Positive Normalized Solutions to a Kind of Fractional Kirchhoff Equation with Critical Growth

Abstract

In this paper, we have investigated the existence of normalized solutions for a class of fractional Kirchhoff equations involving nonlinearity and critical nonlinearity. The nonlinearity satisfies $L^2$-supercritical conditions. We transform the problem into an extremal problem within the framework of Lagrange multipliers by utilizing the energy functional of the equation in the fractional Sobolev space and applying the mass constraint condition (i.e., for given $m>0,$${\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=m^2$). We introduced a new set and proved that it is a natural constraint. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions.

1. Introduction

In this paper, for given $m>0$, we shall investigate the existence of $(u,\lambda )\in H^s\left({\mathbb{R}}^N\right)\times \mathbb{R}$ which fulfill the following problem,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(a+b{\int }_{{\mathbb{R}}^N}{|(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u|}^2{dx)(-\Delta )}^{s}u+\lambda u=f\left(u\right)+{|u|}^{2_s^{\ast }-2}u\mathrm{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=m^2,\hfill \end{array}\right.\end{array}$$

(1)

where $s\in (0,1)$, $1\le N<4s$, and $a,b>0$, $2_s^{\ast }={\displaystyle \frac{2N}{N-2s}}$ is the fractional Sobolev critical exponent. ${(-\Delta )}^s$ is the pseudo-differential operator defined by

$${(-\Delta )}^{s}u\left(x\right)=-{\displaystyle \frac{C(N,s)}{2}}\mathrm{P}.\mathrm{V}.{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{u(x+y)+u(x-y)-2u\left(x\right)}{{|y|}^{N+2s}}}dy$$

$$=C(N,s)\mathrm{P}.\mathrm{V}.{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{u\left(x\right)-u\left(y\right)}{{|x-y|}^{N+2s}}}dy$$

with a positive constant $C(N,s)$, and $P.V.$ is the Cauchy principal value on the integral.

Equation (1) can be regarded as the fractional version of the following classical Kirchhoff-type equation

$$\begin{array}{c}\hfill -\left(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^{2}dx\right)\Delta u+V\left(x\right)u=f(x,u)\mathrm{in}{\mathbb{R}}^N,\end{array}$$

(2)

which is a variant type of the Dirichlet problem of the Kirchhoff type:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-\left(a+b{\int }_{\Omega }{|\nabla u|}^{2}dx\right)\Delta u=f(x,u)\mathrm{in}\Omega ,\hfill \\ u=0\mathrm{on}\partial \Omega .\hfill \end{array}\right.\end{array}$$

(3)

Equation (3) is related to the steady-state analogues of the following equation,

$$\begin{array}{c}\hfill u_{tt}-\left(a+b{\int }_{\Omega }{|\nabla u|}^{2}dx\right)\Delta u=f(t,x,u),\end{array}$$

(4)

which was firstly proposed by Kirchhoff [1] in 1883 as an extension of the classical D’Alembert’s wave equations

$$\begin{array}{c}\hfill \rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\left({\displaystyle \frac{p_0}{h}}+{\displaystyle \frac{E}{2L}}{\int }_0^L{\left|{\displaystyle \frac{\partial u}{\partial x}}\right|}^{2}dx\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=f(t,x,u)\end{array}$$

(5)

for free vibration of elastic strings, where $\rho$ denotes the mass density, u the lateral displacement, h the cross section area, ${\rho }_0$ the initial axial tension, E the Young modulus, L the length of the string, and f the external force. Equation (5) takes into account the changes in the length of the string produced by transverse vibrations.

There are two very different points of views as far as frequency $\lambda$ is concerned in (1). One is to treat the frequency $\lambda$ as a given constant. At this time, u solves (1) if it is a critical point of the corresponding energy functional on the working space (see, e.g., [2,3]). The other point of view is to regard $\lambda$ as an unknown quantity to problem (1). At this time, $\lambda$ can be interpreted as the Lagrange multiplier.

The operator ${(-\Delta )}^s$ can be seen as the infinitesimal generators of Lévy stable diffusion processes; see [4] for example. This operator arises in several areas such as physics, biology, chemistry, and finance (see, e.g., [4,5]). Recently, by employing the constraint variational method, the Poho$\check{\mathrm{z}}$aev method based on the Br$\acute{\mathrm{e}}$zis–Lieb, and monotonicity trick, Kong and Chen [6] studied the existence and asymptotic behavior of positive ground states for the following fractional Kirchhoff equation,

$$\left\{\begin{array}{c}\left(a+b{\int }_{{\mathbb{R}}^3}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|}^2\mathrm{d}x\right){(-\Delta )}^{s}u=\lambda u+{\mu |u|}^{q-2}u+{|u|}^{2_s^{\ast }-2}u,\hfill \\ {\int }_{{\mathbb{R}}^3}{|u|}^{2}dx=c^2.\hfill \end{array}\right.$$

(6)

In addition, Liu and Jin [7] replaced $f\left(u\right)$ with ${\mu |u|}^{q-2}u+{|u|}^{2_s^{\ast }-2}u$ and proved the existence of solutions to (6) without the Ambrosetti–Rabinowitz condition (see, e.g, [8]) by using a perturbation approach. Furthermore, they also studied the asymptotic behavior of solutions for (6). Kong et al. [9] obtained the equivalent system of (6) by using appropriate transform. With the equivalence result, they obtained the nonexistence, existence, and multiplicity of normalized solutions when $s\in (0,1)$. By replacing $\lambda u$ with $V\left(x\right)u$, He and Zou [10] proved that (6) has positive solutions by adapting the indirect approach and finding a minimum mountain threshold level. Liu et al. [11] considered Equation (6) when $2<p<2+{\displaystyle \frac{4s}{3}},2+{\displaystyle \frac{8s}{3}}<q<2_s^{\ast }$ and $2+{\displaystyle \frac{8s}{3}}<p<q<2_s^{\ast }$, respectively. In [12], Li et al. obtained some similar existence results and derived some asymptotic results on the obtained normalized solutions.

In [13], Zhang, Tang, and Chen studied the following equation,

$$\begin{array}{c}\hfill \left(a+b{\int }_{{\mathbb{R}}^3}{\left|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u\right|}^{2}dx\right){(-\Delta )}^{s}u+V\left(x\right)u=f(x,u)+\lambda {|u|}^{p-2}u,\mathrm{in}{\mathbb{R}}^3,\end{array}$$

(7)

where $s\in ({\displaystyle \frac{3}{4}},1),p\ge 2_s^{\ast }.$ By assuming that f satisfies appropriate conditions and $V\left(x\right)$ satisfies the following conditions,

(V1) $V\in C({\mathbb{R}}^3,\mathbb{R})$ satisfies ${inf}_{x\in {\mathbb{R}}^3}V\left(x\right)\ge V_0>0$;

(V2) There exists $h>0$ such that ${lim}_{|y|\to +\infty }meas\left(\{x\in B_h\left(y\right):V\left(x\right)\le c\}\right)=0$ for all $c>0$, and by considering the mountain pass level

$$\begin{array}{c}\hfill c^{\ast }=\underset{\gamma \in \Gamma }{inf}\underset{t\in [0,1]}{max}{J}_{\lambda }\left(\gamma \left(t\right)\right),\Gamma =\left\{\gamma \in \mathcal{C}\left(\right[0,1],H):\gamma \left(0\right)=0,\gamma \left(1\right)=e\right\},\end{array}$$

they proved that there exists a ground state solution for (7). Li, Zhang, Wang, and Teng [14] obtained the existence of non-trivial solutions to (7), under the assumption (V1) and $V\left(x\right)$ are coercive. Bartsch and Wang [15] first proposed the condition (V2) to overcome the lack of compactness and proved the existence and multiplicity for a kind of elliptic equation.

For the case $a=1$ and $b=0$, problem (1) turns into a fractional Schr$\ddot{\mathrm{o}}$dinger equation, which has also been widely studied. By the constrained minimization method, Cazenave and Lions [16] demonstrated the existence of solutions of the following equation in the subcritical case.

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\Delta u+\lambda u=g\left(u\right)\hfill & \mathrm{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{\left|u\right|}^2\mathrm{d}x=a.\hfill \end{array}\right.\end{array}$$

(8)

When $\lambda u$ is replaced by $(\lambda +V(x\left)\right)u$, Zou and Liu [17] obtained the existence of the normalized solution for (8). When (8) exhibited critical nonlinearity ${|u|}^{2_s^{\ast }-2}u$, by using iterative techniques, Zhang and Han [18] proved the existence of ground state solutions in the $L^2$-subcritical perturbation case, $L^2$-critical perturbation case, and $L^2$-supercritical perturbation case, respectively.

When $s=1$, i.e., for the classical Laplacian, there have been many works on this problem. Ye [19] obtained global constraint minimizers for the following equation,

$$\begin{array}{c}\hfill -\left(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x\right)\Delta u+V\left(x\right)u=\mu {|u|}^{p-2}u+\lambda u,\end{array}$$

(9)

where $2<p<2^{\ast }$ and $2+{\displaystyle \frac{8}{N}}$ is the $L^2$-critical exponent. Later, in [20], Ye studied the existence of normalized solutions to (9) with $p=2+{\displaystyle \frac{8}{N}}$ and $V=0$. Li et al. [21] have studied the existence and asymptotic of normalized solutions for a kind of Kirchhoff equation with Sobolev critical growth by using the Sobolev subcritical approximation method. For more results on this topics, we refer the readers to [22,23,24] for normalized solutions in ${\mathbb{R}}^N$. Moreover, [25] is for coupled fractional Kirchhoff-type systems and [26] is for a fractional Choquard system.

2. Preliminaries

In this section, we introduce some notations and useful preliminary conclusions. To treat the problem in (1), we will use a method by He, Lv, Zhang, and Zhong [27] to study an extension problem.

In this paper, the work space $H^s\left({\mathbb{R}}^N\right)$ is defined by

$$\begin{array}{c}\hfill H^s\left({\mathbb{R}}^N\right):=\{u\in L^2\left({\mathbb{R}}^N\right):{\int }_{{\mathbb{R}}^N}{(1+|\xi |}^{2s})|\mathcal{F}\left(\xi \right){|}^{2}d\xi <\infty \},\end{array}$$

equipped with the norm

$$\begin{array}{c}\hfill {\parallel u\parallel }_{H^s}={\left({\int }_{{\mathbb{R}}^N}{(|\mathcal{F}\left(\xi \right)|}^2+{|\xi |}^{2s}{|\mathcal{F}\left(\xi \right)|}^2)d\xi \right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

$H_r^s\left({\mathbb{R}}^N\right)$ denotes the space of radial functions in $H^s\left({\mathbb{R}}^N\right)$, i.e., $H_r^s\left({\mathbb{R}}^N\right)=\{u\in H^s\left({\mathbb{R}}^N\right):u\left(x\right)=u(|x|)\}$ (see [28]). For $s\in (0,1)$, the fractional Sobolev space $D^{s,2}\left({\mathbb{R}}^N\right)$ is defined by

$$\begin{array}{c}\hfill D^{s,2}\left({\mathbb{R}}^N\right)=\{u\in L^{2s^{\ast }}\left({\mathbb{R}}^N\right){:|\xi |}^s\mathcal{F}\left(\xi \right)\in L^2\left({\mathbb{R}}^N\right)\},\end{array}$$

which is the completion of $C_0^{\infty }\left({\mathbb{R}}^N\right)$ under the norm

$$\begin{array}{c}\hfill {\parallel u\parallel }_{D^{s,2}}={\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u\parallel }_2^2={\left({\int }_{{\mathbb{R}}^N}{|\xi |}^{2s}{|\mathcal{F}\left(\xi \right)|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

Let $u\in H^s\left({\mathbb{R}}^N\right)$ and $2<t<2_s^{\ast }$; there exists an optimal constant $C(s,N,t)$ such that the following fractional Gagliardo–Nirenberg–Sobolev inequality holds

$$\begin{array}{c}\hfill {\parallel u\parallel }_t^t\le {C(s,N,t)\parallel u\parallel }_2^{t-{\displaystyle \frac{N(t-2)}{2s}}}{\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u\parallel }_2^{{\displaystyle \frac{N(t-2)}{2s}}}.\end{array}$$

(10)

The best Sobolev constant is given by

$$\begin{array}{c}\hfill S_c:=\underset{u\in D^{s,2}\left({\mathbb{R}}^N\right)\setminus \left\{0\right\}}{inf}{\displaystyle \frac{{\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{\frac{s}{2}}u|}^{2}dx}{{\left({\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\right)}^{\frac{2}{2_s^{\ast }}}}}.\end{array}$$

(11)

We introduce the following fiber map which is used in this paper:

$$\begin{array}{c}\hfill (\varsigma \ast u)\left(x\right):={\varsigma }^{{\displaystyle \frac{N}{2}}}u\left(\varsigma x\right).\end{array}$$

By direct calculation, we have

$$\begin{array}{c}\hfill {\parallel \varsigma \ast u\parallel }_2^2={\parallel u\parallel }_2^2{,\parallel \varsigma \ast u\parallel }_q^q={\varsigma }^{{\displaystyle \frac{N(q-2)}{2}}}{\parallel u\parallel }_q^q{\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{(\varsigma \ast u)\parallel }_2^2={\varsigma }^{2s}{\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}(\varsigma \ast u)\parallel }_2^2.\end{array}$$

It is meaningful to study the solution of problem $\left(P_m\right)$ with a given $L^2$ norm, that is, for a given $m>0$, to study the solution of problem (1) under the $L^2$-norm constrained manifold

$$\begin{array}{c}\hfill S\left(m\right):=\{u\in H^s\left({\mathbb{R}}^N\right):{\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=m^2\}.\end{array}$$

Physically, these kinds of solutions are called the normalized solution of $\left(P_m\right)$, which are the critical points of the energy functional

$$\begin{array}{cc}\hfill \mathcal{J}\left(u\right)& ={\displaystyle \frac{a}{2}}{\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|}^{2}dx+{\displaystyle \frac{b}{4}}{\left({\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|}^{2}dx\right)}^2-{\int }_{{\mathbb{R}}^N}F\left(u\right)dx\hfill \\ \hfill & -{\displaystyle \frac{1}{2_s^{\ast }}}{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\hfill \end{array}$$

restricted to the manifold $S\left(m\right)$.

Consider the following functionals $\mathcal{J}\left(u\right)$ and ${\mathcal{L}}_u\left(\varsigma \right)$,

$$\begin{array}{cc}\hfill {\mathcal{L}}_u\left(\varsigma \right)& =\mathcal{J}(\varsigma \ast u)={\displaystyle \frac{a}{2}}{\varsigma }^{2s}{\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|}^{2}dx+{\displaystyle \frac{b}{4}}{\varsigma }^{4s}{\left({\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|}^{2}dx\right)}^2\hfill \\ \hfill & -{\varsigma }^{-N}{\int }_{{\mathbb{R}}^N}F\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right)dx-{\displaystyle \frac{1}{2_s^{\ast }}}{\varsigma }^{2_s^{\ast }s}{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx.\hfill \end{array}$$

(12)

3. Main Results

In this part, we present the principal results of this work. To perform this, we make the following assumptions.

(F1) $f:\mathbb{R}\to \mathbb{R}$ is continuous and odd.

(F2) There exists some $(\alpha ,\beta )\in {\mathbb{R}}^{+}$ that fulfill $2+{\displaystyle \frac{8s}{N}}<\alpha \le \beta <2_s^{\ast }$ such that

$$\begin{array}{c}\hfill 0<\alpha F\left(t\right)\le f\left(t\right)t\le \beta F\left(t\right)\mathrm{for}t\ne 0,\mathrm{where}F\left(t\right)={\int }_0^{t}f\left(s\right)ds.\end{array}$$

(F3) The function defined by $\tilde{F}\left(t\right):={\displaystyle \frac{1}{2}}f\left(t\right)t-F\left(t\right)$ is of class $C^1$ and

$$\begin{array}{c}\hfill {\tilde{F}}^{\prime }\left(t\right)t\ge \alpha \tilde{F}\left(t\right),\mathrm{for} \mathrm{any}t\in \mathbb{R},\end{array}$$

where $\alpha$ is defined in (F2).

We are now prepared to present and demonstrate the key results.

Remark 1. 

Under the assumptions (F1) and (F2), we can deduce that for arbitrary $t\ge 0$ and for all $s\in \mathbb{R}$, we have

$$\left\{\begin{array}{cc}{t}^{\beta }F\left(s\right)\le F\left(ts\right)\le t^{\alpha }F\left(s\right),\hfill & \mathrm{if}t\le 1,\hfill \\ t^{\alpha }F\left(s\right)\le F\left(ts\right)\le t^{\beta }F\left(s\right),\hfill & \mathrm{if}t\ge 1.\hfill \end{array}\right.$$

(13)

Furthermore, there exists a non-negative constant $C_F$ such that for arbitrary $t\in \mathbb{R}$, and we have

$$\begin{array}{c}\hfill F\left(t\right)\le C_F\left({|t|}^{\alpha }+{|t|}^{\beta }\right).\end{array}$$

We are now in a position to state the main results of this paper.

Theorem 1. 

Let $s\in (0,1)$ and $1\le N<4s$. Assuming (F1)–(F3) hold, problem $\left(P_m\right)$ possesses at least a couple of solutions $(u,\lambda )\in S\left(m\right)\times \mathbb{R}$ with $\lambda >0.$

Remark 2. 

In the present paper, we consider the case of non-homogeneous and mass supercritical general nonlinearities. It is also blank even for the existence. For $r\in (2,2_s^{\ast })$, the embedding $H_r^s\left({\mathbb{R}}^N\right)\hookrightarrow L^r\left({\mathbb{R}}^N\right)$ is compact. Nevertheless, the Sobolev term makes the problem more difficult, and the f leads to the relevant functional being unbounded on S(m). Motivated by [29,30], we prove that the corresponding functions satisfy mountain geometry and, using compactness argument and minimax approaches, we prove the existence of ground state solutions.

Now, we shall introduce the Pohozaev manifold and aim to prove that it is a natural constraint.

Lemma 1. 

If $u\in S\left(m\right)$ is a solution to problem (1), then $u\in {\mathcal{P}}_f$, where

$${\mathcal{P}}_f:=\{u\in H^s\left({\mathbb{R}}^N\right):P_f\left(u\right)=0\}$$

and

$$\begin{array}{cc}\hfill P_f\left(u\right)& =as{\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|}^{2}dx+bs{\left({\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}u|}^{2}dx\right)}^2\hfill \\ \hfill & -N{\int }_{{\mathbb{R}}^N}\left[{\displaystyle \frac{1}{2}}f\left(u\right)u-F\left(u\right)\right]dx-s{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx.\hfill \end{array}$$

(14)

Remark 3. 

From Lemma 1, we can obtain that ${\mathcal{L}}_u^{\prime }\left(\varsigma \right)={\displaystyle \frac{1}{\varsigma }}{P}_f(\varsigma \ast u)$. Particularly, $P_f\left(u\right)={\mathcal{L}}_u^{\prime }\left(1\right)$.

Lemma 2. 

For arbitrary $u\in S\left(m\right),\varsigma \in {\mathbb{R}}^{+}$ is a critical point of ${\mathcal{L}}_u\left(\varsigma \right)$ iff $\varsigma \ast u\in \rho$, where

$$\rho :={\mathcal{P}}_f\cap S\left(m\right).$$

(15)

Proof. 

From Remark 3, by the fact ${\mathcal{L}}_u^{\prime }\left(\varsigma \right)={\displaystyle \frac{1}{\varsigma }}{P}_f(\varsigma \ast u)$, we can directly obtain Lemma 2. □

Lemma 3. 

For arbitrary critical point of ${\mathcal{J}|}_{\rho }$, if ${\mathcal{L}}_u^{\prime \prime }\left(1\right)\ne 0$, there exists $\lambda \in \mathbb{R}$ such that

$$\begin{array}{c}\hfill {\mathcal{J}}^{\prime }\left(u\right)+\lambda u=0in{\mathbb{R}}^N.\end{array}$$

Proof. 

Take $u\in \rho$ be a critical point for $\mathcal{J}\left(u\right)$. Then, there exist $\lambda ,\mu \in \mathbb{R}$ such that

$${\mathcal{J}}^{\prime }\left(u\right)+\lambda u+\mu P_f^{\prime }\left(u\right)=0\mathrm{in}{\mathbb{R}}^N.$$

(16)

We claim that $\mu =0$. Define the energy functional of (16):

$$\mathcal{I}\left(u\right):=\mathcal{J}\left(u\right)+{\displaystyle \frac{1}{2}}\lambda {\parallel u\parallel }_2^2+\mu P_f\left(u\right).$$

Given that u solves (16), it must fulfill the corresponding Poho$\check{\mathrm{z}}$aev identity,

$${\displaystyle \frac{d}{dt}}{\mathcal{I}(\varsigma \ast u)|}_{\varsigma =1}=0.$$

Observing that

$$\begin{array}{cc}\hfill \mathcal{I}(\varsigma \ast u)& =\mathcal{J}(\varsigma \ast u)+{\displaystyle \frac{1}{2}}\lambda {\parallel u\parallel }_2^2+\mu P_f(\varsigma \ast u)\hfill \\ \hfill & ={\mathcal{L}}_u\left(\varsigma \right)+{\displaystyle \frac{1}{2}}\lambda {\parallel u\parallel }_2^2+\mu \varsigma {\mathcal{L}}_u^{\prime }\left(\varsigma \right),\hfill \end{array}$$

hence

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\mathcal{I}(\varsigma \ast u)=(1+\mu ){\mathcal{L}}_u^{\prime }\left(\varsigma \right)+\mu \varsigma {\mathcal{L}}_u^{\prime \prime }\left(\varsigma \right).\end{array}$$

Consequently,

$$\begin{array}{cc}\hfill 0& ={\displaystyle \frac{d}{dt}}{\mathcal{I}(\varsigma \ast u)|}_{\varsigma =1}=(1+\mu ){\mathcal{L}}_u^{\prime }\left(1\right)+\mu {\mathcal{L}}_u^{\prime \prime }\left(1\right)\hfill \\ \hfill & =(1+\mu )P_f\left(u\right)+\mu {\mathcal{L}}_u^{\prime \prime }\left(1\right)=\mu {\mathcal{L}}_u^{\prime \prime }\left(1\right).\hfill \end{array}$$

Since ${\mathcal{L}}_u^{\prime \prime }\left(1\right)\ne 0$, so we conclude that $\mu =0$. □

Lemma 4. 

Assume that (F1) and (F2) hold. Then, for arbitrary $m>0$, there exists some ${\delta }_m>0$ such that

$$\begin{array}{c}\hfill inf\{\varsigma >0:\exists u\in S_r{\left(m\right)with\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^2=1suchthat\varsigma \ast u\in {\rho }_r\}\ge {\delta }_m,\end{array}$$

(17)

where $S_r\left(m\right)=S\left(m\right)\cap H_r^s\left({\mathbb{R}}^N\right)$ and ${\rho }_r=\rho \cap H_r^s\left({\mathbb{R}}^N\right)$. That is,

$$\begin{array}{c}\hfill \underset{u\in {\rho }_r}{inf}{\{\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^2\}\ge {\delta }_m^{2s}.\end{array}$$

Proof. 

From $\varsigma \ast u\in {\rho }_r$, we obtain that $P_f(\varsigma \ast u)=0$. By ${\mathcal{L}}_u^{\prime }\left(\varsigma \right)={\displaystyle \frac{1}{\varsigma }}{P}_f(\varsigma \ast u)$, we obtain

$$\begin{array}{c}\hfill as{\varsigma }^{2s-1}{\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+bs{\varsigma }^{4s-1}{\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^4=N{\varsigma }^{-N-1}{\int }_{{\mathbb{R}}^N}\tilde{F}\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right)dx+s{\varsigma }^{2_s^{\ast }-1}{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx.\end{array}$$

From (F2), we can assume that there exists some suitable constant $C_c>0$ such that

$$\begin{array}{c}\hfill f\left(t\right)t\le C_c\left({|t|}^{\alpha }+{|t|}^{\beta }\right),\mathrm{for} \mathrm{any}t\in \mathbb{R}.\end{array}$$

(18)

According to (11), we can obtain

$$\begin{array}{c}\hfill {\parallel u\parallel }_{2_s^{\ast }}^{2_s^{\ast }}\le S_c^{-{\displaystyle \frac{2_s^{\ast }}{2}}}\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^{2_s^{\ast }}.\end{array}$$

(19)

Then, by fractional Gagliardo–Nirenberg–Sobolev inequality, there exists a non-negative $C_1$ such that

$$\begin{array}{c}\hfill {\parallel u\parallel }_{\alpha }^{\alpha }\le C_1,{\parallel u\parallel }_{\beta }^{\beta }\le C_1.\end{array}$$

(20)

By (F2), for ${\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2=1$, we have

$$\begin{array}{c}\hfill as{\varsigma }^{2s-1}<as{\varsigma }^{2s-1}+bs{\varsigma }^{4s-1}\le N\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta }}\right){\varsigma }^{-N-1}{\int }_{{\mathbb{R}}^N}f\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right){\varsigma }^{{\displaystyle \frac{N}{2}}}udx+s{\varsigma }^{2_s^{\ast }-1}{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx.\end{array}$$

(21)

Then, by (18)–(21), we obtain

$$\begin{array}{c}\hfill a<C_c{C}_{1}N\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta }}\right)\left({\varsigma }^{{\displaystyle \frac{N\alpha }{2}}-N-1}+{\varsigma }^{{\displaystyle \frac{N\beta }{2}}-N-1}\right)+S_c^{-{\displaystyle \frac{2_s^{\ast }}{2}}}s{\varsigma }^{2_s^{\ast }s-1}.\end{array}$$

Since $2+{\displaystyle \frac{8s}{N}}<\alpha <\beta$, we imply the existence of lower bound ${\delta }_m>0$. □

Lemma 5. 

Assume that (F1)–(F3) hold. Then, for arbitrary $u\in \rho$, we have ${\mathcal{L}}_u^{\prime \prime }\left(1\right)<0$ and ρ is a natural constraint of ${\mathcal{J}|}_{S\left(m\right)}$.

Proof. 

For arbitrary $u\in \rho$, we obtain

$$\begin{array}{c}\hfill {as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2{+bs\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^4-s{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx=N{\int }_{{\mathbb{R}}^N}\tilde{F}\left(u\right)dx.\end{array}$$

(22)

Hence

$$\begin{array}{cc}\hfill {\mathcal{L}}_u^{\prime \prime }\left(1\right)& {=as(2s-1)\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+bs(4s-1)\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4-N(N+1){\int }_{{\mathbb{R}}^N}F\left(u\right)dx\hfill \\ \hfill & +\left({\displaystyle \frac{3}{4}}{N}^2+{\displaystyle \frac{1}{2}}N\right){\int }_{{\mathbb{R}}^N}f\left(u\right)udx-{\displaystyle \frac{1}{4}}{N}^2{\int }_{{\mathbb{R}}^N}{f}^{\prime }\left(u\right)u^{2}dx-s(2_s^{\ast }-1){\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\hfill \\ \hfill & {=as(2s-1)\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+bs(4s-1)\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4+N(N+1){\int }_{{\mathbb{R}}^N}\tilde{F}\left(u\right)dx\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{N}^2{\int }_{{\mathbb{R}}^N}{\tilde{F}}^{\prime }\left(u\right)udx-s(2_s^{\ast }s-1){\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx.\hfill \end{array}$$

By (F3), (22), and Lemma 4, we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}_u^{\prime \prime }\left(1\right)& {\le as(2s-1)\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+bs(4s-1)\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4+\left(2N-\alpha N+2\right){\displaystyle \frac{N}{2}}{\int }_{{\mathbb{R}}^N}\tilde{F}\left(u\right)dx\hfill \\ \hfill & -s(2_s^{\ast }-1){\int }_{{\mathbb{R}}^N}{|u|}^{2_s\ast }dx\hfill \\ \hfill & =\left(N+2S-{\displaystyle \frac{N\alpha }{2}}\right){as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+\left(N+4S-{\displaystyle \frac{N\beta }{2}}\right)bs\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4\hfill \\ \hfill & -\left(2_s^{\ast }-\alpha \right){\displaystyle \frac{Ns}{2}}{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\hfill \\ \hfill & \le \left(N+2S-{\displaystyle \frac{N\alpha }{2}}\right){as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+\left(N+4S-{\displaystyle \frac{N\beta }{2}}\right)bs\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4<0.\hfill \end{array}$$

(23)

Therefore, according to Lemma 3, we conclude that $\rho$ is a natural constraint of ${\mathcal{J}|}_S\left(m\right)$. □

Lemma 6. 

Assume that (F1)–(F3) hold. Then, for arbitrary $u\in S\left(m\right)$, there exists a unique ${\varsigma }_{\tau }>0$ such that ${\varsigma }_{\tau }\ast u\in \rho$. Moreover,

$$\begin{array}{c}\hfill \mathcal{J}({\varsigma }_{\tau }\ast u)=\underset{\varsigma >0}{max}\mathcal{J}(\varsigma \ast u).\end{array}$$

(24)

Proof. 

By (18), we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}\left[F\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right)-{\displaystyle \frac{1}{2}}f\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right){\varsigma }^{{\displaystyle \frac{N}{2}}}u\right]dx& \ge \left({\displaystyle \frac{1}{\beta }}-{\displaystyle \frac{1}{2}}\right){\int }_{{\mathbb{R}}^N}f\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right){\varsigma }^{{\displaystyle \frac{N}{2}}}udx\hfill \\ \hfill & \ge \left({\displaystyle \frac{1}{\beta }}-{\displaystyle \frac{1}{2}}\right)C_c{\int }_{{\mathbb{R}}^N}\left(|{\varsigma }^{{\displaystyle \frac{N}{2}}}{u|}^{\alpha }+{|{\varsigma }^{{\displaystyle \frac{N}{2}}}u|}^{\beta }\right)dx.\hfill \end{array}$$

(25)

For $\varsigma \ge 1$, it can be inferred from (F2) and (13) that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}\left[F\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right)-{\displaystyle \frac{1}{2}}f\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right){\varsigma }^{{\displaystyle \frac{N}{2}}}u\right]dx\le \left(1-{\displaystyle \frac{\alpha }{2}}\right){\int }_{{\mathbb{R}}^N}{\varsigma }^{{\displaystyle \frac{N\alpha }{2}}}F\left(u\right)dx.\end{array}$$

(26)

It can be easily determined that

$$\begin{array}{cc}\hfill {\mathcal{L}}_u^{\prime }\left(\varsigma \right)& =as{\varsigma }^{2s-1}{\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+bs{\varsigma }^{4s-1}\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4+{\varsigma }^{-N-1}N{\int }_{{\mathbb{R}}^N}F\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right)dx\hfill \\ \hfill & -{\varsigma }^{-N-1}{\displaystyle \frac{N}{2}}{\int }_{{\mathbb{R}}^N}f\left({\varsigma }^{{\displaystyle \frac{N}{2}}}u\right){\varsigma }^{{\displaystyle \frac{N}{2}}}udx-{\varsigma }^{2_s^{\ast }s-1}s{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx.\hfill \end{array}$$

From (25) and (26), one can see that there exists sufficiently small $\varsigma >0$ such that ${\mathcal{L}}_u^{\prime }\left(\varsigma \right)>0$, and large enough $\varsigma >0$ such that ${\mathcal{L}}_u^{\prime }\left(\varsigma \right)<0$. Hence, there exists ${\varsigma }_{{\tau }_1}>0$ such that ${\mathcal{L}}_u^{\prime }\left({\varsigma }_{{\tau }_1}\right)=0$, and combining Lemma 2, one notes that ${\varsigma }_{{\tau }_1}\ast u\in \rho$. On the other hand, assume that there exists ${\varsigma }_{{\tau }_2}>0$ such that ${\varsigma }_{{\tau }_2}\ast u\in \rho$. Then by Lemma 5, we derive that ${\varsigma }_{{\tau }_1},{\varsigma }_{{\tau }_2}$ are the strict local maximum of ${\mathcal{L}}_u\left(\varsigma \right)$. Next, we suppose that ${\varsigma }_{{\tau }_1}<{\varsigma }_{{\tau }_2}$, then there exists ${\varsigma }_{{\tau }_0}\in ({\varsigma }_{{\tau }_1},{\varsigma }_{{\tau }_2})$ such that

$$\begin{array}{c}\hfill {\mathcal{L}}_u\left({\varsigma }_{{\tau }_0}\right)=\underset{\varsigma \in [{\varsigma }_{{\tau }_1},{\varsigma }_{{\tau }_2}]}{min}{\mathcal{L}}_u\left(\varsigma \right).\end{array}$$

In other words, ${\varsigma }_{{\tau }_0}$ is a local minimum of ${\mathcal{L}}_u\left(\varsigma \right)$. Consequently, we obtain that ${\mathcal{L}}_u^{\prime }\left({\varsigma }_{{\tau }_0}\right)=0$ and hence ${\varsigma }_{{\tau }_0}\ast u\in \rho$ with ${\mathcal{L}}_u^{\prime \prime }\left({\varsigma }_{{\tau }_0}\right)\ge 0$. This contradicts with Lemma 5. □

Remark 4. 

Assume that (F1)–(F3) hold. Then, for arbitrary $u\in \rho$, according to Lemma 6, we obtain

$$\begin{array}{c}\hfill \mathcal{J}\left(u\right)=\underset{\varsigma >0}{max}\mathcal{J}(\varsigma \ast u)>0.\end{array}$$

Next, we give characterizations of the mountain pass levels for $\mathcal{J}\left(u\right)$ and ${\mathcal{L}}_u\left(\varsigma \right)$. Let ${\mathcal{J}}^d$ denote the closed set $\{u\in S_r\left(m\right):\mathcal{J}\left(u\right)\le d\}$.

Lemma 7. 

$$\begin{array}{c}\hfill {\sigma }_r\left(m\right)=\underset{\gamma \in \Gamma \left(m\right)}{inf}\underset{ϵ\in [0,1]}{max}\mathcal{J}\left(\gamma \left(ϵ\right)\right),\end{array}$$

where

$$\begin{array}{c}\hfill \Gamma \left(m\right)=\{\gamma \in C\left([0,1],S_r\left(m\right)\right):\gamma \left(0\right)\in \mathcal{D},\gamma \left(1\right)\in {\mathcal{J}}^0\},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{\sigma }}_r\left(m\right)=\underset{\tilde{\gamma }\in \tilde{\Gamma }\left(m\right)}{inf}\underset{ϵ\in [0,1]}{max}{\mathcal{L}}_u\left(\tilde{\gamma }\left(ϵ\right)\right),\end{array}$$

where

$$\begin{array}{c}\hfill \tilde{\Gamma }\left(m\right)=\{\tilde{\gamma }\in C\left([0,1],S_r\left(m\right)\times \mathbb{R}\right):\tilde{\gamma }\left(0\right)\in (\mathcal{D},1),\gamma \left(1\right)\in ({\mathcal{J}}^0,1)\}.\end{array}$$

Then, one has

$$\begin{array}{c}\hfill {\sigma }_r\left(m\right)={\tilde{\sigma }}_r\left(m\right).\end{array}$$

Lemma 8. 

Assume that (F1), (F2) hold. Then, there exists a sequence $\left\{u_n\right\}\subset S_r\left(m\right)$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathcal{J}\left(u_n\right)\to {\sigma }_r\left(m\right),\hfill & \mathrm{as} n\to \infty ,\hfill \\ \mathcal{J}\prime \left(u_n\right){|}_{S_r\left(m\right)}\to 0,\hfill & \mathrm{as} n\to \infty ,\hfill \\ P_f\left(u_n\right)\to 0,\hfill & \mathrm{as} n\to \infty .\hfill \end{array}\right.\end{array}$$

(27)

Proof. 

By Ekeland’s variational principle [31], the first two properties hold. By Lemma 9 in [32], we can obtain the third property in (27). □

For given $m>0$, set

$$\begin{array}{c}\hfill E_f:=\underset{u\in \rho }{inf}\mathcal{J}\left(u\right).\end{array}$$

Since the solution u to problem (1) must belong to $\rho$, it can be inferred that if u satisfies $E_f$, then u is indeed a normalized solution.

Lemma 9. 

Assume that (F1)–(F3) hold. Then, for arbitrary $m>0$, we have $E_f>0.$

Proof. 

For any $u\in {\rho }_r$, we obtain $P_f\left(u\right)=0$ and

$$\begin{array}{c}\hfill {\displaystyle \frac{N}{2}}{\int }_{{\mathbb{R}}^N}f\left(u\right)udx-N{\int }_{{\mathbb{R}}^N}{F\left(u\right)dx\le as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+bs\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4.\end{array}$$

(28)

It can be seen from (F2) that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}f\left(u\right)u-F\left(u\right)\ge {\displaystyle \frac{\alpha -2}{2}}F\left(u\right).\end{array}$$

(29)

Combining (28) and (29), we can conclude that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}F\left(u\right)dx\le {\displaystyle \frac{2s}{N(\alpha -2)}}\left({a\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+b\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4\right).\end{array}$$

(30)

Therefore, from (29) and (30), we derive that

$$\begin{array}{cc}\hfill \mathcal{J}\left(u\right)& =\mathcal{J}\left(u\right)-{\displaystyle \frac{1}{4s}}{P}_f\left(u\right)\hfill \\ \hfill & \ge {\displaystyle \frac{a}{4}}{\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2-\left(1-{\displaystyle \frac{N(\alpha -2)}{8s}}\right){\int }_{{\mathbb{R}}^N}F\left(u\right)dx+\left({\displaystyle \frac{s}{N}}-{\displaystyle \frac{1}{4}}\right){\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\hfill \\ & \ge {\displaystyle \frac{a}{4}}{\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u\parallel }_2^2+\left({\displaystyle \frac{1}{4}}-{\displaystyle \frac{2s}{N(\alpha -2)}}\right)\left({a\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2+b\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4\right)\hfill \\ \hfill & +\left({\displaystyle \frac{s}{N}}-{\displaystyle \frac{1}{4}}\right){\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\hfill \\ \hfill & =\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{N(\alpha -2)-8s}{4N(\alpha -2)}}\right){a\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2++\left({\displaystyle \frac{1}{4}}-{\displaystyle \frac{2s}{N(\alpha -2)}}\right)b\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^4\hfill \\ \hfill & +\left({\displaystyle \frac{s}{N}}-{\displaystyle \frac{1}{4}}\right){\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx.\hfill \end{array}$$

(31)

Since $\alpha >2+{\displaystyle \frac{8s}{N}}$ and $N<4s$, we can conclude that ${\displaystyle \frac{N(\alpha -2)-8s}{4(\alpha -2)}}>0$, ${\displaystyle \frac{1}{4}}-{\displaystyle \frac{2s}{N(\alpha -2)}}>0$ and ${\displaystyle \frac{s}{N}}-{\displaystyle \frac{1}{4}}>0$. By Lemma 4, we have $E_f>0.$ □

Define

$$\begin{array}{c}\hfill E_f^r:=\underset{S_r\left(m\right)}{inf}\mathcal{J}\left(u\right).\end{array}$$

(32)

Then, it is easy to see that

$$\begin{array}{c}\hfill E_f^r=\underset{u\in {\rho }_r}{inf}\mathcal{J}\left(u\right).\end{array}$$

Lemma 10. 

Moreover, we have

$$\begin{array}{c}\hfill E_f^r=E_f.\end{array}$$

Proof. 

For any $u\in \rho$, let $u^{\star }$ be the symmetric decreasing rearrangement of u. Then we obtain $u^{\star }={|u|}^{\star }$. It can be inferred from the Schwarz rearrangement that $\mathcal{J}\left(u^{\star }\right)\le \mathcal{J}\left(u\right)$. It is clear that $E_f^r\ge E_f$. In addition, for arbitrary $t>0$, we have

$$\begin{array}{cc}\hfill & |\{x:(\varsigma \ast u^{\star })\left(x\right)>t\}|=|\{x:{\varsigma }^{{\displaystyle \frac{N}{2}}}{u}^{\star }\left(\varsigma x\right)>t\}|=|\{y:{\varsigma }^{{\displaystyle \frac{N}{2}}}{u}^{\star }\left(y\right)>t\}|{\varsigma }^{-N}\hfill \\ \hfill & =|\{u^{\star }\left(y\right)>{\varsigma }^{-{\displaystyle \frac{N}{2}}}t\}|{\varsigma }^{-N}=|\{u\left(y\right)>{\varsigma }^{-{\displaystyle \frac{N}{2}}}t\}|{\varsigma }^{-N}=|\{{\varsigma }^{{\displaystyle \frac{N}{2}}}u\left(y\right)>t\}|{\varsigma }^{-N}\hfill \\ \hfill & =|\{{\varsigma }^{{\displaystyle \frac{N}{2}}}u\left(\varsigma x\right)>t\}|=|\{(\varsigma \ast u)\left(x\right)>t\}|=|\{{(\varsigma \ast u)}^{\star }\left(x\right)>t\}|.\hfill \end{array}$$

Consequently,

$$\begin{array}{c}\hfill \varsigma \ast u^{\star }={(\varsigma \ast u)}^{\star },\mathrm{for} \mathrm{any}\varsigma \in {\mathbb{R}}^{+}.\end{array}$$

Then, for arbitrary $\varsigma \in {\mathbb{R}}^{+}$,

$$\begin{array}{c}\hfill \mathcal{J}(\varsigma \ast u^{\star })=\mathcal{J}\left({(\varsigma \ast u)}^{\star }\right)\le \mathcal{J}(\varsigma \ast u)\le \underset{t>0}{max}\mathcal{J}(t\ast u)=\mathcal{J}\left(u\right).\end{array}$$

Then, we derive that $E_f^r\le E_f$. Hence, $E_f^r=E_f$. □

Define

$$\begin{array}{c}\hfill \mathcal{D}:=\{u\in S\left(m\right):\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2<k\},\end{array}$$

where $k>0.$ Then combining (F2), (10), (11), and (18) by the scaling technique, we conclude that for $u\in \mathcal{D}$, there exists a suitable k such that $\mathcal{J}\left(u\right)>0.$

Lemma 11. 

${\sigma }_r\left(m\right)=E_f^r$.

Proof. 

First, we claim that $\mathcal{J}\left(u\right)\le 0$ can deduce $P_f\left(u\right)<0$. By Lemma 6, we have that for any $u\in S_r\left(m\right)$, there exists a unique ${\varsigma }_u\in {\mathbb{R}}^{+}$ such that ${\varsigma }_u\ast u\in {\rho }_r$ and $\mathcal{L}\left(\varsigma \right)$ is decreasing on $({\varsigma }_u,+\infty )$. Since $\mathcal{L}\left(1\right)=\mathcal{J}(1\ast u)=\mathcal{J}\left(u\right)\le 0$ and $\mathcal{L}\left({\varsigma }_u\right)>0,$ we infer that ${\varsigma }_u<1.$ Again by Lemma 6, we obtain the conclusion that

$$\begin{array}{c}\hfill P_f\left(u\right)=P_f(1\ast u)={\mathcal{L}}^{\prime }\left(1\right)<0.\end{array}$$

(33)

Next, let $u\in S_r\left(m\right)$. Take $0<{\varsigma }^{-}<1$ and ${\varsigma }^{+}>1$ such that ${\varsigma }^{-}\ast u\in \mathcal{D}$ and $\mathcal{J}({\varsigma }^{+}\ast u)<0$, respectively. Then define

$$\begin{array}{c}\hfill {\gamma }_u:ϵ\in [0,1]\mapsto \left((1-ϵ){\varsigma }^{-}+ϵ{\varsigma }^{+}\right)\ast u\in \Gamma .\end{array}$$

(34)

According to the definition of ${\sigma }_m$, we can determine that

$$\begin{array}{c}\hfill \underset{ϵ\in [0,1]}{max}\mathcal{J}\left({\gamma }_u\left(ϵ\right)\right)\ge {\sigma }_r\left(m\right).\end{array}$$

Hence, we obtain $E_f^r\ge {\sigma }_r\left(m\right)$. Furthermore, for arbitrary $\tilde{\gamma }\left(ϵ\right)=({\tilde{\gamma }}_1\left(ϵ\right),{\tilde{\gamma }}_2\left(ϵ\right))\in \tilde{\Gamma }$, consider the following function,

$$\begin{array}{c}\hfill \widetilde{P_f}\left(ϵ\right)=\widetilde{P_f}({\tilde{\gamma }}_2\left(ϵ\right)\ast {\tilde{\gamma }}_1\left(ϵ\right)).\end{array}$$

By (F2), (10), (11), and (18), we derive

$$\begin{array}{cc}\hfill P_f\left(u\right)& {\ge as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2{+bs\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^4-C_c\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta }}\right){N(\parallel u\parallel }_{\alpha }^{\beta }+{\parallel u\parallel }_{\beta }^{\beta }{)-s\parallel u\parallel }_{2_s^{\ast }}^{2_s^{\ast }}\hfill \\ \hfill & {\ge as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2{+bs\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^4-C_{c}C(s,N,\alpha )m^{\alpha -{\displaystyle \frac{N(\alpha -2)}{2s}}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta }}\right)N\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^{{\displaystyle \frac{N(\alpha -2)}{2s}}}\hfill \\ \hfill & -C_{c}C(s,N,\beta )m^{\beta -{\displaystyle \frac{N(\beta -2)}{2s}}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\beta }}\right){N\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^{{\displaystyle \frac{N(\beta -2)}{2s}}}-{\displaystyle \frac{s}{S_c^{\frac{2_s^{\ast }}{2}}}}\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^{2_s^{\ast }}.\hfill \end{array}$$

Since ${\displaystyle \frac{N(\alpha -2)}{2s}},{\displaystyle \frac{N(\beta -2)}{2s}}>2$, we can take a suitable k such that $P_f\left(u\right)>0.$ Then, since $({\tilde{\gamma }}_2\left(0\right)\ast {\tilde{\gamma }}_1\left(0\right))={\tilde{\gamma }}_1\left(0\right)\in \mathcal{D}$, we derive that

$$\begin{array}{c}\hfill \widetilde{P_f}\left(0\right)=\widetilde{P_f}\left({\tilde{\gamma }}_1\left(0\right)\right)>0.\end{array}$$

Since $({\tilde{\gamma }}_2\left(1\right)\ast {\tilde{\gamma }}_1\left(1\right)){\tilde{\gamma }}_1\left(1\right)\in {\mathcal{J}}^0$, one has

$$\begin{array}{c}\hfill \widetilde{P_f}\left(1\right)=\widetilde{P_f}\left({\tilde{\gamma }}_1\left(1\right)\right)<0.\end{array}$$

Hence, there exists $\overline{ϵ}\in (0,1)$ such that $\widetilde{P_f}\left(\overline{ϵ}\right)=0$, one has $({\tilde{\gamma }}_2\left(\overline{ϵ}\right)\ast {\tilde{\gamma }}_1\left(\overline{ϵ}\right))\in {\rho }_r.$ Then,

$$\begin{array}{c}\hfill \underset{s\in [0,1]}{max}\mathcal{J}({\tilde{\gamma }}_2\left(ϵ\right)\ast {\tilde{\gamma }}_1\left(ϵ\right))=\underset{ϵ\in [0,1]}{max}\mathcal{J}\left(\tilde{\gamma }\left(ϵ\right)\right)\ge \underset{u\in \rho }{inf}\mathcal{J}\left(u\right).\end{array}$$

Hence, we conclude that $E_f^r\le {\sigma }_r\left(m\right)$. We complete the proof. □

Proof of Theorem 1. 

Take a sequence $\left\{u_n\right\}\subset {\rho }_r$ such that $\mathcal{J}\left(u_n\right)\to E_f>0$. By (31), we know that $\left\{u_n\right\}$ is bounded in $H_r^s\left({\mathbb{R}}^N\right)$. Hence, up to a subsequence, we may assume that $u_n\rightharpoonup u$ in $H_r^s\left({\mathbb{R}}^N\right)$. Noting that ${\mathcal{L}}_u^{\prime \prime }\left(1\right)<0$ is an open constraint, there exist $\left\{{\lambda }_n\right\},\left\{{\mu }_n\right\}\subset \mathbb{R}$ such that

$$\begin{array}{c}\hfill {\mathcal{J}}^{\prime }\left(u_n\right)+{\lambda }_n{u}_n+{\mu }_n{\mathcal{P}}^{\prime }\left(u_n\right)\to 0\mathrm{as}n\to \infty .\end{array}$$

Applying a similar argument as Lemma 3, we have

$$\begin{array}{c}\hfill {\mu }_n{\mathcal{L}}_u^{\prime \prime }\left(1\right)\to 0.\end{array}$$

By (23), we obtain

$$\begin{array}{c}\hfill {\mu }_n{\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n\parallel }_2^2\to 0.\end{array}$$

Furthermore, from Lemma 4, we derive that ${\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^2\ge {\delta }_m^{2s}>0$. Therefore, ${\mu }_n\to 0$. Since $\left\{u_n\right\}$ is bounded, we have

$$\begin{array}{c}\hfill {\mathcal{J}}^{\prime }\left(u_n\right)u_n+{\lambda }_n\parallel u_n\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(35)

It can be concluded from (35) and $P_f\left(u_n\right)=0$ that

$$\begin{array}{cc}\hfill {\lambda }_n{\parallel u_n\parallel }_2^2& ={\int }_{{\mathbb{R}}^N}f\left(u_n\right)u_{n}dx+{\int }_{{\mathbb{R}}^N}|u_n{|}^{2_s^{\ast }}dx-a\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^2-b\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^4\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}f\left(u_n\right)u_{n}dx+{\int }_{{\mathbb{R}}^N}|u_n{|}^{2_s^{\ast }}dx-{\displaystyle \frac{N}{s}}{\int }_{{\mathbb{R}}^N}\tilde{F}\left(u_n\right)dx-{\int }_{{\mathbb{R}}^N}{|u_n|}^{2_s^{\ast }}dx\hfill \\ \hfill & ={\displaystyle \frac{N}{s}}{\int }_{{\mathbb{R}}^N}F\left(u_n\right)dx-{\displaystyle \frac{N-2s}{2s}}{\int }_{{\mathbb{R}}^N}f\left(u_n\right)u_{n}dx\hfill \\ \hfill & \ge \left(N-{\displaystyle \frac{\beta (N-2s)}{2s}}\right){\int }_{{\mathbb{R}}^N}F\left(u_n\right)dx.\hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill {as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^2+bs\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^4& =N{\int }_{{\mathbb{R}}^N}\tilde{F}\left(u_n\right)dx+s{\int }_{{\mathbb{R}}^N}|u_n|dx\hfill \\ \hfill & \le {\displaystyle \frac{N(\beta -2)}{2}}{\int }_{{\mathbb{R}}^N}F\left(u_n\right)dx+s{\int }_{{\mathbb{R}}^N}|u_n|dx.\hfill \end{array}$$

Since $2+{\displaystyle \frac{8s}{N}}<\alpha \le \beta <2_s^{\ast }$, by Lemma 4, ${\lambda }_n$ is bounded. Then, we assume up to a subsequence, ${\lambda }_n\to \lambda >0.$

Next, we claim $u\ne 0$. Assume by contradiction that $u=0$. Under (F2), by the compact embedding $H_r^s\left({\mathbb{R}}^N\right)\hookrightarrow L^t\left({\mathbb{R}}^N\right)$ for $t\in (2,2_s^{\ast })$, we derive that ${\int }_{{\mathbb{R}}^N}F\left(u_n\right)dx\to {\int }_{{\mathbb{R}}^N}F\left(u\right)dx$ as $n\to \infty .$ Then by $u=0$, we have ${\int }_{{\mathbb{R}}^N}\tilde{F}\left(u_n\right)dx=o\left(1\right)$ and ${\parallel u\parallel }_{2_s^{\ast }}^{2_s^{\ast }}=o\left(1\right)$. By $\left\{u_n\right\}\subset {\rho }_r$, we deduce that

$$\begin{array}{c}\hfill {as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^2+bs\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^4=o\left(1\right),\end{array}$$

a contradiction to Lemma 4.

We may assume that (up to a subsequence)

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}^N}{|{(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n|}^{2}dx=\mathcal{A}>0.\end{array}$$

Then, one can see that $u\in H_r^s\left({\mathbb{R}}^N\right)$ satisfies

$$\begin{array}{c}\hfill -(a+b\mathcal{A}){(-\Delta )}^{s}u+\lambda u=f\left(u\right)+{|u|}^{2_s^{\ast }-2}u\mathrm{in}{\mathbb{R}}^N.\end{array}$$

(36)

Then, we have $u\in {\mathcal{P}}^r$, where ${\mathcal{P}}^r=\mathcal{P}\cap H_r^s\left({\mathbb{R}}^N\right).$ Then, we deduce

$$\begin{array}{cc}\hfill {as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^2+bs\mathcal{A}\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^2& =N{\int }_{{\mathbb{R}}^N}\tilde{F}\left(u\right)dx+s{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\hfill \\ \hfill & =\underset{n\to \infty }{lim}N{\int }_{{\mathbb{R}}^N}\tilde{F}\left(u_n\right)dx+s{\int }_{{\mathbb{R}}^N}{|u_n|}^{2_s^{\ast }}dx\hfill \\ \hfill & =\underset{n\to \infty }{lim}\left({as\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^2+bs\mathcal{A}{\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n\parallel }_2^2\right)\hfill \\ \hfill & =as\mathcal{A}+bs{\mathcal{A}}^2.\hfill \end{array}$$

Hence, $(as+bs\mathcal{A})(\parallel {(-\Delta )}^{{\displaystyle \frac{s}{2}}}u{\parallel }_2^2-\mathcal{A})=0.$ Since $a>0,b>0,\mathcal{A}>0,s\in (0,1)$, we obtain ${\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2=\mathcal{A}$. Consequently, we derive that $u_n\to u$ in $D^{s,2}\left({\mathbb{R}}^N\right)$. Hence, by (36), we determine that u satisfies

$$\begin{array}{c}\hfill -(a+b{\int }_{{\mathbb{R}}^N}{|(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u|}^2{dx)(-\Delta )}^{s}u+\lambda u=f\left(u\right)+{|u|}^{2_s^{\ast }-2}u\mathrm{in}{\mathbb{R}}^N.\end{array}$$

(37)

Furthermore, we have

$$\begin{array}{cc}\hfill {a\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^2{+b\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\parallel }_2^4+\lambda {\parallel u\parallel }_2^2& ={\int }_{{\mathbb{R}}^N}f\left(u\right)udx+{\int }_{{\mathbb{R}}^N}{|u|}^{2_s^{\ast }}dx\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}f\left(u_n\right)u_{n}dx+{\int }_{{\mathbb{R}}^N}{|u_n|}^{2_s^{\ast }}dx+o\left(1\right)\hfill \\ \hfill & {=a\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^2{+b\parallel (-\Delta )}^{{\displaystyle \frac{s}{2}}}{u}_n{\parallel }_2^4+\lambda {\parallel u_n\parallel }_2^2+o\left(1\right).\hfill \end{array}$$

Then, we imply that

$$\begin{array}{c}\hfill \lambda (m-\parallel u{\parallel }_2^2)=0.\end{array}$$

We deduce that $u\in S\left(m\right)$ and $u\in \rho$. Therefore,

$$\begin{array}{c}\hfill E_f\le \mathcal{J}\left(u\right)=\underset{n\to \infty }{lim}\mathcal{J}\left(u_n\right)=E_f.\end{array}$$

That is to say, $(u,\lambda )\in H^s\left({\mathbb{R}}^N\right)\times \mathbb{R}$ is a couple of solutions to (1). □

4. Conclusions

In this paper, we have investigated the existence of ground state solutions for a class of fractional Kirchhoff equations involving $L^2$-supercritical nonlinearity and critical nonlinearity. By introducing the fractional Laplacian, we extend the classical Kirchhoff theory. By combining variational methods, critical point theory, and mountain pass theorem, we established the existence of normalized solutions under suitable assumptions on the nonlinearity f. Combining the Pohozaev manifold with mass constraint, we introduced a new set “ $\rho :={\mathcal{P}}_f\cap S\left(m\right)$" and proved that it is a natural constraint. To overcome the lack of compactness caused by the unbounded domain ${\mathbb{R}}^N$, we employed the symmetric decreasing rearrangement technique. By proving $E_f=E_f^r$ (Lemma 10), it means that radial symmetry does not introduce additional complexity or alter the energy characteristics of the system when solving the problem. Through rigorous analysis of the fibering functional ${\mathcal{L}}_u\left(\varsigma \right)$, we establish a direct relationship between the mountain pass level ${\sigma }_r\left(m\right)$ and the constrained energy minimum $E_f^r$ (Lemma 11), reconciling geometric and topological approaches to variational problems. These approaches effectively address the technical challenges posed by the fractional non-local operator. The conclusions obtained here and the techniques used in the paper can be used to study similar fractional equations.