde{c}$;
(ii) 
$\lim\limits_{n\rightarrow\infty}\|\psi'(v_n)\|_*=0$;
(iii) 
$\lim\limits_{n\rightarrow\infty}dist(v_n,A_n)=0$.
We observe that 
$B=\emptyset$ is admissible, it is sufficient to follow the usual convention of defining 
$\sup\psi(\emptyset)=-\infty$. Since 
$G(u):=\|u\|_2^2-c^2$ is of class C 1 and for any 
$u\in S(c)$, we have 
$\langle G'(u),u\rangle=2c^2 \gt 0$. Therefore, by the implicit function theorem, S(c) is a C 1-Finsler manifold.
Lemma 2.6. Assume that 
$u_n\rightarrow u$ in 
$H^2(\mathbb{R}^4)$, and 
$s_n\rightarrow s$ in 
$\mathbb{R}$ as 
$n\rightarrow\infty$, then 
$\mathcal{H}(u_n,s_n)\rightarrow \mathcal{H}(u,s)$ in 
$H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$.
Proof. We first prove that 
$\mathcal{H}(u_n,s_n)\rightharpoonup\mathcal{H}(u,s)$ in 
$H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$. Taking any 
$\phi\in C_0^\infty(\mathbb{R}^4)$, and for a compact set K containing the support of 
$\phi(e^{-s_n}\cdot)$ for any n large enough, using the Lebesgue dominated convergence theorem, we get
\begin{align*}
\int_{\mathbb{R}^4}e^{2s_n}u_n(e^{s_n}x)\phi(x)dx&= \int_{K}e^{-2s_n}u_n(y)\phi(e^{-s_n}y)dy\\
&\rightarrow \int_{K}e^{-2s}u(y)\phi(e^{-s}y)dy\\& =\int_{\mathbb{R}^4}e^{2s}u(e^{s}x)\phi(x)dx,\quad\text{as}\,\, n\rightarrow\infty.
\end{align*} Similarly, we can show that for any 
$i=1,2,3,4$ and 
$\phi\in C_0^\infty(\mathbb{R}^4)$
\begin{align*}
\int_{\mathbb{R}^4}\phi \frac{\partial (\mathcal{H}(u_n,s_n))}{\partial {x_i}}dx\rightarrow \int_{\mathbb{R}^4}\phi \frac{\partial (\mathcal{H}(u,s))}{\partial {x_i}}dx,\quad\text{as}\,\, n\rightarrow\infty,
\end{align*}and
\begin{align*}
\int_{\mathbb{R}^4}\phi \frac{\partial ^2 (\mathcal{H}(u_n,s_n))}{\partial {x_i}^2}dx\rightarrow \int_{\mathbb{R}^4}\phi \frac{\partial ^2(\mathcal{H}(u,s))}{\partial {x_i}^2}dx,\quad\text{as}\,\, n\rightarrow\infty.
\end{align*} As a consequence, 
$\mathcal{H}(u_n,s_n)\rightharpoonup\mathcal{H}(u,s)$ in 
$H^2(\mathbb{R}^4)$. Furthermore,
\begin{align*}
\|\mathcal{H}(u_n,s_n)\|^2&= \int_{\mathbb{R}^4}|u_n|^2dx+2e^{2s_n}\int_{\mathbb{R}^4}|\nabla u_n|^2dx+e^{4s_n}\int_{\mathbb{R}^4}|\Delta u_n|^2dx\\
&\rightarrow \int_{\mathbb{R}^4}|u|^2dx+2e^{2s}\int_{\mathbb{R}^4}|\nabla u|^2dx+e^{4s}\int_{\mathbb{R}^4}|\Delta u|^2dx\\&=\|\mathcal{H}(u,s)\|^2,\quad\text{as}\,\, n\rightarrow\infty,
\end{align*} thus we can get 
$\|\mathcal{H}(u_n,s_n)-\mathcal{H}(u,s)\|\rightarrow0$ in 
$H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$.
Lemma 2.7. Assume that 
$(f_1)$-
$(f_4)$ hold, let 
$\{u_n\}\subset S(c)$ be a bounded sequence in 
$H^2(\mathbb{R}^4)$, if 
$u_n\rightharpoonup u$ in 
$H^2(\mathbb{R}^4)$ and
\begin{equation}
\int_{\mathbb{R}^4} (I_\mu*F(u_n))f(u_n)u_n dx\leq K_0
\end{equation} for some constant 
$K_0 \gt 0$, then for any 
$\phi\in C_0^\infty(\mathbb{R}^4)$, we have
\begin{equation}
\int_{\mathbb{R}^4}\Delta u_n\Delta\phi dx \rightarrow \int_{\mathbb{R}^4}\Delta u\Delta\phi dx,\,\,\,\text{as}\,\,\, n\rightarrow\infty,
\end{equation}
\begin{equation}
\int_{\mathbb{R}^4}\nabla u_n\cdot\nabla\phi dx \rightarrow \int_{\mathbb{R}^4}\nabla u\cdot\nabla\phi dx,\,\,\,\text{as}\,\,\, n\rightarrow\infty,
\end{equation}
\begin{equation}
\int_{\mathbb{R}^4}u_n\phi dx \rightarrow \int_{\mathbb{R}^4}u\phi dx,\,\,\,\text{as}\,\,\, n\rightarrow\infty,
\end{equation}and
\begin{equation}
\int_{\mathbb{R}^4} (I_\mu*F(u_n))f(u_n)\phi dx\rightarrow \int_{\mathbb{R}^4} (I_\mu*F(u))f(u)\phi dx,\,\,\,\text{as}\,\,\, n\rightarrow\infty.
\end{equation}Proof. For any fixed 
$v\in H^2(\mathbb{R}^4)$, define
\begin{align*}
f_v(u)&:=\int_{\mathbb{R}^4}\Delta u\Delta v dx,\quad g_v(u):=\int_{\mathbb{R}^4}\nabla u\nabla v dx,\quad h_v(u)\\&:=\int_{\mathbb{R}^4}u v dx\quad \text{for every}\,\, u \in H^2(\mathbb{R}^4).
\end{align*}Then, by the Hölder inequality, we have
\begin{equation*}
|f_v(u)|,|g_v(u)|,|h_v(u)|\leq\|v\|\|u\|,
\end{equation*} this yields that fv, gv, and hv are continuous linear functionals on 
$H^2(\mathbb{R}^4)$. Thus, by 
$u_n\rightharpoonup u$ in 
$H^2(\mathbb{R}^4)$ and 
$C_0^\infty(\mathbb{R}^4)$ is dense in 
$H^2(\mathbb{R}^4)$, we prove that (2.5)–(2.7) hold. Now, adopting some ideas of lemma 4.8 in [Reference Qin and Tang42], we will prove (2.8). By Fatou lemma and (2.4), we have
\begin{equation}
\int_{\mathbb{R}^4} (I_\mu*F(u))f(u)u dx\leq K_0.
\end{equation} Let 
$\Omega=supp \phi$. For any given ɛ > 0, let 
$M_\varepsilon=\frac{K_0\|\phi\|_\infty}{\varepsilon}$, then it follows from 
$(f_3)$, (2.4) and (2.9) that,
\begin{equation}
\int_{\{|u_n|\geq M_\varepsilon\}\cup \{|u|=M_\varepsilon\}} (I_\mu*F(u_n))|f(u_n)\phi| dx\leq \frac{2\varepsilon}{K_0}\int_{|u_n|\geq \frac{M_\varepsilon}{2}} (I_\mu*F(u_n))f(u_n)u_n dx\leq 2\varepsilon,
\end{equation}and
\begin{equation}
\int_{|u|\geq M_\varepsilon} (I_\mu*F(u))|f(u)\phi| dx\leq \frac{\varepsilon}{K_0}\int_{|u|\geq M_\varepsilon} (I_\mu*F(u))f(u)u dx\leq \varepsilon.
\end{equation} Since 
$|f(u_n)|{\chi_{|u_n|\leq M_\varepsilon}}\rightarrow |f(u)|{\chi_{|u|\leq M_\varepsilon}}$ a.e. in 
$\Omega\backslash D_\varepsilon$, where 
$D_\varepsilon=\{x\in \Omega:|u(x)|=M_\varepsilon\}$, and for any 
$x\in \Omega$,
\begin{equation*}
|f(u_n)|{\chi_{|u_n|\leq M_\varepsilon}}\leq \max\limits_{|t|\leq M_\varepsilon}|f(t)| \lt \infty,
\end{equation*}using the Lebesgue dominated convergence theorem, we get
\begin{equation}
\lim\limits_{n\rightarrow\infty}\int_{\{\Omega\backslash D_\varepsilon\}\cap\{|u_n|\leq M_\varepsilon\}}|f(u_n)|^{\frac{8}{8-\mu}}dx=\int_{\{\Omega\backslash D_\varepsilon\}\cap\{|u|\leq M_\varepsilon\}}|f(u)|^{\frac{8}{8-\mu}}dx.
\end{equation} Choosing 
$K_\varepsilon \gt R_0$ such that
\begin{equation}
\|\phi\|_\infty\Big(\frac{M_0K_0}{K_\varepsilon}\Big)^{\frac{1}{2}}\Big(\int_{\Omega}|f(u)|^{\frac{8}{8-\mu}}dx\Big)^{\frac{8-\mu}{8}} \lt \varepsilon,
\end{equation}and
\begin{equation}
\int_{|u|\leq M_\varepsilon} (I_\mu*(F(u)\chi _{|u|\geq K_\varepsilon}))|f(u)\phi| dx \lt \varepsilon.
\end{equation} Then from 
$(f_4)$, (2.1), (2.2), (2.12) and (2.13), one has
\begin{align}
&\int_{\{|u_n|\leq M_\varepsilon\}\cap \{|u|\neq M_\varepsilon\}} (I_\mu*(F(u_n)\chi_{|u_n|\geq K_\varepsilon}))|f(u_n)\phi| dx\nonumber\\
&\leq \|\phi\|_\infty \int_{\Omega\backslash D_\varepsilon} (I_\mu*(F(u_n)\chi_{|u_n|\geq K_\varepsilon}))|f(u_n)|\chi_{|u_n|\leq M_\varepsilon} dx\nonumber\\
& \leq\|\phi\|_\infty\Big(\int_{\mathbb{R}^4} (I_\mu*(F(u_n)\chi_{|u_n|\geq K_\varepsilon}))F(u_n)\chi_{|u_n|\geq K_\varepsilon} dx\Big)^{\frac{1}{2}}\nonumber\\
&\quad\times \Big(\int_{\mathbb{R}^4} (I_\mu*(|f(u_n)|\chi_{\{\Omega\backslash D_\varepsilon\}\cap\{|u_n|\leq M_\varepsilon\}} ))|f(u_n)|\chi_{\{\Omega\backslash D_\varepsilon\}\cap\{|u_n|\leq M_\varepsilon\}} dx\Big)^{\frac{1}{2}}\nonumber\\
&\leq \|\phi\|_\infty\Big(\int_{|u_n|\geq K_\varepsilon} (I_\mu*F(u_n))F(u_n) dx\Big)^{\frac{1}{2}}\nonumber\\
&\quad\times \Big(\int_{\mathbb{R}^4} (I_\mu*(|f(u_n)|\chi_{\{\Omega\backslash D_\varepsilon\}\cap\{|u_n|\leq M_\varepsilon\}} ))|f(u_n)|\chi_{\{\Omega\backslash D_\varepsilon\}\cap\{|u_n|\leq M_\varepsilon\}} dx\Big)^{\frac{1}{2}}\nonumber\\
&\leq C\|\phi\|_\infty\Big(\int_{|u_n|\geq K_\varepsilon} (I_\mu*F(u_n))F(u_n) dx\Big)^{\frac{1}{2}}\Big(\int_{\{\Omega\backslash D_\varepsilon\}\cap\{|u_n|\leq M_\varepsilon\}}|f(u_n)|^{\frac{8}{8-\mu}}dx\Big)^{\frac{8-\mu}{8}}\nonumber\\
&\leq C\|\phi\|_\infty\Big(\frac{M_0}{K_\varepsilon}\int_{|u_n|\geq K_\varepsilon} (I_\mu*F(u_n))f(u_n)u_n dx\Big)^{\frac{1}{2}}\Big(\int_{\Omega}|f(u)|^{\frac{8}{8-\mu}}dx\Big)^{\frac{8-\mu}{8}}\nonumber\\
& \leq C\|\phi\|_\infty\Big(\frac{M_0K_0}{K_\varepsilon}\Big)^{\frac{1}{2}}\Big(\int_{\Omega}|f(u)|^{\frac{8}{8-\mu}}dx\Big)^{\frac{8-\mu}{8}} \lt C\varepsilon.
\end{align} For any 
$x\in \mathbb{R}^4$, set
\begin{equation*}
\varpi_n(x):=\int_{\mathbb{R}^4}\frac{|F(u_n)|\chi_{|u_n|\leq K_\varepsilon}}{|x-y|^\mu}dy\quad \text{and}\quad \varpi(x):=\int_{\mathbb{R}^4}\frac{|F(u)|\chi_{|u|\leq K_\varepsilon}}{|x-y|^\mu}dy.
\end{equation*} Then from (1.7), for any 
$x\in \mathbb{R}^4$ and R > 0, one has
\begin{align*}
&|\varpi_n(x)-\varpi(x)|\leq \int_{\mathbb{R}^4}\frac{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}-|F(u)|\chi_{|u|\leq K_\varepsilon}\Big|}{|x-y|^\mu}dy\\
&\quad \leq \Big( \int_{|x-y|\leq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}-|F(u)|\chi_{|u|\leq K_\varepsilon}\Big|^{\frac{4+\mu}{4-\mu}}}dy\Big)^{\frac{4-\mu}{4+\mu}} \Big(\int_{|x-y| \leq R}|x-y|^{-\frac{4+\mu}{2}}dy\Big)^{\frac{2\mu}{4+\mu}}\\
&\qquad +\Big( \int_{|x-y|\geq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}-|F(u)|\chi_{|u|\leq K_\varepsilon}\Big|^{\frac{8-\mu}{8-2\mu}}}dy\Big)^{\frac{8-2\mu}{8-\mu}} \Big(\int_{|x-y| \geq R}|x-y|^{\mu-8}dy\Big)^{\frac{\mu}{8-\mu}}\\
&\quad \leq \Big(\frac{4\pi^2}{4-\mu}R^{\frac{4-\mu}{2}}\Big)^{\frac{2\mu}{4+\mu}}\Big( \int_{|x-y|\leq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}-|F(u)|\chi_{|u|\leq K_\varepsilon}\Big|^{\frac{4+\mu}{4-\mu}}}dy\Big)^{\frac{4-\mu}{4+\mu}}\\ &\qquad+
\Big(\frac{2\pi^2}{(4-\mu)R^{4-\mu}}\Big)^{\frac{\mu}{8-\mu}}\Big( \int_{|x-y|\geq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}-|F(u)|\chi_{|u|\leq K_\varepsilon}\Big|^{\frac{8-\mu}{8-2\mu}}}dy\Big)^{\frac{8-2\mu}{8-\mu}}.
\end{align*}Similar to (2.12), we can prove that
\begin{equation*}
\int_{|x-y|\leq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}-|F(u)|\chi_{|u|\leq K_\varepsilon}\Big|^{\frac{4+\mu}{4-\mu}}}dy\rightarrow0,\quad\text{as}\,\, n\rightarrow\infty.
\end{equation*}By (1.7), we have
\begin{align*}
&\Big( \int_{|x-y|\geq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}-|F(u)|\chi_{|u|\leq K_\varepsilon}\Big|^{\frac{8-\mu}{8-2\mu}}}dy\Big)^{\frac{8-2\mu}{8-\mu}}\\\leq& C\Big(\|u_n\|_{\frac{(8-\mu)(\nu+1)}{8-2\mu}}^{\nu+1}+\|u_n\|_{\frac{(8-\mu)(q+1)}{8-2\mu}}^{q+1}+\|u\|_{\frac{(8-\mu)(\nu+1)}{8-2\mu}}^{\nu+1}+\|u\|_{\frac{(8-\mu)(q+1)}{8-2\mu}}^{q+1}\Big)\leq C.
\end{align*}Choosing R > 0 large enough, then we have
\begin{equation}
\varpi_n(x)\rightarrow \varpi(x) \quad \text{for any}\,\, x\in \mathbb{R}^4.
\end{equation} From (1.7), for any 
$x\in \mathbb{R}^4$, one also has
\begin{align}
\varpi_n(x)&\leq \Big( \int_{|x-y|\leq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}\Big|^{\frac{4+\mu}{4-\mu}}}dy\Big)^{\frac{4-\mu}{4+\mu}}\Big(\int_{|x-y| \leq R}|x-y|^{-\frac{4+\mu}{2}}dy\Big)^{\frac{2\mu}{4+\mu}}\nonumber\\
&\quad+\Big( \int_{|x-y|\geq R}{\Big||F(u_n)|\chi_{|u_n|\leq K_\varepsilon}\Big|^{\frac{8-\mu}{8-2\mu}}}dy\Big)^{\frac{8-2\mu}{8-\mu}}\Big(\int_{|x-y| \geq R}|x-y|^{\mu-8}dy\Big)^{\frac{\mu}{8-\mu}}\nonumber\\
&\leq \Big(\frac{4\pi^2}{4-\mu}R^{\frac{4-\mu}{2}}\Big)^{\frac{2\mu}{4+\mu}}\Big(\frac{\pi^2R^4}{2}\Big)^{\frac{4-\mu}{4+\mu}}\max\limits_{|t|\leq K_\varepsilon}|F(t)|\nonumber\\ &
\quad+
C\Big(\frac{2\pi^2}{(4-\mu)R^{4-\mu}}\Big)^{\frac{\mu}{8-\mu}}\Big(\|u_n\|_{\frac{(8-\mu)(\nu+1)}{8-2\mu}}^{\nu+1}+\|u_n\|_{\frac{(8-\mu)(q+1)}{8-2\mu}}^{q+1}\Big)\leq C.
\end{align}Then
\begin{equation*}
|\varpi_n(x)f(u_n(x))\chi_{|u_n| \leq M_\varepsilon}\phi(x)|\leq C \|\phi\|_\infty\max\limits_{|t|\leq M_\varepsilon}|f(t)|,\quad \text{for any}\,\, x\in \Omega.
\end{equation*}This together with (2.16), using the Lebesgue dominated convergence theorem, yields that
\begin{align}
&\lim\limits_{n\rightarrow\infty}\int_{\{|u_n|\leq M_\varepsilon\}\cap \{|u|\neq M_\varepsilon\}} (I_\mu*(F(u_n)\chi_{|u_n|\leq K_\varepsilon}))|f(u_n)\phi| dx\nonumber\\ &\quad =\int_{|u|\leq M_\varepsilon} (I_\mu*(F(u)\chi_{|u|\leq K_\varepsilon}))|f(u)\phi| dx.
\end{align}By the arbitrariness of ɛ > 0, from (2.10), (2.11), (2.14), (2.15) and (2.18), we get (2.8).
Lemma 2.8. If 
$u\in H^2(\mathbb{R}^4)$ is a critical point of 
$\mathcal J(u)$ on S(c), then 
$u\in \mathcal{P}(c)$.
Proof. If 
$u\in H^2(\mathbb{R}^4)$ is a critical point of 
$\mathcal J(u)$ on S(c), there is 
$\lambda\in\mathbb{R}$ such that
\begin{equation}
\mathcal J'(u)-\lambda u=0\ \ \mbox{in}\ (H^{2}(\mathbb{R}^4))^*.
\end{equation}Testing (2.19) with u, we have
\begin{equation}
\int_{\mathbb{R}^4}|\Delta u |^2\ dx+\beta\int_{\mathbb{R}^4}|\nabla u|^2 dx-\int_{\mathbb{R}^4} (I_\mu*F(u))f(u)u dx-\lambda \int_{\mathbb{R}^4} |u|^2 dx=0.
\end{equation} On the other hand, consider a cut-off function 
$\varphi\in C_0^\infty(\mathbb{R}^4,[0,1])$ such that 
$\varphi(x)=1$ if 
$|x|\leq1$, 
$\varphi(x)=0$ if 
$|x|\geq2$. For any fixed ρ > 0, set 
$\widetilde{u}_\rho(x)=\varphi(\rho x)x\cdot\nabla u(x)$ as a test function of (1.1) to obtain
\begin{equation*}
\int_{\mathbb{R}^4}\Delta u \Delta\widetilde{u}_\rho dx+ \beta\int_{\mathbb{R}^4}\nabla u \cdot\nabla \widetilde{u}_\rho dx-\int_{\mathbb{R}^4} (I_\mu*F(u))f(u)\widetilde{u}_\rho dx-\lambda \int_{\mathbb{R}^4} u\widetilde{u}_\rho dx=0.
\end{equation*}By [Reference Moroz and Van Schaftingen36], we know
\begin{equation*}
\lim_{\rho\rightarrow0}\int_{\mathbb{R}^4}\nabla u\cdot \nabla \widetilde{u}_\rho dx=-\int_{\mathbb{R}^4}|\nabla u|^2 dx,\quad\lim_{\rho\rightarrow0}\int_{\mathbb{R}^4} u\widetilde{u}_\rho dx=-2\int_{\mathbb{R}^4}| u|^2 dx,
\end{equation*}and
\begin{equation*}
\lim_{\rho\rightarrow0} \int_{\mathbb{R}^4} (I_\mu*F(u))f(u)\widetilde{u}_\rho dx=-\frac{8-\mu}{2} \int_{\mathbb{R}^4} (I_\mu*F(u))F(u)dx.
\end{equation*}Integrating by parts, we find that
\begin{align*}
\int_{\mathbb{R}^4}\Delta u \Delta\widetilde{u}_\rho dx&=2\int_{\mathbb{R}^4}\varphi(\rho x)|\Delta u|^2 dx+\int_{\mathbb{R}^4}\varphi(\rho x)\Delta u(x\cdot \nabla (\Delta u))dx\\
&=2\int_{\mathbb{R}^4}\varphi(\rho x)|\Delta u|^2 dx+\int_{\mathbb{R}^4}\varphi(\rho x)(x\cdot \nabla (\frac{|\Delta u|^2}{2}))dx
\\ &=2\int_{\mathbb{R}^4}\varphi(\rho x)|\Delta u|^2 dx-\int_{\mathbb{R}^4}\rho x\cdot\nabla \varphi(\rho x)\frac{|\Delta u|^2}{2}dx-\int_{\mathbb{R}^4}4\varphi(\rho x)\frac{|\Delta u|^2}{2}dx.
\end{align*}The Lebesgue dominated convergence theorem implies that
\begin{equation*}
\lim_{\rho\rightarrow0} \int_{\mathbb{R}^4}\Delta u \Delta\widetilde{u}_\rho dx=0.
\end{equation*}Thus
\begin{equation}
-\beta\int_{\mathbb{R}^4}|\nabla u|^2 dx=-2\lambda\int_{\mathbb{R}^4} |u|^2 dx-\frac{8-\mu}{2} \int_{\mathbb{R}^4} (I_\mu*F(u))F(u)dx.
\end{equation}From (2.20) and (2.21), we deduce that
\begin{align*}
&2\int_{\mathbb{R}^4}|\Delta u |^2\ dx+\beta\int_{\mathbb{R}^4}|\nabla u|^2 dx+\frac{8-\mu}{2} \int_{\mathbb{R}^4} (I_\mu*F(u))F(u)dx\\&\quad -2\int_{\mathbb{R}^4} (I_\mu*F(u))f(u)u dx=0.
\end{align*} That is, 
$u\in \mathcal{P}(c)$.
3. The estimation for the upper bound of E(c)
In this section, we introduce the modified Adams functions and estimate the upper bound of E(c).
Lemma 3.1. Assume that 
$(f_1)$-
$(f_3)$ hold, then for any fixed 
$u\in S(c)$, we have
(i) 
$\mathcal{J}(\mathcal{H}(u,s))\rightarrow0^+$ as 
$s\rightarrow-\infty$;
(ii) 
$\mathcal{J}(\mathcal{H}(u,s))\rightarrow-\infty$ as 
$s\rightarrow+\infty$.
Proof. A straightforward calculation shows that for any q > 2,
\begin{align*}
\|\mathcal{H}(u,s)\|_2& =c,\,\,\, \|\Delta \mathcal{H}(u,s)\|_2=e^{2s} \|\Delta u\|_2,\,\,\, \|\nabla \mathcal{H}(u,s)\|_2=e^{s} \|\nabla u\|_2,\,\,\,\|\mathcal{H}(u,s)\|_q\\&=e^{\frac{2(q-2)s}{q}}\|u\|_q.
\end{align*} So there exists 
$s_1 \lt \lt 0$ such that
\begin{equation*}
\|\Delta \mathcal{H}(u,s)\|_2^2= e^{4s} \|\Delta u\|_2^2 \lt \frac{8-\mu}{8},\quad \text{for all}\,\, s\leq s_1.
\end{equation*} Fix 
$\alpha \gt 32\pi^2$ close to 
$32\pi^2$ and m > 1 close to 1 with 
$\frac{8\alpha m \|\Delta \mathcal{H}(u,s)\|^2}{8-\mu}\leq 32\pi^2$ for all 
$s\leq s_1$. For 
$m'=\frac{m}{m-1}$, using proposition 2.1, Lemma 2.3, the Hölder and Sobolev inequalities, we have
\begin{align}
&\int_{\mathbb{R}^4}(I_\mu*F(\mathcal{H}(u,s)))F(\mathcal{H}(u,s))dx\leq C\|F(\mathcal{H}(u,s))\|_{\frac{8}{8-\mu}}^2\nonumber\\
&\quad \leq C\bigg(\int_{\mathbb{R}^4}\Big(\xi|\mathcal{H}(u,s)|^{\nu+1}+C_\xi|\mathcal{H}(u,s)|^{q+1}(e^{\alpha \mathcal{H}^2(u,s)}-1)\Big)^{\frac{8}{8-\mu}}dx\bigg)^{\frac{8-\mu}{4}}\nonumber\\
&\quad \leq C \bigg(\int_{\mathbb{R}^4}\Big(|\mathcal{H}(u,s)|^{\frac{8(\nu+1)}{8-\mu}}+|\mathcal{H}(u,s)|^{\frac{8(q+1)}{8-\mu}}(e^{\frac{8\alpha \mathcal{H}^2(u,s)}{8-\mu}}-1)\Big)dx\bigg)^{\frac{8-\mu}{4}}\nonumber\\
&\quad \leq C\|\mathcal{H}(u,s)\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+C\bigg(\int_{\mathbb{R}^4}|\mathcal{H}(u,s)|^{\frac{8(q+1)}{8-\mu}}(e^{\frac{8\alpha \mathcal{H}^2(u,s)}{8-\mu}}-1)dx\bigg)^{\frac{8-\mu}{4}}\nonumber\\
&\quad \leq C\|\mathcal{H}(u,s)\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+C\Big(\int_{\mathbb{R}^4}|\mathcal{H}(u,s)|^{\frac{8(q+1)m'}{8-\mu}}dx\Big)^{\frac{8-\mu}{4m'}}\nonumber\\
&\qquad \times \Big(\int_{\mathbb{R}^4}(e^{\frac{8\alpha m \|\Delta \mathcal{H}(u,s)\|^2}{8-\mu}(\frac{\mathcal{H}(u,s)}{\|\Delta\mathcal{H}(u,s)\|})^2}-1)dx\Big)^{\frac{8-\mu}{4m}}\nonumber\\
&\quad \leq C\|\mathcal{H}(u,s)\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+C\|\mathcal{H}(u,s)\|_{\frac{8(q+1)m'}{8-\mu}}^{2(q+1)}\nonumber\\
& \quad =Ce^{(4\nu+\mu-4)s}\|u\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+Ce^{(4q+4-\frac{8-\mu}{m'})s}\|u\|_{\frac{8(q+1)m'}{8-\mu}}^{2(q+1)},\quad \text{for all}\,\, s\leq s_1.
\end{align} Since 
$\beta\geq0$, 
$\nu \gt 2-\frac{\mu}{4}$, q > 2, and mʹ large enough, it follows from (3.1) that
\begin{align*}
\mathcal{J}(\mathcal{H}(u,s))&=\frac{e^{4s}}{2}\int_{\mathbb{R}^4}|\Delta u|^2dx-Ce^{(4\nu+\mu-4)s}\|u\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}\\
&\quad -Ce^{(4q+4-\frac{8-\mu}{m'})s}\|u\|_{\frac{8(q+1)m'}{8-\mu}}^{2(q+1)}\rightarrow 0^+, \quad\text{as}\,\, s\rightarrow-\infty.
\end{align*} For any fixed 
$s \gt \gt 0$, set
\begin{equation*}
\mathcal{M}(t)=\frac{1}{2}\int_{\mathbb{R}^4}(I_\mu*F(tu))F(tu)dx,\quad\text{for}\,\, t \gt 0.
\end{equation*} It follows from 
$(f_3)$ that
\begin{equation*}
\frac{\frac{d\mathcal{M}(t)}{dt}}{\mathcal{M}(t)}\geq\frac{2\theta}{t},\quad\text{for}\,\, t \gt 0.
\end{equation*} Thus, integrating this over 
$[1,e^{2s}]$, we have
\begin{equation}
\frac{1}{2}\int_{\mathbb{R}^4}(I_\mu*F(e^{2s}u))F(e^{2s}u)dx\geq \frac{e^{4\theta s}}{2}\int_{\mathbb{R}^4}(I_\mu*F(u))F(u)dx.
\end{equation}Therefore,
\begin{align*}
\mathcal J(\mathcal{H}(u,s))
\leq&
\frac{e^{4s}}{2}\int_{\mathbb{R}^4}|\Delta u|^2dx+\frac{\beta e^{2s}}{2}\int_{\mathbb{R}^4}|\nabla u|^2dx-\frac{e^{(4\theta+\mu-8) s}}{2}\int_{\mathbb{R}^4}(I_\mu*F(u)F(u)dx.
\end{align*} Since 
$\theta \gt 3-\frac{\mu}{4}$, the above inequality yields that 
$\mathcal{J}(\mathcal{H}(u,s))\rightarrow-\infty$ as 
$s\rightarrow+\infty$.
Lemma 3.2. Assume that f satisfies 
$(f_1)$-
$(f_3)$ and 
$(f_6)$, then for any fixed 
$u\in S(c)$, the following statements hold.
(i) The function 
$g_u(s)={\mathcal J}(\mathcal{H}(u,s))$ achieves its maximum with positive level at a unique point 
$s_u\in \mathbb{R}$ such that 
$\mathcal{H}(u,s_u) \in \mathcal{P}(c)$.
(ii) The mapping 
$u\mapsto s_u$ is continuous in 
$u\in S(c)$.
(iii) 
$s_{-u}=s_u$.
Proof. (i) The proof can be found from [Reference Chen and Wang13, lemma 3.9], we give it here for the completeness. From Lemma 3.1, there exists 
$s_u\in \mathbb{R}$ such that 
$P(\mathcal{H}(u,s_u))=\frac{d}{ds}\mathcal{J}(\mathcal{H}(u,s))|_{s=s_u}=0$, and 
$\mathcal{J}(\mathcal{H}(u,s_u)) \gt 0$.
In the following, we prove the uniqueness of su for any 
$u\in S(c)$. Since
\begin{align*}
P(\mathcal{H}(u,s))=&e^{4s}\Big[2 \int_{\mathbb{R}^4}|\Delta u|^2dx+ \frac{\beta}{e^{2s}}\int_{\mathbb{R}^4}|\nabla u|^2dx -2\int_{\mathbb{R}^4}\Big(I_\mu*\frac{F(e^{2s}u)}{(e^{2s})^{3-\frac{\mu}{4}}}\Big)\frac{\overline{F}(e^{2s}u)}{(e^{2s})^{3-\frac{\mu}{4}}}\Big]dx.
\end{align*}Denote
\begin{equation*}
\psi(s)=\int_{\mathbb{R}^4}\Big(I_\mu*\frac{F(e^{2s}u)}{(e^{2s})^{3-\frac{\mu}{4}}}\Big)\frac{\overline{F}(e^{2s}u)}{(e^{2s})^{3-\frac{\mu}{4}}}dx.
\end{equation*} For any 
$t\in \mathbb{R}\backslash \{0\}$, from 
$(f_3)$ and 
$(f_6)$, we see that 
$\frac{F(st)}{s^{3-\frac{\mu}{4}}}$ is increasing in 
$s\in (0,+\infty)$ and 
$\frac{\overline{F}(st)}{s^{3-\frac{\mu}{4}}}$ is non-decreasing in 
$s\in (0,+\infty)$. This fact implies 
$\psi(s)$ is increasing in 
$s\in \mathbb{R}$ and there is at most one 
$s_u\in \mathbb{R}$ such that 
$\mathcal{H}(u,s_u) \in \mathcal{P}(c)$.
(ii) By (i), the mapping 
$u\mapsto s_u$ is well defined. Let 
$\{u_n\}\subset S(c)$ be any sequence such that 
$u_n\rightarrow u$ in 
$H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$. We only need to prove that up to a subsequence, 
$s_{u_n}\rightarrow s_u$ in 
$\mathbb{R}$ as 
$n\rightarrow\infty$.
We first show that 
$\{s_{u_n}\}$ is bounded. If up to a subsequence, 
$s_{u_n}\rightarrow+\infty$ as 
$n\rightarrow\infty$, then by (3.2) and 
$u_n\rightarrow u\neq0$ in 
$ H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$, we have
\begin{align*}
0&\leq \lim\limits_{n\rightarrow\infty}e^{-4s_{u_n}}\mathcal J(\mathcal{H}(u_n,s_{u_n}))\\
&\leq \lim\limits_{n\rightarrow\infty}\frac{1}{2}\Big[\int_{\mathbb{R}^4}|\Delta u_n|^2dx+\frac{\beta}{e^{2s_{u_n}}}\int_{\mathbb{R}^4}|\nabla u_n|^2dx-e^{(4\theta+\mu-12)s_{u_n}}\\
&\int_{\mathbb{R}^4}(I_\mu*F(u_n))F(u_n)dx\Big]=-\infty,
\end{align*} which is a contradiction. Therefore, 
$\{s_{u_n}\}$ is bounded from above. On the other hand, by Lemma 2.6, 
$\mathcal{H}(u_n,s_{u})\rightarrow \mathcal{H}(u,s_{u})$ in 
$H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$, it follows from (i) that
\begin{equation*}
\mathcal J(\mathcal{H}(u_n,s_{u_n}))\geq \mathcal J(\mathcal{H}(u_n,s_{u}))=\mathcal J(\mathcal{H}(u,s_{u}))+o_n(1),
\end{equation*}and thus
\begin{equation*}
\liminf\limits_{n\rightarrow\infty}\mathcal J(\mathcal{H}(u_n,s_{u_n}))\geq \mathcal J(\mathcal{H}(u,s_{u})) \gt 0.
\end{equation*} If up to a subsequence, 
$s_{u_n}\rightarrow-\infty$ as 
$n\rightarrow\infty$, using 
$(f_3)$, we get
\begin{equation*}
\mathcal J(\mathcal{H}(u_n,s_{u_n}))\leq \frac{e^{4s_{u_n}}}{2}\|\Delta u\|_2^2+\frac{\beta e^{2s_{u_n}}}{2}\|\nabla u\|_2^2\rightarrow0, \quad\text{as}\,\,\, n\rightarrow\infty,
\end{equation*} which is an absurd. Therefore, 
$\{s_{u_n}\}$ is bounded from below. Up to a subsequence, we assume that 
$s_{u_n}\rightarrow s_*$ as 
$n\rightarrow\infty$. Recalling that 
$u_n\rightarrow u$ in 
$H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$, then 
$\mathcal{H}(u_n,s_{u_n})\rightarrow \mathcal{H}(u,s_{*})$ in 
$H^2(\mathbb{R}^4)$ as 
$n\rightarrow\infty$. Since 
$P(\mathcal{H}(u_n,s_{u_n}))=0$ for any 
$n\in \mathbb{N}^+$, it follows that 
$P(\mathcal{H}(u,s_{*}))=0$. By the uniqueness of su, we get 
$s_u=s_*$, thus (ii) is proved.
(iii) Since f is odd or even in 
$\mathbb{R}$, it is easy to find that 
$\int_{\mathbb{R}^4}(I_\mu*F(u))F(u)dx$ and 
$\int_{\mathbb{R}^4}(I_\mu*F(u))f(u)udx$ are even in 
$H^2(\mathbb{R}^4)$, thus
\begin{equation*}
P(\mathcal{H}(-u,s_u))=P(-\mathcal{H}(u,s_u))=P(\mathcal{H}(u,s_u)),
\end{equation*} which implies that 
$s_{-u}=s_u$.
Lemma 3.3. Assume that 
$(f_1)$-
$(f_3)$ hold, then there exists δ > 0 small enough such that
\begin{equation*}
\mathcal J(u)
\geq\frac{1}{4}\int_{\mathbb{R}^4}|\Delta u|^2dx+\frac{\beta}{2}\int_{\mathbb{R}^4}|\nabla u|^2dx\quad\text{and}\quad P(u)
\geq\int_{\mathbb{R}^4}|\Delta u|^2dx+\beta\int_{\mathbb{R}^4}|\nabla u|^2dx
\end{equation*} for all 
$u\in S(c)$ satisfying 
$\|\Delta u\|_2\leq \delta$.
Proof. If 
$\delta \lt \sqrt{\frac{8-\mu}{8}}$, then 
$\|\Delta u\|_2^2 \lt \frac{8-\mu}{8}$. Fix 
$\alpha \gt 32\pi^2$ close to 
$32\pi^2$ and m > 1 close to 1 such that
\begin{equation*}
\frac{8\alpha m\| \Delta u\|_2^2}{8-\mu}\leq 32\pi^2.
\end{equation*} Arguing as (3.1), for 
$m'=\frac{m}{m-1}$, by (2.3), we have
\begin{align*}
\int_{\mathbb{R}^4}(I_\mu*F(u))F(u)dx&\leq C\|u\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+C\|u\|_{\frac{8(q+1)m'}{8-\mu}}^{2(q+1)}\\ &\leq Cc^{4-\frac{\mu}{2}}\|\Delta u\|_2^{2\nu+\frac{\mu}{2}-2}+C c^{\frac{8-\mu}{2m'}} \|\Delta u\|_2^{2(q+1)-\frac{8-\mu}{2m'}}\\
&\leq\Big [Cc^{4-\frac{\mu}{2}}\delta^{2\nu+\frac{\mu}{2}-4}+C c^{\frac{8-\mu}{2m'}} \delta^{2q-\frac{8-\mu}{2m'}}\Big]\|\Delta u\|_2^2
\end{align*}and
\begin{align*}
\int_{\mathbb{R}^4}(I_\mu*F(u))f(u)udx&\leq C\|u\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+C\|u\|_{\frac{8(q+1)m'}{8-\mu}}^{2(q+1)}
\\ &\leq Cc^{4-\frac{\mu}{2}}\|\Delta u\|_2^{2\nu+\frac{\mu}{2}-2}+C c^{\frac{8-\mu}{2m'}} \|\Delta u\|_2^{2(q+1)-\frac{8-\mu}{2m'}}\\
&\leq\Big [Cc^{4-\frac{\mu}{2}}\delta^{2\nu+\frac{\mu}{2}-4}+C c^{\frac{8-\mu}{2m'}} \delta^{2q-\frac{8-\mu}{2m'}}\Big]\|\Delta u\|_2^2.
\end{align*} Since 
$\nu \gt 2-\frac{\mu}{4}$, q > 2 and mʹ large enough, choosing 
$\delta \lt \sqrt{\frac{8-\mu}{8}}$ small enough, we conclude the result.
Lemma 3.4. Assume that f satisfies 
$(f_1)$-
$(f_3)$ and 
$(f_6)$, then we have 
$\inf\limits_{u\in \mathcal{P}(c)}\|\Delta u\|_2 \gt 0$ and 
$E(c) \gt 0$.
Proof. By Lemma 3.2, we obtain 
$\mathcal{P}(c)\neq\emptyset$. Suppose that there exists a sequence 
$\{u_n\}\subset \mathcal{P}(c)$ such that 
$\|\Delta u_n\|_2\rightarrow0$ as 
$n\rightarrow\infty$, then by Lemma 3.3, up to a subsequence,
\begin{equation*}
0=P(u_n)\geq\int_{\mathbb{R}^4}|\Delta u_n|^2dx+\beta\int_{\mathbb{R}^4}|\nabla u_n|^2dx\geq0,
\end{equation*} which implies that 
$\int_{\mathbb{R}^4}|\Delta u_n|^2dx=\beta\int_{\mathbb{R}^4}|\nabla u_n|^2dx=0$ for any 
$n\in \mathbb{N}^+$. Hence, by 
$(f_3)$ and 
$P(u_n)=0$, we have
\begin{align*}
0&=\int_{\mathbb{R}^4}(I_\mu*F(u_n))\Big(\frac{8-\mu}{2}F(u_n)-2f(u_n)u_n\Big)dx\\
&\quad \leq \int_{\mathbb{R}^4}(I_\mu*F(u_n))\Big(\frac{8-\mu}{2\theta}-2\Big)f(u_n)u_ndx\leq0.
\end{align*} So 
$u_n\rightarrow0$ a.e. in 
$\mathbb{R}^4$, which is contradict to c > 0.
For any 
$u\in \mathcal{P}(c)$, by Lemma 3.2, we have
\begin{equation*}
\mathcal{J}(u)=\mathcal{J}(\mathcal{H}(u,0))\geq \mathcal{J}(\mathcal{H}(u,s))\quad\text{for all}\,\, s\in \mathbb{R}.
\end{equation*} Let δ > 0 be the number given by Lemma 3.3 and 
$2s:=\ln\frac{\delta}{\|\Delta u\|_2}$. Then 
$\|\Delta(\mathcal{H}(u,s))\|_2=\delta$, by 
$\beta\geq0$ and Lemma 3.3, we deduce that
\begin{equation*}
\mathcal{J}(u)\geq \mathcal{J}(\mathcal{H}(u,s))\geq \frac{1}{4}\int_{\mathbb{R}^4}|\Delta \mathcal{H}(u,s)|^2dx+\frac{\beta}{2}\int_{\mathbb{R}^4}|\nabla \mathcal{H}(u,s)|^2dx\geq\frac{\delta^2}{4} \gt 0.
\end{equation*} By the arbitrariness of 
$u\in \mathcal{P}(c)$, we derive the conclusion.
The modified Adams functions
In order to estimate the upper bound of E(c), let us introduce the modified Adams functions as follows. For any 
$\varphi(t)\in C_0^\infty([0,\infty),[0,1])$ such that 
$\varphi(t)=1$ if 
$0\leq t\leq1$, 
$\varphi(t)=0$ if 
$t\geq2$. Define a sequence of functions 
$\tilde{\omega}_n$ by
\begin{align*}
\begin{split}
\tilde{\omega}_n(x)=\left\{
\begin{array}{ll}
\sqrt{\frac{\log n}{8\pi^2}}+\frac{1-n^2|x|^2}{\sqrt{32\pi^2\log n}},&\quad \text{for} \,\,\,|x|\leq \frac{1}{n},\\
-\frac{\log |x|}{\sqrt{8\pi^2\log n}},&\quad \text{for} \,\,\,\frac{1}{n} \lt |x|\leq1,\\
-\frac{\varphi(|x|)\log |x|}{\sqrt{8\pi^2\log n}},&\quad \text{for} \,\,\,1 \lt |x| \lt 2,\\
0,&\quad \text{for} \,\,\,|x|\geq2.
\end{array}
\right.
\end{split}
\end{align*} One can check that 
$\tilde{\omega}_n\in H^2(\mathbb{R}^4)$. A straightforward calculation shows that
\begin{equation*}
\|\tilde{\omega}_n\|_2^2=\frac{1+32M_1}{128\log n}-\frac{1}{96 n^4}-\frac{1}{192 n^4 \log n}=\frac{1+32M_1}{128\log n}+o(\frac{1}{\log ^4n}),
\end{equation*}
\begin{equation*}
\|\nabla\tilde{\omega}_n\|_2^2=\frac{1+2M_2}{8\log n}-\frac{1}{12n^2 \log n}=\frac{1+2M_2}{8\log n}+o(\frac{1}{\log ^3n}),
\end{equation*}and
\begin{equation*}
\|\Delta\tilde{\omega}_n\|_2^2=1+\frac{4+M_3}{4\log n},
\end{equation*}where
\begin{equation*}
M_1=\int_1^2\varphi^2(r)r^3\log ^2rdr,
\end{equation*}
\begin{equation*}
M_2=\int_1^2\Big(\varphi'(r)\log r+\frac{\varphi(r)}{r}\Big)^2r^3dr,
\end{equation*}and
\begin{equation*}
M_3=\int_1^2\Big(\varphi''(r)\log r+\frac{-\varphi(r)+3\varphi'(r)+2r\varphi'(r)+3r\varphi'(r)\log r}{r^2}\Big)r^3dr.
\end{equation*} Let 
$\omega_n=\frac{c\tilde{\omega}_n}{\|\tilde{\omega}_n\|_2}$. Then 
$\omega_n\in S(c)$ and
\begin{equation}
\|\nabla \omega_n\|_2^2=\frac{c^2(\frac{1+2M_2}{8\log n}+o(\frac{1}{\log ^3n}))}{\frac{1+32M_1}{128\log n}+o(\frac{1}{\log ^4n})}=\frac{16c^2(1+2M_2)}{1+32M_1}\Big(1+o(\frac{1}{\log ^2n})\Big),
\end{equation}
\begin{equation}
\|\Delta \omega_n\|_2^2=\frac{c^2(1+\frac{4+M_3}{4\log n})}{\frac{1+32M_1}{128\log n}+o(\frac{1}{\log ^4n})}=\frac{128c^2}{1+32M_1}\Big(\frac{4+M_3}{4}+\log n+o(\frac{1}{\log ^2n})\Big).
\end{equation}Furthermore, we have
\begin{align}
\begin{split}
\omega_n(x)=\left\{
\begin{array}{ll}
\frac{c(1+o(\frac{1}{\log ^3n}))}{\sqrt{\frac{1+32M_1}{128}}}\Big( \frac{\log n}{\sqrt{8\pi^2}}+\frac{1-n^2|x|^2}{\sqrt{32\pi^2}}\Big),&\quad \text{for} \,\,\,|x|\leq \frac{1}{n},\\
-\frac{c(1+o(\frac{1}{\log ^3n}))}{\sqrt{\frac{1+32M_1}{128}}}\frac{\log |x|}{\sqrt{8\pi^2}},&\quad \text{for} \,\,\,\frac{1}{n} \lt |x|\leq1,\\
-\frac{c(1+o(\frac{1}{\log ^3n}))}{\sqrt{\frac{1+32M_1}{128}}}\frac{\varphi(|x|)\log |x|}{\sqrt{8\pi^2}},&\quad \text{for} \,\,\,1 \lt |x| \lt 2,\\
0,&\quad \text{for} \,\,\,|x|\geq2.
\end{array}
\right.
\end{split}
\end{align}For any t > 0, let
\begin{align*}
g_n(t)&:=
\mathcal J(t\omega_n(t^{\frac{1}{2}}x))=\frac{t^2}{2}\int_{\mathbb{R}^4}|\Delta \omega_n|^2dx+\frac{\beta t}{2}\int_{\mathbb{R}^4}|\nabla \omega_n|^2dx-\frac{t^{\frac{\mu}{2}-4}}{2}\\&\quad \int_{\mathbb{R}^4}(I_\mu*F(t\omega_n))F(t\omega_n)dx.
\end{align*} By lemmas 3.2 and 3.4, we know 
$E(c)=\inf\limits_{u\in S(c)}\max\limits_{s\in \mathbb{R}}\mathcal J(\mathcal{H}(u,s)) \gt 0$, this together with 
$\omega_n\in S(c)$ yields that
\begin{equation*}
0 \lt E(c)\leq\max\limits_{s\in \mathbb{R}}\mathcal J(\mathcal{H}(\omega_n,s))=\max\limits_{t \gt 0}g_n(t).
\end{equation*}Lemma 3.5. Assume that 
$(f_1)$-
$(f_3)$ hold, then for any fixed 
$n\in \mathbb{N}^+$, 
$\max\limits_{t \gt 0}g_n(t)$ is attained at some 
$t_n \gt 0$.
Proof. For any fixed 
$n\in \mathbb{N}^+$, as t > 0 small, fix 
$\alpha \gt 32\pi^2$ close to 
$32\pi^2$ and m > 1 close to 1 such that
\begin{equation*}
\frac{8\alpha m\| \Delta(t\omega_n)\|_2^2}{8-\mu}\leq 32\pi^2.
\end{equation*} Arguing as (3.1), for 
$m'=\frac{m}{m-1}$, we have
\begin{align*}
\frac{t^{\frac{\mu}{2}-4}}{2}\int_{\mathbb{R}^4}(I_\mu*F(t\omega_n))F(t\omega_n)dx&\leq
Ct^{\frac{\mu}{2}-4}\|t\omega_n\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+Ct^{\frac{\mu}{2}-4}\|t\omega_n\|_{\frac{8(q+1)m'}{8-\mu}}^{2(q+1)}\\
&=Ct^{2\nu-2+\frac{\mu}{2}}\|\omega_n\|_{\frac{8(\nu+1)}{8-\mu}}^{2(\nu+1)}+Ct^{2q-2+\frac{\mu}{2}}\|\omega_n\|_{\frac{8(q+1)m'}{8-\mu}}^{2(q+1)},
\end{align*} where 
$\nu \gt 2-\frac{\mu}{4}$ and q > 2. So 
$g_n(t) \gt 0$ for t > 0 small enough. By (3.2), we obtain
\begin{equation*}
\frac{t^{\frac{\mu}{2}-4}}{2}\int_{\mathbb{R}^4}(I_\mu*F(t\omega_n))F(t\omega_n)dx\geq \frac{t^{2\theta +\frac{\mu}{2}-4 }}{2}\int_{\mathbb{R}^4}(I_\mu*F(\omega_n))F(\omega_n)dx.
\end{equation*} Since 
$\theta \gt 3-\frac{\mu}{4}$, we have that 
$g_n(t) \lt 0$ for t > 0 large enough. Thus 
$\max\limits_{t \gt 0}g_n(t)$ is attained at some 
$t_n \gt 0$.
Lemma 3.6. Assume that f satisfies 
$(f_1)$-
$(f_3)$ and 
$(f_5)$, then 
$\max\limits_{t \gt 0}g_n(t) \lt \frac{8-\mu}{16}$ for 
$n\in \mathbb{N}^+$ large enough.
Proof. By Lemma 3.5, 
$\max\limits_{t \gt 0}g_n(t)$ is attained at some 
$t_n \gt 0$ and thus 
$g'_n(t_n)=0$. By 
$(f_3)$,
\begin{align}
t_n^2\int_{\mathbb{R}^4}|\Delta \omega_n|^2dx+\frac{\beta }{2}t_n\int_{\mathbb{R}^4}|\nabla \omega_n|^2dx&=\frac{\mu-8}{4}t_n^{\frac{\mu}{2}-4}\int_{\mathbb{R}^4}(I_\mu*F(t_n\omega_n))F(t_n\omega_n)dx\nonumber\\
&\quad+t_n^{\frac{\mu}{2}-4}\int_{\mathbb{R}^4}(I_\mu*F(t_n\omega_n))f(t_n\omega_n)t_n\omega_ndx\nonumber\\
&\geq \frac{4\theta+\mu-8}{4\theta}t_n^{\frac{\mu}{2}-4}\int_{\mathbb{R}^4}(I_\mu*F(t_n\omega_n))f(t_n\omega_n)t_n\omega_ndx.
\end{align}Note that
\begin{equation*}
\liminf\limits_{t\rightarrow\infty}\frac{tF(t)}{e^{32\pi^2t^2}}\geq \liminf\limits_{t\rightarrow\infty}\frac{\int_0^tsf(s)ds}{e^{32\pi^2 t^2}}=\liminf\limits_{t\rightarrow\infty}\frac{f(t)}{64\pi^2 e^{32\pi^2 t^2}}.
\end{equation*} This with 
$(f_5)$ yields that, for any ɛ > 0. there exists 
$R_\varepsilon \gt 0$ such that for any 
$t\geq R_\varepsilon$,
\begin{equation}
f(t)\geq (\varrho-\varepsilon)e^{32\pi^2 t^2},\quad tF(t)\geq \frac{\varrho-\varepsilon}{64\pi^2}e^{32\pi^2 t^2}.
\end{equation} Case 1: 
$\lim\limits_{n\rightarrow\infty}t_n^2\log n=0$. Then 
$\lim\limits_{n\rightarrow\infty}t_n=0$, by (3.3) and (3.4), we have that 
$\frac{t_n^2}{2}\int_{\mathbb{R}^4}|\Delta \omega_n|^2dx\rightarrow 0$, 
$\frac{\beta t_n}{2}\int_{\mathbb{R}^4}|\nabla \omega_n|^2dx\rightarrow0$ as 
$n\rightarrow\infty$. Noting that 
$F(t_n\omega_n) \gt 0$ by 
$(f_3)$, so we have
\begin{equation*}
0 \lt g_n(t_n)\leq\frac{t_n^2}{2}\int_{\mathbb{R}^4}|\Delta \omega_n|^2dx+\frac{\beta t_n}{2}\int_{\mathbb{R}^4}|\nabla \omega_n|^2dx,
\end{equation*} thus 
$\lim\limits_{n\rightarrow\infty}g_n(t_n)=0$, and we conclude.
Case 2: 
$\lim\limits_{n\rightarrow\infty}t_n^2\log n=l\in (0,\infty]$. We claim that 
$l \lt \infty$. Otherwise, if 
$l=\infty$, then 
$\lim\limits_{n\rightarrow\infty}(t_n\log n)=\infty$. By (3.3)–(3.5) and (3.6)–(3.7), we have
\begin{align*}
&\frac{128c^2t_n^2}{1+32M_1}\Big(\frac{4+M_3}{4}+\log n+o(\frac{1}{\log ^2n})\Big)+\frac{8\beta t_nc^2(1+2M_2)}{1+32M_1}\Big(1+o(\frac{1}{\log ^2n})\Big)\\
&\geq \frac{4\theta+\mu-8}{4\theta}t_n^{\frac{\mu}{2}-4}\int_{B_{\frac{1}{n}}(0)}\int_{B_{\frac{1}{n}}(0)}\frac{F(t_n\omega_n(y))f(t_n\omega_n (x)) t_n\omega_n(x)}{|x-y|^\mu}dxdy\\
&\geq \frac{(4\theta+\mu-8)(\varrho-\varepsilon)^2}{256\theta \pi^2}t_n^{\frac{\mu}{2}-4}e^{\frac{1024 c^2 t_n^2 \log^2 n(1+o(\frac{1}{\log^3 n}))}{1+32M_1}}\int_{B_{\frac{1}{n}}(0)}\int_{B_{\frac{1}{n}}(0)}\frac{dxdy}{|x-y|^\mu}.
\end{align*} Since 
$B_{\frac{1}{n}-|x|}(0)\subset B_{\frac{1}{n}}(x)$ for any 
$|x|\leq \frac{1}{n}$, the last integral can be estimated as follows
\begin{align*}
\int_{B_{\frac{1}{n}}(0)}\int_{B_{\frac{1}{n}}(0)}\frac{dxdy}{|x-y|^\mu}&=\int_{B_{\frac{1}{n}}(0)}dx\int_{B_{\frac{1}{n}}(x)}\frac{dz}{|z|^\mu}\\
&\geq\int_{B_{\frac{1}{n}}(0)}dx\int_{B_{\frac{1}{n}-|x|}(0)}\frac{dz}{|z|^\mu}\\
&=\frac{2\pi^2}{4-\mu} \int_{B_{\frac{1}{n}}(0)}\Big(\frac{1}{n}-|x|\Big)^{4-\mu}dx\\
&=\frac{4\pi^4}{4-\mu} \int_{0}^{\frac{1}{n}}\Big(\frac{1}{n}-r\Big)^{4-\mu}r^3dr\\
&=\frac{24\pi^4}{(4-\mu)(5-\mu)(6-\mu)(7-\mu)(8-\mu)n^{8-\mu}}=\frac{C_\mu}{n^{8-\mu}},
\end{align*}where
\begin{equation*}
C_\mu=\frac{24\pi^4}{(4-\mu)(5-\mu)(6-\mu)(7-\mu)(8-\mu)}.
\end{equation*}Consequently, we obtain
\begin{align}
&\frac{128 c^2t_n^2}{1+32M_1}\Big(\frac{4+M_3}{4}+\log n+o(\frac{1}{\log ^2n})\Big)+\frac{8\beta t_nc^2(1+2M_2)}{1+32M_1}\Big(1+o(\frac{1}{\log ^2n})\Big)\nonumber\\
&\geq \frac{(4\theta+\mu-8)C_\mu(\varrho-\varepsilon)^2}{256\theta \pi^2}t_n^{\frac{\mu}{2}-4}e^{\Big(\frac{1024 c^2 t_n^2 \log n(1+o(\frac{1}{\log^3 n}))}{1+32M_1}-(8-\mu)\Big)\log n},
\end{align} which is a contradiction. Thus 
$l\in(0,\infty)$, and 
$\lim\limits_{n\rightarrow\infty}t_n=0$, 
$\lim\limits_{n\rightarrow\infty}(t_n\log n)=\infty$. By (3.8), letting 
$n\rightarrow\infty$, we have that
\begin{equation*}
0 \lt l \lt \frac{(1+32M_1)(8-\mu)}{1024c^2}.
\end{equation*} Otherwise, if 
$l \gt \frac{(1+32M_1)(8-\mu)}{1024c^2}$, the right side of (3.8) approaches to infinity as 
$n\rightarrow\infty$, which is impossible. If 
$l=\frac{(1+32M_1)(8-\mu)}{1024c^2}$, by the definition of ωn, we can find that
\begin{equation*}
A_n:=\frac{1024 c^2 t_n^2 \log n(1+o(\frac{1}{\log^3 n}))}{1+32M_1}-(8-\mu)\rightarrow 0^+,\quad \text{as}\,\, n\rightarrow\infty
\end{equation*}and using the Taylor’s formula, we have
\begin{equation*}
n^{A_n}=1+A_n\log n+{\frac{A_n^2\log^2 n}{2}}+\cdots\geq1.
\end{equation*}Thus
\begin{equation*}
\frac{128 c^2t_n^2 \log n}{1+32M_1}\geq \frac{(4\theta+\mu-8)C_\mu(\varrho-\varepsilon)^2}{256\theta \pi^2} t_n^{\frac{\mu}{2}-4},
\end{equation*}which is a contradiction. Therefore,
\begin{align*}
\lim\limits_{n\rightarrow\infty}g_n(t_n)&\leq \lim\limits_{n\rightarrow\infty}\Big(\frac{ t_n^2}{2}\int_{\mathbb{R}^4}|\Delta \omega_n|^2dx+\frac{\beta t_n}{2}\int_{\mathbb{R}^4}|\nabla \omega_n|^2dx\Big)=\frac{64 c^2 l }{1+32M_1} \lt \frac{8-\mu}{16}.
\end{align*}This completes the proof.
4. The monotonicity of the function 
$c\mapsto E(c)$
To guarantee the weak limit of a 
$(PS)_{E(c)}$ sequence is a ground state solution of (1.1) and (1.2), in this section, we study the monotonicity of the function 
$c\mapsto E(c)$.
Lemma 4.1. Assume that f satisfies 
$(f_1)$-
$(f_3)$ and 
$(f_6)$, then the function 
$c\mapsto E(c)$ is non-increasing on 
$(0,+\infty)$.
Proof. For any given c > 0, if 
$\tilde{c} \gt c$, we prove that 
$E(\tilde{c})\leq E(c)$. By the definition of E(c), for any ɛ > 0, there exists 
$u\in \mathcal{P}(c)$ such that
\begin{equation}
\mathcal J(u)\leq E(c)+\frac{\varepsilon}{3}.
\end{equation} Consider a cut-off function 
$\phi\in C_0^\infty(\mathbb{R}^4,[0,1])$ such that 
$\phi(x)=1$ if 
$|x|\leq1$, 
$\phi(x)=0$ if 
$|x|\geq2$. For any small δ > 0, define 
$u_\delta(x)=\phi(\delta x)u(x)\in H^2(\mathbb{R}^4)\backslash\{0\}$, then 
$u_\delta\rightarrow u$ in 
$H^2(\mathbb{R}^4)$ as 
$\delta\rightarrow0^+$. By lemmas 2.6 and 3.2, we have 
$s_{u_\delta}\rightarrow s_u=0$ in 
$\mathbb{R}$ as 
$\delta\rightarrow0^+$ and
\begin{equation*}
\mathcal{H}(u_\delta,s_{u_\delta})\rightarrow \mathcal{H}(u,s_{u})=u \quad\text{in}\,\, H^2(\mathbb{R}^4)\,\, \text{as}\,\, \delta\rightarrow0^+.
\end{equation*} Fix 
$\delta_0 \gt 0$ small enough such that
\begin{equation}
\mathcal J(\mathcal{H}(u_{\delta_0},s_{u_{\delta_0}}))\leq \mathcal J(u)+\frac{\varepsilon}{3}.
\end{equation} Let 
$v\in C_0^\infty(\mathbb{R}^4)$ satisfy 
$supp(v)\subset B_{1+\frac{4}{\delta_0}}(0)\backslash B_{\frac{4}{\delta_0}}(0)$, and set
\begin{equation*}
v_{\delta_0}=\frac{\sqrt{\tilde{c}^2-\|u_{\delta_0}\|_2^2}}{\|v\|_2}v.
\end{equation*} Define 
$\omega_h:=u_{\delta_0}+\mathcal{H}(v_{\delta_0},h)$ for any λ < 0, then 
$\omega_h\rightarrow u_{\delta_0}$ in 
$H^2(\mathbb{R}^4)$ as 
$h\rightarrow-\infty$. Since
\begin{equation*}dist(u_{\delta_0},\mathcal{H}(v_{\delta_0},h))\geq \frac{2}{\delta_0} \gt 0,\end{equation*} we have 
$\|\omega_h\|_2^2=\tilde{c}^2$, i.e., 
$\omega_h \in S(\tilde{c})$. Then
\begin{equation*}
E(\tilde{c})\leq \mathcal J(\mathcal{H}(\omega_h,s_{\omega_h})).
\end{equation*} By lemmas 2.6 and 3.2, we know 
$s_{\omega_h}\rightarrow s_{u_{\delta_0}}$ in 
$\mathbb{R}$ as 
$h\rightarrow-\infty$, thus
\begin{equation*}
\mathcal{H}(\omega_h,s_{\omega_h})\rightarrow \mathcal{H}(u_{\delta_0},s_{u_{\delta_0}}), \quad \text{in}\,\, H^2(\mathbb{R}^4)\,\, \text{as}\,\, h\rightarrow-\infty.
\end{equation*}This with (4.1) and (4.2) yields that
\begin{equation*}
E(\tilde{c})\leq \lim\limits_{h\rightarrow-\infty}\mathcal J(\mathcal{H}(\omega_h,s_{\omega_h}))=\mathcal J(\mathcal{H}(u_{\delta_0},s_{u_{\delta_0}}))\leq E(c)+\varepsilon.
\end{equation*} By the arbitrariness of ɛ > 0, we deduce that 
$E(\tilde{c})\leq E(c)$ for any 
$\tilde{c} \gt c$.
Lemma 4.2. Assume that f satisfies 
$(f_1)$-
$(f_3)$ and 
$(f_6)$. Suppose that (1.1) and (1.2) possesses a ground state solution u with λ < 0, then 
$E(c') \lt E(c)$ for any 
$c' \gt c$ close to c.
Proof. For any t > 0 and 
$s\in \mathbb{R}$, we know 
$\mathcal{H}(tu,s)\in S(tc)$ and
\begin{align*}
\mathcal J(\mathcal{H}(tu,s))&=\frac{t^2 e^{4s}}{2}\int_{\mathbb{R}^4}|\Delta u|^2dx+\frac{\beta t^2e^{2s}}{2}\int_{\mathbb{R}^4}|\nabla u|^2dx\\
&\quad -\frac{e^{(\mu-8)s}}{2}\int_{\mathbb{R}^4}(I_\mu*F(te^{2s}u))F(te^{2s}u)dx.
\end{align*} Denote 
$\alpha(t,s):=\mathcal J(\mathcal{H}(tu,s))$, then
\begin{align*}
\frac{\partial \alpha(t,s) }{\partial t}&= t e^{4s}\int_{\mathbb{R}^4}|\Delta u|^2dx+\beta te^{2s}\\&\quad\int_{\mathbb{R}^4}|\nabla u|^2dx-e^{(\mu-8)s}\int_{\mathbb{R}^4}(I_\mu*F(te^{2s}u))f(te^{2s}u)e^{2s}udx\\
&=\frac{\langle\mathcal{J}'(\mathcal{H}(tu,s)),\mathcal{H}(tu,s)\rangle}{t}.
\end{align*} By Lemma 2.6, 
$\mathcal{H}(tu,s)\rightarrow u$ in 
$H^2(\mathbb{R}^4)$ as 
$(t,s)\rightarrow (1,0)$. Since λ < 0, 
$\langle\mathcal{J}'(u),u\rangle=\lambda \|u\|_2^2=\lambda c^2 \lt 0$. Hence, one can fix δ > 0 small enough such that
\begin{equation*}
\frac{\partial \alpha(t,s) }{\partial t} \lt 0\quad\text{for any}\,\, (t,s)\in (1,1+\delta]\times [-\delta,\delta].
\end{equation*} For any 
$t\in (1,1+\delta]$ and 
$s\in [-\delta,\delta]$, using the mean value theorem, we obtain
\begin{equation*}
\alpha(t,s)=\alpha(1,s)+(t-1)\cdot \frac{\partial \alpha(t,s) }{\partial t}\Big|_{t=\xi} \lt \alpha(1,s)
\end{equation*} for some 
$\xi\in (1,t)$. By Lemma 3.2, 
$s_{tu}\rightarrow s_u=0$ in 
$\mathbb{R}$ as 
$t \rightarrow1^+$. For any 
$c' \gt c$ close to c, let 
$t_0=\frac{c'}{c}$, then
\begin{equation*}
t_0\in (1,1+\delta]\quad \text{and}\quad s_{t_0u}\in [-\delta,\delta]
\end{equation*}and thus, using Lemma 3.2 again,
\begin{equation*}
E(c')\leq \alpha(t_0,s_{t_0u}) \lt \alpha(1,s_{t_0u})=\mathcal J(\mathcal{H}(u,s_{t_0u}))\leq \mathcal J(u)=E(c).
\end{equation*}From lemmas 4.1 and 4.2, we can directly obtain the following result.
5. Palais-Smale sequence
In this section, using a minimax principle based on the homotopy stable family of compact subsets of S(c), we will construct a 
$(PS)_{E(c)}$ sequence on 
$\mathcal{P}(c)$ for 
$\mathcal J$.
Proposition 5.1. There exists a non-negative 
$(PS)_{E(c)}$ sequence 
$\{u_n\}\subset \mathcal{P}(c)$ for 
$\mathcal J$.
Following by [Reference Willem52], we recall that for any c > 0, the tangent space of S(c) at u is defined by
\begin{equation*}
T_u=\Big\{v\in H^2(\mathbb{R}^4):\int_{\mathbb{R}^4}uvdx=0\Big\}.
\end{equation*} To prove proposition 5.1, we borrow some arguments from [Reference Bartsch and Soave4, Reference Bartsch and Soave5] and consider the functional 
$\mathcal{I}:S(c)\rightarrow\mathbb{R}$ defined by
\begin{equation}
\mathcal{I}(u)=\mathcal J(\mathcal{H}(u,s_u)),
\end{equation} where 
$s_u\in \mathbb{R}$ is the unique number obtained in Lemma 3.2 for any 
$u\in S(c)$. By Lemma 3.2, we know that su is continuous as a mapping of 
$u\in S(c)$. However, it remains unknown that whether su is of class C 1. Inspired by [Reference Szulkin and Weth50, proposition 2.9], we observe that
Lemma 5.1. Assume that f satisfies 
$(f_1)$-
$(f_3)$ and 
$(f_6)$, then the functional 
$\mathcal{I}:S(c)\rightarrow\mathbb{R}$ is of class C 1 and
\begin{align*}
\langle\mathcal{I}'(u),v\rangle&=e^{4s_u}\int_{\mathbb{R}^4}\Delta u \Delta v dx+\beta e^{2s_u}\\
&\quad \int_{\mathbb{R}^4}\nabla u \cdot\nabla v dx-e^{(\mu-8)s_u}\int_{\mathbb{R}^4}(I_\mu*F(e^{2s_u}u))f(e^{2s_u}u)e^{2s_u}v dx\\
&=\langle\mathcal J'(\mathcal{H}(u,s_u)),\mathcal{H}(v,s_u)\rangle
\end{align*} for any 
$u\in S(c)$ and 
$v\in T_u$.
Proof. Let 
$u\in S(c)$ and 
$v\in T_u$, then for any 
$|t|$ small enough, by Lemma 3.2,
\begin{align*}
\mathcal{I}(u+tv)-\mathcal{I}(u)=&\mathcal J(\mathcal{H}(u+tv,s_{u+tv}))-\mathcal J(\mathcal{H}(u,s_u))\\
&\leq \mathcal J(\mathcal{H}(u+tv,s_{u+tv}))-\mathcal J(\mathcal{H}(u,s_{u+tv}))\\
&=\frac{e^{4s_{u+tv}}}{2}\int_{\mathbb{R}^4}\Big[|\Delta (u+tv)|^2-|\Delta u|^2\Big]dx\\
&+\frac{\beta e^{2s_{u+tv}}}{2}\int_{\mathbb{R}^4}\Big[|\nabla (u+tv)|^2-|\nabla u|^2\Big]dx\\ &-\frac{e^{(\mu-8)s_{u+tv}}}{2}\int_{\mathbb{R}^4}\Big[(I_\mu*F(e^{2s_{u+tv}}(u+tv)))F(e^{2s_{u+tv}}(u+tv))\\
&-(I_\mu*F(e^{2s_{u+tv}}u))F(e^{2s_{u+tv}}u)\Big] dx\\
&=\frac{e^{4s_{u+tv}}}{2}\int_{\mathbb{R}^4}\Big(t^2|\Delta v|^2+2t\Delta u \Delta v\Big)dx\\
&+\frac{\beta e^{2s_{u+tv}}}{2}\int_{\mathbb{R}^4}\Big(t^2|\nabla v|^2+2t\nabla u \cdot \nabla v\Big)dx\\
&-\frac{e^{(\mu-8)s_{u+tv}}}{2}\int_{\mathbb{R}^4}(I_\mu*F(e^{2s_{u+tv}}(u+tv)))\\
&f(e^{2s_{u+tv}}(u+\xi_ttv))e^{2s_{u+tv}}tv dx\\
&-\frac{e^{(\mu-8)s_{u+tv}}}{2}\int_{\mathbb{R}^4}(I_\mu*F(e^{2s_{u+tv}}u))f(e^{2s_{u+tv}}(u+\xi_ttv))e^{2s_{u+tv}}tv dx,
\end{align*} where 
$\xi_t\in (0,1)$. Analogously, we have
\begin{align*}
\mathcal{I}(u+tv)-\mathcal{I}(u)
\geq &
\frac{e^{4s_{u}}}{2}\int_{\mathbb{R}^4}\Big(t^2|\Delta v|^2+2t\Delta u\Delta v\Big)dx+\frac{\beta e^{2s_{u}}}{2}\int_{\mathbb{R}^4}\Big(t^2|\nabla v|^2+2t\nabla u\cdot \nabla v\Big)dx\\ &-\frac{e^{(\mu-8)s_{u}}}{2}\int_{\mathbb{R}^4}(I_\mu*F(e^{2s_{u}}(u+tv)))f(e^{2s_{u}}(u+\zeta_ttv))e^{2s_{u}}tv dx\\ &-\frac{e^{(\mu-8)s_{u}}}{2}\int_{\mathbb{R}^4}(I_\mu*F(e^{2s_{u}}u))f(e^{2s_{u}}(u+\zeta_ttv))e^{2s_{u}}tv dx,
\end{align*} where 
$\zeta_t\in (0,1)$. By Lemma 3.2, 
$\lim\limits_{t\rightarrow0}s_{u+tv}=s_u$, from the above inequalities, we conclude
\begin{align*}
\lim\limits_{t\rightarrow0}\frac{\mathcal{I}(u+tv)-\mathcal{I}(u)}{t}=&e^{4s_u}\int_{\mathbb{R}^4}\Delta u \Delta v dx+\beta e^{2s_u}\int_{\mathbb{R}^4}\nabla u \cdot\nabla v dx\\
&-e^{(\mu-8)s_u}\int_{\mathbb{R}^4}(I_\mu*F(e^{2s_u}u))f(e^{2s_u}u)e^{2s_u}v dx.
\end{align*} Using Lemma 3.2, we find that the Gâteaux derivative of 
$\mathcal{I}$ is continuous linear in v and continuous in u. Therefore, by [Reference Willem52, proposition 1.3], we get 
$\mathcal{I}$ is of class C 1. Changing variables in the integrals, we prove the rest.
Lemma 5.2. Assume that f satisfies 
$(f_1)$-
$(f_3)$ and 
$(f_6)$, let 
$\mathcal{F}$ be a homotopy stable family of compact subsets of S(c) without boundary (i.e., 
$B=\emptyset$) and set
\begin{equation*}
E_{\mathcal{F}}:=\inf\limits_{A\in \mathcal{F}}\max\limits_{u\in A}\mathcal I(u),
\end{equation*} where 
$\mathcal I(u)$ defined by (5.1). If 
$E_{\mathcal{F}} \gt 0$, then there exists a non-negative 
$(PS)_{E_{\mathcal{F}}}$ sequence 
$\{u_n\}\subset \mathcal{P}(c)$ for 
$\mathcal J$.
Proof. Let 
$\{A_n\}\subset \mathcal{F}$ be a minimizing sequence of 
$E_{\mathcal{F}}$. We define the mapping
\begin{equation*}
\eta:[0,1]\times S(c)\rightarrow S(c), \quad\eta(t,u)=\mathcal{H}(u,ts_u).
\end{equation*} By Lemma 3.2, 
$\eta(t,u)$ is continuous in 
$[0,1]\times S(c)$, and satisfies 
$\eta(t,u)=u$ for all 
$(t,u)\in \{0\}\times S(c)$. Thus by the definition of 
$\mathcal{F}$, one has
\begin{equation*}
D_n:=\eta(1,A_n)=\{\mathcal{H}(u,s_u):u\in A_n\}\subset \mathcal{F}.
\end{equation*} Obviously, 
$D_n\subset \mathcal{P}(c)$ for any 
$n\in \mathbb{N}^+$. Since 
$\mathcal{I}(\mathcal{H}(u,s))=\mathcal{I}(u)$ for any 
$u\in S(c)$ and 
$s\in \mathbb{R}$, then
\begin{equation*}
\max\limits_{u\in D_n}\mathcal{I}(u)=\max\limits_{u\in A_n}\mathcal{I}(u)\rightarrow E_{\mathcal{F}},\quad\text{as}\,\, n\rightarrow\infty,
\end{equation*} which implies that 
$\{D_n\}\subset \mathcal{F}$ is another minimizing sequence of 
$E_{\mathcal{F}}$. By Lemma 2.5, we obtain a 
$(PS)_{E_{\mathcal{F}}}$ sequence 
$\{v_n\}\subset \mathcal{S}(c)$ for 
$\mathcal I$ such that 
$\lim\limits_{n\rightarrow\infty}dist(v_n,D_n)=0$. By Lemma 3.2, we know 
$s_{-u}=s_u$ for any 
$u\in S(c)$, thus 
$\mathcal{I}(u)$ is even in S(c). Without loss of generality, we assume that 
$v_n\geq0$ for all 
$n\in \mathbb{N}^+$. Let
\begin{equation*}
u_n:=\mathcal{H}(v_n,s_{v_n}),
\end{equation*} then 
$u_n\geq0$ for all 
$n\in \mathbb{N}^+$, and we prove that 
$\{u_n\}\subset \mathcal{P}(c)$ is the desired sequence.
We claim that there exists C > 0 such that 
$e^{-4s_{v_n}}\leq C$ for any 
$n\in \mathbb{N}^+$. Indeed, we observe that
\begin{equation*}
e^{-4s_{v_n}}=\frac{\int_{\mathbb{R}^4}|\Delta v_n|^2dx}{\int_{\mathbb{R}^4}|\Delta u_n|^2dx}.
\end{equation*} Since 
$\{u_n\}\subset \mathcal{P}(c)$, by Lemma 3.4, there exists a constant 
$C_1 \gt 0$ such that 
$\int_{\mathbb{R}^4}|\Delta u_n|^2dx\geq C_1$ for any 
$n\in \mathbb{N}^+$. Regarding the term of 
$\{v_n\}$, since 
$D_n\subset \mathcal{P}(c)$ for any 
$n\in \mathbb{N}^+$ and for any 
$u\in \mathcal{P}(c)$, one has 
$\mathcal{J}(u)=\mathcal{I}(u)$, thus
\begin{equation*}
\max\limits_{u\in D_n}\mathcal{J}(u)= \max\limits_{u\in D_n}\mathcal{I}(u)\rightarrow E_{\mathcal{F}},\quad\text{as}\,\, n\rightarrow\infty.
\end{equation*} This fact together with 
$D_n\subset \mathcal{P}(c)$ and 
$(f_3)$ yields that 
$\{D_n\}$ is uniformly bounded in 
$H^2(\mathbb{R}^4)$, thus from 
$\lim\limits_{n\rightarrow\infty}dist(v_n,D_n)=0$, we obtain 
$\sup\limits_{n\geq1}\|v_n\|^2 \lt \infty$. This proves the claim.
From 
$\{u_n\}\subset \mathcal{P}(c)$, it follows that
\begin{equation*}
\mathcal{J}(u_n)=\mathcal{I}(u_n)=\mathcal{I}(v_n)\rightarrow E_{\mathcal{F}},\quad\text{as}\,\, n\rightarrow\infty.
\end{equation*} For any 
$\varphi \in T_{u_n}$, we have
\begin{equation*}
\int_{\mathbb{R}^4}v_n\mathcal{H}(\varphi,-s_{v_n})dx=\int_{\mathbb{R}^4}\mathcal{H}(v_n,s_{v_n})\varphi dx=\int_{\mathbb{R}^4}u_n\varphi dx=0,
\end{equation*} which means 
$\mathcal{H}(\varphi,-s_{v_n})\in T_{v_n}$. Also, by the claim, we have
\begin{align*}
\|\mathcal{H}(\varphi,-s_{v_n})\|^2&=e^{-4s_{v_n}}\|\Delta \varphi\|_2^2+2e^{-2s_{v_n}}\|\nabla \varphi\|_2^2+\|\varphi\|_2^2\leq C\|\Delta \varphi\|_2^2\\&
\quad +2\sqrt{C}\|\nabla \varphi\|_2^2+\|\varphi\|_2^2\leq \max\{1,C\}\|\varphi\|^2.
\end{align*} By Lemma 5.1, for any 
$\varphi\in T_{u_n}$, we deduce that
\begin{align*}
|\langle\mathcal{J}'(u_n),\varphi\rangle|&=|\langle\mathcal{J}'(\mathcal{H}(v_n,s_{v_n}),\mathcal{H}(\mathcal{H}(\varphi,-s_{v_n}),s_{v_n})\rangle|=|\langle\mathcal{I}'(v_n),\mathcal{H}(\varphi,-s_{v_n})\rangle|\\
&\leq \|\mathcal{I}'(v_n)\|_*\cdot \|\mathcal{H}(\varphi,-s_{v_n})\|\leq \max\{1,\sqrt{C}\}\|\mathcal{I}'(v_n)\|_*\cdot \|\varphi\|,
\end{align*} which implies that 
$\|\mathcal{J}'(u_n)\|_*\leq \max\{1,\sqrt{C}\}\|\mathcal{I}'(v_n)\|_*\rightarrow0$ as 
$n\rightarrow\infty$. Thus 
$\{u_n\}\subset \mathcal{P}(c)$ is a non-negative 
$(PS)_{E_{\mathcal{F}}}$ sequence for 
$\mathcal J$.
Proof of proposition 5.1
Note that the class 
$\mathcal{F}$ of all singletons included in S(c) is a homotopy stable family of compact subsets of S(c) without boundary. By Lemma 5.2, if 
$E_{\mathcal{F}} \gt 0$, then there exists a non-negative 
$(PS)_{E_{\mathcal{F}}}$ sequence 
$\{u_n\}\subset \mathcal{P}(c)$ for 
$\mathcal J$. By Lemma 3.4, we know 
$E(c) \gt 0$, so if we can prove that 
$E_{\mathcal{F}}=E(c)$, then we complete the proof. In fact, by the definition of 
$\mathcal{F}$, we have
\begin{equation*}
E_{\mathcal{F}}=\inf\limits_{A\in \mathcal{F}}\max\limits_{u\in A}\mathcal I(u)=\inf\limits_{u\in S(c)}\mathcal I(u)=\inf\limits_{u\in S(c)}\mathcal I(\mathcal{H}(u,s_u))=\inf\limits_{u\in S(c)}\mathcal J(\mathcal{H}(u,s_u)).
\end{equation*} For any 
$u\in S(c)$, it follows from 
$\mathcal{H}(u,s_u)\in \mathcal{P}(c)$ that 
$\mathcal J(\mathcal{H}(u,s_u))\geq E(c)$, by the arbitrariness of 
$u\in S(c)$, we get 
$E_{\mathcal{F}}\geq E(c)$. On the other hand, for any 
$u\in \mathcal{P}(c)$, by Lemma 3.2, we deduce that 
$s_u=0$ and 
$\mathcal{J}(u)=\mathcal J(\mathcal{H}(u,0))\geq \inf\limits_{u\in S(c)}\mathcal J(\mathcal{H}(u,s_u))$, by the arbitrariness of 
$u\in \mathcal{P}(c)$, we get 
$E(c)\geq E_{\mathcal{F}}$.
For the sequence 
$\{u_n\}$ obtained in proposition 5.1, by 
$(f_3)$, we know that 
$\{u_n\}$ is bounded in 
$H^2(\mathbb{R}^4)$, up to a subsequence, we assume that 
$u_n\rightharpoonup u_c$ in 
$H^2(\mathbb{R}^4)$. Furthermore, by 
${\mathcal J}'|_{S(c)}(u_n)\rightarrow0$ as 
$n\rightarrow\infty$ and Lagrange multiplier rule, there exists 
$\{\lambda_n\}\subset \mathbb{R}$ such that
\begin{equation}
\Delta^2u_n-\beta\Delta u_n=\lambda_n u_n+(I_\mu*F(u_n))f(u_n)+o_n(1).
\end{equation}Lemma 5.3. Assume that f satisfies 
$(f_1)$-
$(f_6)$, then up to a subsequence and up to translations in 
$\mathbb{R}^4$, 
$u_n\rightharpoonup u_c\neq0$ in 
$H^2(\mathbb{R}^4)$.
Proof. We claim that
\begin{equation*}
\kappa:=\limsup\limits_{n\rightarrow\infty}\Big(\sup\limits_{y\in \mathbb{R}^4}\int_{B(y,r)}|u_n|^2dx\Big) \gt 0
\end{equation*} for any r > 0. Otherwise, by the Lions Lemma [Reference Willem52, Lemma 1.21], 
$u_n\rightarrow0$ in 
$L^p(\mathbb{R}^4)$ for any p > 2.
By 
${\mathcal J}(u_n)=E(c)+o_n(1)$, 
$P(u_n)=0$ and 
$(f_3)$, we get
\begin{align*}
{\mathcal J}(u_n)-\frac{1}{4}P(u_n)&=\frac{\beta }{4}\int_{\mathbb{R}^4}|\nabla u_n|^2dx\\
&\quad +\frac{\mu-12}{8}\int_{\mathbb{R}^4}(I_\mu*F(u_n))F(u_n)dx+\frac{1}{2}\int_{\mathbb{R}^4}(I_\mu*F(u_n))f(u_n)u_n dx\nonumber\\
&\geq\frac{\beta }{4}\int_{\mathbb{R}^4}|\nabla u_n|^2dx+\frac{4\theta+\mu-12}{8}\int_{\mathbb{R}^4}(I_\mu*F(u_n))F(u_n)dx.
\end{align*} Since 
$\beta\geq0$, by 
$\theta \gt 3-\frac{\mu}{4}$, we have
\begin{equation}
\limsup\limits_{n\rightarrow\infty} \int_{\mathbb{R}^4} (I_\mu*F(u_n))F(u_n) dx\leq \frac{8 E(c)}{4\theta+\mu-12}.
\end{equation} This together with 
$P(u_n)=0$ and the boundedness of 
$\{u_n\}$ implies that, up to a subsequence, there exists constant 
$L_0 \gt 0$ such that
\begin{equation}
\int_{\mathbb{R}^4} (I_\mu*F(u_n))f(u_n)u_n dx\leq L_0.
\end{equation} For any 
$\delta\in (0,\frac{M_0L_0}{R_0})$, from 
$(f_3)$, 
$(f_4)$, we can choose 
$M_\delta \gt \frac{M_0L_0}{\delta} \gt R_0$, then
\begin{align*}
&\int_{|u_n|\geq M_\delta} (I_\mu*F(u_n))F(u_n)dx\leq M_0\int_{|u_n|\geq M_\delta} (I_\mu*F(u_n))|f(u_n)|dx\\
&\leq \frac{M_0}{M_\delta}\int_{|u_n|\geq M_\delta} (I_\mu*F(u_n))f(u_n)u_ndx \lt \delta
.
\end{align*}On the other hand, by (1.7) and (2.1),
\begin{equation*}
\int_{|u_n|\leq M_\delta} (I_\mu*F(u_n))F(u_n)dx\leq C (\|u_n\|_{\nu+1}^{\nu+1}+\|u_n\|_{q+1}^{q+1})=o_n(1).
\end{equation*}Due to the arbitrariness of δ > 0, we deduce that
\begin{equation*}
\int_{\mathbb{R}^4} (I_\mu*F(u_n))F(u_n)dx=o_n(1).
\end{equation*}Hence, by Lemma 3.6, we have
\begin{equation*}
\limsup\limits_{n\rightarrow\infty} \|\Delta u_n\|_2^2\leq 2E(c) \lt \frac{8-\mu}{8}.
\end{equation*} Up to a subsequence, we assume that 
$\sup\limits_{n\in \mathbb{N}^+} \|\Delta u_n\|_2^2 \lt \frac{8-\mu}{8}$. Using (2.1) again, we have
\begin{equation*}
\int_{\mathbb{R}^4} (I_\mu*F(u_n))f(u_n)u_ndx\leq C\|F(u_n)\|_{\frac{8}{8-\mu}}\|f(u_n)u_n\|_{\frac{8}{8-\mu}}.
\end{equation*} Fix 
$\alpha \gt 32\pi^2$ close to 
$32\pi^2$ and m > 1 close to 1 such that
\begin{equation*}
\sup\limits_{n\in \mathbb{N}^+} \frac{8\alpha m \|\Delta u_n\|_2^2}{8-\mu}\leq 32\pi^2.
\end{equation*} Arguing as (3.1), for 
$m'=\frac{m}{m-1}$, we have
\begin{equation*}
\|F(u_n)\|_{\frac{8}{8-\mu}}\leq C\|u_n\|_{\frac{8(\nu+1)}{8-\mu}}^{\nu+1}+C\|u_n\|_{\frac{8(q+1)m'}{8-\mu}}^{q+1}\rightarrow0,\quad\text{as}\,\, n\rightarrow\infty.
\end{equation*}Similarly, we can prove that
\begin{equation*}
\|f(u_n)u_n\|_{\frac{8}{8-\mu}}\rightarrow0,\quad\text{as}\,\, n\rightarrow\infty.
\end{equation*}Therefore, we get
\begin{equation*}
\int_{\mathbb{R}^4} (I_\mu*F(u_n))f(u_n)u_ndx=o_n(1).
\end{equation*} Recalling that 
$P(u_n)=0$, so 
$\|\Delta u_n\|_2=o_n(1)$, which implies 
$E(c)=0$, this is impossible, since 
$E(c) \gt 0$. According to κ > 0, there exists 
$\{y_n\}\subset \mathbb{R}^4$ such that 
$\int_{B(y_n,1)}|u_n|^2dx \gt \frac{\kappa}{2}$, i.e., 
$\int_{B(0,1)}|u_n(x-y_n)|^2dx \gt \frac{\kappa}{2}$. Up to a subsequence and up to translations in 
$\mathbb{R}^4$, 
$u_n\rightharpoonup u_c\neq0$ in 
$H^2(\mathbb{R}^4)$.
Lemma 5.4. Assume that f satisfies 
$(f_1)$-
$(f_6)$, then 
$\{\lambda_n\}$ is bounded in 
$\mathbb{R}$. Furthermore, there exists 
$\beta_* \gt 0$, such that 
$\lambda_n\rightarrow\lambda_c$ with some 
$\lambda_c \lt 0$ as 
$n\rightarrow\infty$ for any 
$\beta\in [0,\beta_*)$.
Proof. Testing (5.2) with un, we have
\begin{equation*}
\int_{\mathbb{R}^4}|\Delta u_n|^2dx+\beta\int_{\mathbb{R}^4}|\nabla u_n|^2dx=\lambda_n \int_{\mathbb{R}^4}|u_n|^2dx+\int_{\mathbb{R}^4}(I_\mu*F(u_n))f(u_n)u_ndx+o_n(1).
\end{equation*} Combing this with 
$P(u_n)=0$ leads to
\begin{equation*}
\lambda_n c^2=\frac{\beta}{2}\int_{\mathbb{R}^4}|\nabla u_n|^2dx-\frac{8-\mu}{4}\int_{\mathbb{R}^4}(I_\mu*F(u_n))F(u_n)dx+o_n(1).
\end{equation*}Thus
\begin{equation}
\limsup\limits_{n\rightarrow\infty}|\lambda_n| \leq \limsup\limits_{n\rightarrow\infty}\Big(\frac{\beta}{2c^2}\int_{\mathbb{R}^4}|\nabla u_n|^2dx+\frac{8-\mu}{4c^2}\int_{\mathbb{R}^4}(I_\mu*F(u_n))F(u_n)dx\Big),
\end{equation}and
\begin{equation}
\limsup\limits_{n\rightarrow\infty}\lambda_n = \limsup\limits_{n\rightarrow\infty}\frac{\beta}{2c^2}\int_{\mathbb{R}^4}|\nabla u_n|^2dx- \liminf\limits_{n\rightarrow\infty}\frac{8-\mu}{4c^2}\int_{\mathbb{R}^4}(I_\mu*F(u_n))F(u_n)dx.
\end{equation} From (5.3) and (5.5), it follows that 
$\{\lambda_n\}$ is bounded in 
$\mathbb{R}$. Since 
$\{u_n\}$ is bounded in 
$H^2(\mathbb{R}^4)$, there exists constant M > 0 such that 
$\|u_n\|^2\leq M$. Choosing
\begin{equation*}
\beta_*=\frac{8-\mu}{2M}\int_{\mathbb{R}^4}(I_\mu*F(u_c))F(u_c)dx,
\end{equation*} then for any 
$\beta\in [0,\beta_*)$, by (5.6), using Fatou lemma, we have 
$\limsup\limits_{n\rightarrow\infty}\lambda_n \lt 0$. Since 
$\{\lambda_n\}$ is bounded, up to a subsequence, we have that 
$\lambda_n\rightarrow\lambda_c$ with some 
$\lambda_c \lt 0$ as 
$n\rightarrow\infty$ for any 
$\beta\in [0,\beta_*)$.
6. Proof of the main result
Proof of theorem 1.1
Under the assumptions of theorem 1.1, from (5.2), (5.4), lemmas 2.7, 5.3 and 5.4, we can see that uc is a nontrivial weak solution of (1.1) with λ < 0, and 
$P(u_c)=0$. Using the Brézis-Lieb Lemma [Reference Willem52, lemma 1.32], we have
\begin{equation*}
\int_{\mathbb{R}^4}|u_n|^2dx=\int_{\mathbb{R}^4}|u_n-u_c|^2dx+\int_{\mathbb{R}^4}|u_c|^2dx+o_n(1).
\end{equation*} Let 
$c_1:=\|u_c\|_2 \gt 0$ and 
$c_{2,n}:=\|u_n-u_c\|_2$, then 
$c^2=c_1^2+c_{2.n}^2+o_n(1)$. Since 
$P(u_c)=0$, using 
$(f_3)$ and Fatou lemma, we have
\begin{align*}
\mathcal{J}(u_c)={\mathcal J}(u_c)-\frac{1}{4}P(u_c)&=\frac{\beta}{4}\int_{\mathbb{R}^4}|\nabla u_c|^2dx+\frac{1}{2}\int_{\mathbb{R}^4}(I_\mu*F(u_c))\nonumber\\
&\Big(f(u_c)u_c-(3-\frac{\mu}{4})F(u_c)\Big) dx\nonumber\\
&\leq\frac{1}{2}\liminf\limits_{n\rightarrow\infty}\Big[\frac{\beta}{2}\int_{\mathbb{R}^4}|\nabla u_n|^2dx+\int_{\mathbb{R}^4}(I_\mu*F(u_n))\nonumber\\
&\Big(f(u_n)u_n-(3-\frac{\mu}{4})F(u_n)\Big) dx\Big]\nonumber\\
&=\liminf\limits_{n\rightarrow\infty}({\mathcal J}(u_n)-\frac{1}{4}P(u_n))=E(c).
\end{align*} On the other hand, it follows from Lemma 4.1 that 
$\mathcal{J}(u_c)\geq E(c_1)\geq E(c)$. Thus 
$\mathcal{J}(u_c)= E(c_1)= E(c)$. By Lemma 4.3, we obtain 
$c=c_1$. This implies uc is a ground state solution of (1.1) and (1.2).
Acknowledgements
The authors were supported by Natural Science Foundation of Chongqing, China CSTB2024NSCQ-LZX0038.
