Fractional Kirchhoff equation with a general critical nonlinearity

TL;DR

This paper investigates the existence and asymptotic behavior of solutions to a fractional Kirchhoff equation with critical nonlinearity in  small, using a perturbation approach without the Ambrosetti-Rabinowitz condition.

Contribution

It introduces a novel perturbation method to prove solution existence for fractional Kirchhoff equations without the Ambrosetti-Rabinowitz condition when parameter b is small.

Findings

Existence of solutions for small b

Asymptotic behavior of solutions as b approaches zero

Solution properties without the Ambrosetti-Rabinowitz condition

Abstract

In this paper, we study the fractional Kirchhoff equation with critical nonlinearity \begin{align*} \left(a+b\int_{\mathbb R^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^su+u=f(u)\ \ \mbox{in}\ \ \mathbb R^N, \end{align*} where $N>2s$ and $(-\Delta)^s$ is the fractional Laplacian with $0<s<1$. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti-Rabinowitz condition when the parameter $b$ small. What's more, we obtain the asymptotic behavior of solutions as $b\to 0$.

Equations

$$\left(a+b{\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{\frac{s}{2}}u{|}^{2}dx\right)(-\Delta {)}^{s}u+u=f(u)\text{mbox}in{\mathbb{R}}^N,$$

$$-\left(a+b{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)\Delta u+u=f(u)\text{mbox}in{\mathbb{R}}^N,$$

$$\displaystyle\left\{\begin{array}[]{ll}M\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}\right)(-\Delta)^{s}u=f(x,u),\ \ \mbox{in}\ \ \Omega,\\ u=0\ \ \ \ \mbox{in}\ \ \mathbb{R}^{N}\backslash\Omega,\end{array}\right.$$

$$a(-\Delta {)}^{s}u+u=f(u),u\in H^s({\mathbb{R}}^N).$$

$$(-\Delta {)}^s\varphi (x)=c_s\text{mbox}P.V.{\int }_{{\mathbb{R}}^N}\frac{\varphi (x)-\varphi (y)}{|x-y{|}^{N+2s}}\mathrm{d}y,x\in {\mathbb{R}}^N,$$

$$H^s({\mathbb{R}}^N)=\left\{u\in L^2({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^N}|\xi {|}^{2s}|\hat{u}{|}^2\mathrm{d}\xi <\infty \right\},$$

$${\mathcal{D}}^{s,2}({\mathbb{R}}^N)=\{u\in L^{2_s^{\ast }}({\mathbb{R}}^N):|\xi {|}^s\hat{u}\in L^2({\mathbb{R}}^N)\},$$

$$I_b(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(a|(-\Delta {)}^{\frac{s}{2}}u{|}^2+u^2\right)\mathrm{d}x+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{\frac{s}{2}}u{|}^2\right)}^2\mathrm{d}x-{\int }_{{\mathbb{R}}^N}F(u)\mathrm{d}x,$$

$$\left(a+b\Vert u{\Vert }_{{\mathcal{D}}^{s,2}}^2\right){\int }_{{\mathbb{R}}^N}(-\Delta {)}^{\frac{s}{2}}u(-\Delta {)}^{\frac{s}{2}}\varphi \mathrm{d}x+{\int }_{{\mathbb{R}}^N}u\varphi \mathrm{d}x={\int }_{{\mathbb{R}}^N}f(u)\varphi \mathrm{d}x.$$

$$L(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(a|(-\Delta {)}^{\frac{s}{2}}u{|}^2+u^2\right)\mathrm{d}x-{\int }_{{\mathbb{R}}^N}F(u)\mathrm{d}x,u\in H^s({\mathbb{R}}^N).$$

$$c=\underset{\gamma \in {\Gamma }_L}{\inf }\underset{t\in [0,1]}{\max }L(\gamma (t))>0,$$

$$\frac{N-2s}{2}{\int }_{{\mathbb{R}}^N}a|(-\Delta {)}^{s/2}u{|}^2\mathrm{d}x+\frac{N}{2}{\int }_{{\mathbb{R}}^N}{u}^2\mathrm{d}x=N{\int }_{{\mathbb{R}}^N}F(u)\mathrm{d}x.$$

$$L(U_{\tau})=\Big{(}\frac{a\tau^{N-2s}}{2}-\frac{N-2s}{2N}\tau^{N}\Big{)}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}U|^{2}.$$

$$I_b(U_{{\tau }_0})<-2,\Vert U_{\tau }\Vert \le {\mathcal{C}}_0,\forall \tau \in (0,{\tau }_0],\Vert u\Vert \le {\mathcal{C}}_0,\forall u\in S_r.$$

$$\displaystyle\Upsilon_{b}=\big{\{}\gamma\in C([0,\tau_{0}],H_{r}^{s}(\mathbb{R}^{N})):\gamma(0)=0,\gamma(\tau_{0})=U_{\tau_{0}},\|\gamma(\tau)\|\leq\mathcal{C}_{0}+1,\tau\in[0,\tau_{0}]\big{\}}.$$

$$I_b^{\alpha }:=\{u\in H_r^s({\mathbb{R}}^N):I_b(u)\le \alpha \}$$

$$S^d:=\left\{u\in H_r^s({\mathbb{R}}^N):\underset{v\in S_r}{\inf }\Vert u-v\Vert \le d\right\}.$$

$$\underset{n\to \infty }{\lim }{I}_{b_n}(u_{b_n})\le E\text{mbox}and\underset{n\to \infty }{\lim }{I}_{b_n}^{\prime }(u_{b_n})=0.$$

$$I_b^{\prime }(u_b)\varphi =L^{\prime }(u_b)\varphi +b\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2{\int }_{{\mathbb{R}}^N}(-\Delta {)}^{s/2}{u}_b(-\Delta {)}^{s/2}\varphi .$$

$$\Vert I_b^{\prime }(u)\Vert \ge \omega$$

$$I_b(\gamma (\tau ))\ge C_b-\alpha \text{mbox}impliesthat\gamma (\tau )\in S^{\frac{d}{2}},$$

$$\left(a+b\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+bA\right)\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+\Vert u_b{\Vert }_2^2={\int }_{{\mathbb{R}}^N}g(u_b)u_b+\Vert u_b{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}.$$

$$\frac{N-2s}{2}\left(a+b\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+bA\right)\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+\frac{N}{2}\Vert u_b{\Vert }_2^2=N{\int }_{{\mathbb{R}}^N}G(u_b)+\frac{N}{2_s^{\ast }}\Vert u_b{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}.$$

$$\left(a+b\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+bA\right)(\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+A)+(\Vert u_b{\Vert }_2^2+\Vert v_n{\Vert }_2^2)={\int }_{{\mathbb{R}}^N}g(u_b)u_b+\Vert u_b{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}+\Vert v_n{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}+o_n(1).$$

$$\left(a+b\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+bA\right)A+\Vert v_n{\Vert }_2^2=\Vert v_n{\Vert }_{2_s^{\ast }}^{2_s^{\ast }}+o_n(1).$$

$$I_b(u_n)$$

$$\ge \left(\frac{1}{2}-\frac{1}{2_s^{\ast }}\right)aA+b\left(\frac{1}{4}-\frac{1}{2_s^{\ast }}\right)(\Vert u_b{\Vert }_{{\mathcal{D}}^{s,2}}^2+A{)}^2+o(1).$$

$$I_b(u_n)\ge \left(\frac{1}{2}-\frac{1}{2_s^{\ast }}\right)aA-4b\left|\frac{1}{4}-\frac{1}{2_s^{\ast }}\right|B^2+o(1).$$

$$\left(\frac{1}{2}-\frac{1}{2_s^{\ast }}\right)aA\le D_b+b\left|\frac{1}{4}-\frac{1}{2_s^{\ast }}\right|B^2.$$

Full text

[TABLE]

where $N>2s$ and $(-\Delta)^{s}$ is the fractional Laplacian with $0<s<1$. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti-Rabinowitz condition when the parameter $b$ small. What’s more, we obtain the asymptotic behavior of solutions as $b\rightarrow 0$.

W. Liu is the corresponding author.

1. Introduction and main result

In this paper, we are concerned with the following fractional Kirchhoff equation

[TABLE]

where $N>2s$ with $0<s<1$, $a,b$ are positive constants and $(-\Delta)^{s}u$ is the fractional Laplacian which arises in the description of various phenomena in the applied science, such as the phase transition [19], Markov processes [1] and fractional quantum mechanics [15]. When $a=1$ and $b=0$, (1.1) becomes the fractional Schrödinger equations which have been studied by many authors. We refer the readers to [2, 5, 6, 7] and the references therein for the details. When $s=1$, the problem (1.1) reduces to the well-known Kirchhoff equation

[TABLE]

which has been studied in the last decade, see [17, 9, 12]. The equation (1.2) is related to the stationary analogue of the Kirchhoff equation $u_{tt}-\left(a+b\int_{\Omega}|\nabla u|^{2}dx\right)\Delta u=f(x,u)$ on $\Omega\subset\mathbb{R}^{N}$ bounded, which was proposed by Kirchhoff [13] in 1883 as a generalization the classic D’Alembert’s wave equation for free vibrations of elastic strings.

Recently, in bounded regular domains of $\mathbb{R}^{N}$, Fiscella and Valdinoci [11] proposed the following fractional stationary Kirchhoff equation

[TABLE]

which models nonlocal aspects of the tension arising from nonlocal measurements of the fractional length of the string. Also in bounded domains, Autuori et al. [4] dealt with the existence and the asymptotic behavior of non-negative solutions of a class of fractional stationary Kirchhoff equation. In the whole of $\mathbb{R}^{N}$, Pucci et al. [18] established the existence and multiplicity of nontrivial non-negative entire solutions of a stationary Kirchhoff eigenvalue problem. In the subcritical case, by using minimax arguments, Ambrosio et al. [3] obtained the multiplicity results for (1.2) in $H_{r}^{s}(\mathbb{R}^{N})$ with $b$ small. Also in the subcritical case, without the (AR)-condition, the authors[20] investigated the existence of radial solutions by using the variational methods combined with a cut-off function technique. More recently, without the (AR)-condition and monotonicity assumptions, in low dimension($N=2,N=3$), Z. Liu et al.[16] studied the existence of ground states in the critical case. To the best of our knowledge, there are few papers on the fractional Kirchhoff equations involving the critical growth in $\mathbb{R}^{N}$ with $N>3$, because of the tough difficulties brought by the nonlocal term and the lack of compactness of the Sobolev embedding $H_{r}^{s}(\mathbb{R}^{N})\rightarrow L^{2_{s}^{*}}(\mathbb{R}^{N})$.

Motivated by the works above, we investigate the existence of the positive solutions of (1.1) in $\mathbb{R}^{N}(N>2s)$ with the critical growth. Precisely, $f$ satisfies the following conditions:

• $(f_{1})$

$f\in C^{1}(\mathbb{R}^{+},\mathbb{R})$, $\lim_{t\to 0}f(t)/t=0$ and $f(t)\equiv 0$ for $t\leq 0$,

• $({f_{2}})$

$\lim_{t\to\infty}{f(t)/t^{2_{s}^{*}-1}}=1$, where $2_{s}^{*}=\frac{2N}{N-2s}$,

• $({f_{3}})$

there exist $D>0$ and $p<2_{s}^{*}$ such that $f(t)\geq t^{2_{s}^{*}-1}+Dt^{p-1}$ for $t\geq 0$.

Our main result can read as

Theorem 1.1**.**

Suppose that $f$ satisfies $(f_{1})-(f_{3})$ and $\max\{2,2_{s}^{*}-2\}<p<2_{s}^{*}$, then for $b$ small, (1.1) admits a nontrivial positive radial solution $u_{b}$. What’s more, along a subsequence, $u_{b}$ converges to $u$ in $H_{r}^{s}(\mathbb{R}^{N})$ as $b\rightarrow 0$, where $u$ is a radial ground state to the limit problem

[TABLE]

Because of the presence of the Kirchhoff term, in high dimension $N>4s$, for the energy functional $I_{b}(u)$ (see section 2), one has $I_{b}(tu)\rightarrow+\infty$ as $t\rightarrow+\infty$ for each $u\neq 0$. That means Mountain pass geometry may not holds and Mountain pass theorem may not be appropriate. To overcome this difficulty, we use the variational method combined with the perturbation approach[21, 22] to get a special bounded (PS)-sequence. On the other hand, because of the presence of the Kirchhoff term, for the bounded (PS)-sequence $\{u_{n}\}$, even $u_{n}\rightarrow u_{0}$ weakly, it doesn’t hold in general that $u_{0}$ is the critical point of the energy functional, which brings us more tough to get the compactness. We use the properties of the special (PS)-sequence and some results of the limit problem (1.6) to recover the compactness. Moreover,we obtain the asymptotic behavior of the solutions of (1.1) as $b\rightarrow 0$.

The paper is organized as follows. Some preliminaries are presented in Section 2. In Section 3, we construct the min-max level. In Section 4, we complete the proof of Theorem 1.1.

2. Preliminaries and functional setting

2.1. Fractional order Sobolev spaces

The fractional Laplacian $(-\Delta)^{s}$ with $s\in(0,1)$ of a function $\phi:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is defined by $\mathcal{F}((-\Delta)^{s}\phi)(\xi)=|\xi|^{2s}\mathcal{F}(\phi)(\xi),$ where $\mathcal{F}$ is the Fourier transform. If $\phi$ is smooth enough, it can be computed by the following singular integral

[TABLE]

where $c_{s}$ is a normalization constant and P.V. stands the principal value. For any $s\in(0,1)$, we consider the fractional order Sobolev space

[TABLE]

endowed with the norm $\|u\|=\left(\int_{\mathbb{R}^{N}}(1+a|\xi|^{2s})|\hat{u}|^{2}\,\mathrm{d}\xi\right)^{1/2}.$ $H_{r}^{s}(\mathbb{R}^{N})$ denotes the space of radial functions in $H^{s}(\mathbb{R}^{N})$, i.e. $H_{r}^{s}(\mathbb{R}^{N})=\{u\in H^{s}(\mathbb{R}^{N}):u(x)=u(|x|)\}.$ The homogeneous Sobolev space $\mathcal{D}^{s,2}(\mathbb{R}^{N})$ is defined by

[TABLE]

which is the completion of $C_{0}^{\infty}(\mathbb{R}^{N})$ under the norm $\|u\|_{\mathcal{D}^{s,2}}^{2}=\|(-\Delta)^{s/2}u\|_{2}^{2}=\int_{\mathbb{R}^{N}}|\xi|^{2s}|\hat{u}|^{2}\,\mathrm{d}\xi.$

For the further introduction on the fractional order Sobolev space, we refer to [10]. Now, we introduce the following Sobolev embedding theorems.

Lemma 2.1** (see [14, 8, 10]).**

For any $s\in(0,1)$, $H^{s}(\mathbb{R}^{N})$ is continuously embedded into $L^{q}(\mathbb{R}^{N})$ for $q\in[2,2^{\ast}_{s}]$ and compactly embedded into $L^{q}_{loc}(\mathbb{R}^{N})$ for $q\in[1,2^{\ast}_{s})$. $H_{r}^{s}(\mathbb{R}^{N})$ is compactly embedded into $L^{q}(\mathbb{R}^{N})$ for $q\in(2,2^{\ast}_{s})$ and $\mathcal{D}^{s,2}(\mathbb{R}^{N})$ is continuously embedded into $L^{2^{\ast}_{s}}(\mathbb{R}^{N})$, i.e., there exists $S_{s}>0$ such that $S_{s}\left(\int_{\mathbb{R}^{N}}|u|^{2^{\ast}_{s}}\,\mathrm{d}x\right)^{2/2^{\ast}_{s}}\leq\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}\,\mathrm{d}x.$

2.2. The variational setting

We define the energy functional $I_{b}:H^{s}(\mathbb{R}^{N})\to\mathbb{R}$ by

[TABLE]

with $F(t)=\int_{0}^{t}f(\zeta)\,\mathrm{d}\zeta$. It is standard to show that $I_{b}$ is of class $C^{1}$.

Definition 2.2**.**

We call $u\in H^{s}(\mathbb{R}^{N})$ a weak solution of (1.1) if for any $\phi\in H^{s}(\mathbb{R}^{N})$,

[TABLE]

Obviously, the critical points of $I_{b}$ are the weak solutions of (1.1).

Similar to the proof of Brezis-Lieb Lemma in [21], we can give the following lemma.

Lemma 2.3**.**

For $s\in(0,1)$, assume $(f_{1})-(f_{2})$ hold. Let $\{u_{n}\}\subset H^{s}(\mathbb{R}^{N})$ such that $u_{n}\to u$ weakly in $H^{s}(\mathbb{R}^{N})$ and a.e. in $\mathbb{R}^{N}$ as $n\to\infty$, then $\int_{\mathbb{R}^{N}}F(u_{n})\to\int_{\mathbb{R}^{N}}F(u_{n}-u)+\int_{\mathbb{R}^{N}}F(u).$

When $b=0$, problem (1.1) becomes the limit problem (1.6) which plays a crucial role in our paper. The energy functional of (1.6) is defined as

[TABLE]

With the same assumptions on $f$ in Theorem 1.1, it is not difficult to check that $L(u)$ satisfies the Mountain pass geometry. The Mountain pass value denoted by $c$ is defined by

[TABLE]

where $\Gamma_{L}=\{\gamma\in C([0,1],H^{s}(\mathbb{R}^{N})),\gamma(0)=0,L(\gamma(1))<0\}.$ In the following, we present some results of the ground states of (1.6) and the proof is similar as that in [22].

Proposition 2.4**.**

Suppose $f$ satisfies $(f_{1})-(f_{3})$ and $\max\{2,2_{s}^{*}-2\}<p<2_{s}^{*}$. Let $S_{r}$ be the set of positive radial ground states of (1.6), then

(i) $S_{r}$ is not empty and $S_{r}$ is compact in $H_{r}^{s}(\mathbb{R}^{N})$,

(ii) $c<\frac{s}{N}(aS_{s})^{\frac{N}{2s}}$ and $c$ agrees with the least energy level denoted by $E$, that is, there exists $\gamma\in\Gamma_{L}$ such that $u\in\gamma(t)$ and $\max_{[0,1]}L(\gamma(t))=E$, where $u\in S_{r}$,

(iii)$u\in S_{r}$ satisfies the Pohozǎev identity

[TABLE]

3. The minimax level

In order to get a bounded (PS)-sequence by the local deformation argument, a different min-max level is needed. Take $U\in S_{r}$ be arbitrary but fixed. By the definition of $\hat{U}=\mathcal{F}(U)$, for U_{\tau}(x)=U\big{(}\frac{x}{\tau}\big{)},\tau>0, we have $\hat{U}_{\tau}(\cdot)=\tau^{N}\hat{U}(\tau\cdot)$. Thus $\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}U_{\tau}|^{2}\,\mathrm{d}x=\tau^{N-2s}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}U|^{2}\,\mathrm{d}x.$ From the Pohozǎev identity (2.1), we obtain

[TABLE]

So, there exists $\tau_{0}>1$ such that $L(U_{\tau})<-2$ for $\tau\geq\tau_{0}$. Set $D_{b}\equiv\max_{\tau\in[0,\tau_{0}]}I_{b}(U_{\tau}).$ Noting that $I_{b}(U_{\tau})=L(U_{\tau})+\frac{b}{4}\|U_{\tau}\|_{\mathcal{D}^{s,2}}^{4}$ and $\max_{\tau\in[0,\tau_{0}]}L(U_{\tau})=E$, we have $D_{b}\rightarrow E$ as $b\rightarrow 0^{+}$.

Lemma 3.1**.**

There exist $b_{1}>0$ and $\mathcal{C}_{0}>0$, such that for any $0<b<b_{1}$ there hold

[TABLE]

Proof.

Since $S_{r}$ is compact, it is easy to verify that there exists $\mathcal{C}_{0}>0$ such that the second and third part of the assertion hold. It follows from $I_{b}(U_{\tau_{0}})\leq L(U_{\tau_{0}})+\frac{b}{4}\mathcal{C}_{0}^{4}$ and $L(U_{\tau_{0}})<-2$ that the first part holds for any $0<b<b_{1}$, where $b_{1}>0$ small. The proof is completed. ∎

Now, for any $b\in(0,b_{1})$, we define a min-max value $C_{b}:=\inf_{\gamma\in\Upsilon_{b}}\max_{\tau\in[0,\tau_{0}]}I_{b}(\gamma(\tau))$, where

[TABLE]

Proposition 3.2**.**

$\lim\limits_{b\rightarrow 0^{+}}C_{b}=E.$**

Proof.

For $\tau>0$, by $\|U_{\tau}\|^{2}=a\tau^{N-2s}\|U\|_{\mathcal{D}^{s,2}}^{2}+\tau^{N}\|U\|_{2}^{2},$ we can define $U_{0}\equiv 0$. So $U_{\tau}\in\Upsilon_{b}$. Moreover, $\limsup_{b\rightarrow 0^{+}}C_{b}\leq\lim_{b\rightarrow 0^{+}}D_{b}=E=c.$ On the other hand, for any $\gamma\in\Upsilon_{b}$, it follows from $L(U_{\tau_{0}})<-2$ that $\tilde{\gamma}(\cdot)=\gamma(\tau_{0}\cdot)\in\Gamma_{L}$. Thus, from the definition of $c$ and $C_{b}$, we obtain $C_{b}\geq E$ for any $b\in(0,b_{1})$. The proof is completed. ∎

4. The Proof of Theorem 1.1

For $\alpha,d>0$, we define

[TABLE]

and

[TABLE]

Proposition 4.1**.**

Let $\{b_{n}\}_{n=1}^{\infty}$ be such that $\lim_{n\rightarrow\infty}b_{n}=0$ and $\{u_{b_{n}}\}\subset S^{d}$ with

[TABLE]

Then for $d$ small, there is $u_{0}\in S_{r}$, up to a subsequence, such that $u_{b_{n}}\rightarrow u_{0}$ strongly in $H_{r}^{s}(\mathbb{R}^{N})$.

Proof.

For convenience, we write $u_{b_{n}}$ for $u_{b}$. Since $u_{b}\in S^{d}$, there exists $\tilde{u}_{b}\in S_{r}$ such that $\|u_{b}-\tilde{u}_{b}\|\leq d$. Let $v_{b}=u_{b}-\tilde{u}_{b}$. By the fact that $S_{r}$ is compact and $\|v_{b}\|\leq d$, up to a subsequence, there exist $\tilde{u}_{0}\in S_{r}$ and $v_{0}\in H^{s}(\mathbb{R}^{N})$, such that $\tilde{u}_{b}\rightarrow\tilde{u}_{0}$ strongly in $H_{r}^{s}(\mathbb{R}^{N})$ and $v_{b}\rightarrow v_{0}$ weakly in $H^{s}(\mathbb{R}^{N})$. Denoting $u_{0}=\tilde{u}_{0}+v_{0}$, then $u_{0}\in S^{d}$ and $u_{b}\rightarrow u_{0}$ weakly in $H_{r}^{s}(\mathbb{R}^{N})$. Next, we show $u_{b}\rightarrow u_{0}$ strongly in $H_{r}^{s}(\mathbb{R}^{N})$. Since $\lim_{n\rightarrow\infty}I_{b}^{\prime}(u_{b})=0$, then for any $\phi\in C_{0}^{\infty}(\mathbb{R}^{N})$,

[TABLE]

It follows from Lemma 2.1 and $u_{b}\in S^{d}$ that $L^{\prime}(u_{0})=0$ as $b\to 0$. Obviously $u_{0}\not\equiv 0$ by $u_{0}\in S^{d}$ with $d$ small. Thus $L(u_{0})\geq E$. Meanwhile, from Lemma 2.3, $I_{b}(u_{b})=L(u_{b})+\frac{b}{4}\|u_{b}\|_{\mathcal{D}^{s,2}}^{4}=L(u_{0})+L(u_{b}-u_{0})+o(1).$ Together with $\lim_{n\rightarrow\infty}I_{b_{n}}(u_{b_{n}})\leq E$, we obtain $L(u_{b}-u_{0})\leq o(1)$. Thus, by $(f_{1})-(f_{2})$ and Sobolev embedding theorem, there exists constant $c_{1}>0$ such that $\|u_{b}-u_{0}\|^{2}\leq c_{1}\|u_{b}-u_{0}\|^{2_{s}^{*}}$. If $\|u_{b}-u_{0}\|\nrightarrow 0$ as $b\rightarrow 0$, there exists constant $c_{2}>0$ such that $\|u_{b}-u_{0}\|\geq c_{2}$ for $b$ small. On the other hand, from $\tilde{u}_{0}\in S_{r}$ and $u_{0}\in S^{d}$, we get $\|\tilde{u}_{0}-u_{0}\|\leq d$. Then $\|u_{b}-u_{0}\|\leq\|u_{b}-\tilde{u}_{b}\|+\|\tilde{u}_{b}-\tilde{u}_{0}\|+\|\tilde{u}_{0}-u_{0}\|\leq 2d+o(1),$ which is a contradiction for $d$ small. The proof is completed. ∎

Remark 4.2**.**

By Proposition 4.1, for small $d\in(0,1)$, there exist $\omega>0,b_{0}>0$ such that

[TABLE]

Thus, we have the following proposition.

Proposition 4.3**.**

There exists $\alpha>0$ such that for small $b>0$ and $\gamma(\tau)=U(\frac{\cdot}{\tau}),\tau\in(0,\tau_{0}]$,

[TABLE]

Proof.

By the Pohozǎev identity (2.1), I_{b}(\gamma(\tau))=\Big{(}\frac{a\tau^{N-2s}}{2}-\frac{N-2s}{2N}\tau^{N}\Big{)}\|U\|_{\mathcal{D}^{s,2}}^{2}+\frac{b}{4}\tau^{2N-4s}\|U\|_{\mathcal{D}^{s,2}}^{4}. Then \lim_{b\rightarrow 0^{+}}\max_{\tau\in[0,\tau_{0}]}I_{b}(\gamma(\tau))=\max_{\tau\in[0,\tau_{0}]}\Big{(}\frac{a\tau^{N-2s}}{2}-\frac{N-2s}{2N}\tau^{N}\Big{)}\|U\|_{\mathcal{D}^{s,2}}^{2}=E. On the other hand, $\lim\limits_{b\rightarrow 0^{+}}C_{b}=E.$ The conclusion follows. ∎

Thanks to (4.1) and Proposition 4.3, we can prove the following proposition, which assures the existence of a bounded (PS)-sequence for $I_{b}$. The proof is similar as that in [21, 22]. We omit the details here.

Proposition 4.4**.**

For $b>0$ small, there exist $\{u_{n}\}_{n}\subset I_{b}^{D_{b}}\cap S^{d}$ such that $I_{b}^{\prime}(u_{n})\rightarrow 0$ as $n\rightarrow\infty$.

**The completion of Proof of Theorem 1.1 **

Proof.

It follows from Proposition 4.4 that there exists $b_{0}>0$ such that for $b\in(0,b_{0})$, there exists $\{u_{n}\}\in I_{b}^{D_{b}}\cap S^{d}$ with $I_{b}^{\prime}(u_{n})\rightarrow 0$ as $n\rightarrow\infty$. Thus, there exists $u_{b}\in H_{r}^{s}(\mathbb{R}^{N})$, up to a subsequence, such that $u_{n}\rightarrow u_{b}$ weakly in $H_{r}^{s}(\mathbb{R}^{N})$, $u_{n}\rightarrow u_{b}$ strongly in $L^{p}(\mathbb{R}^{N})$, $p\in(2,2_{s}^{*})$ and $u_{n}\rightarrow u_{b}$ a.e in $\mathbb{R}^{N}$. Next, we claim that $I_{b}^{\prime}(u_{b})=0$ for $b$ small. Set $f(t)=g(t)+t^{2_{s}^{*}-1}$. By Lemma 2.1, we have $\int_{\mathbb{R}^{N}}g(u_{n})\varphi=\int_{\mathbb{R}^{N}}g(u_{b})\varphi+o_{n}(1)$ for any $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$ and $\int_{\mathbb{R}^{N}}g(u_{n})u_{n}=\int_{\mathbb{R}^{N}}g(u_{b})u_{b}+o_{n}(1).$ Let $v_{n}=u_{n}-u_{b}$ and $\|v_{n}\|_{\mathcal{D}^{s,2}}^{2}\rightarrow A\geq 0$, then $\|u_{n}\|_{\mathcal{D}^{s,2}}^{2}=\|u_{b}\|_{\mathcal{D}^{s,2}}^{2}+A+o_{n}(1)$. From $I_{b}^{\prime}(u_{n})\rightarrow 0$, we have

[TABLE]

The corresponding Pohozǎev identity is

[TABLE]

It follows from $I_{b}^{\prime}(u_{n})u_{n}\rightarrow 0$ and Brezis-Lieb Lemma that

[TABLE]

Together with (4.2), we have

[TABLE]

It follows from Lemma 2.1 that $A\leq\frac{1}{a}\left(\frac{A}{S_{s}}\right)^{\frac{2_{s}^{*}}{2}}+o(1).$ If $A=0$, we have done. If $A>0$, then $A\geq a^{\frac{N-2s}{2s}}S_{s}^{\frac{N}{2s}}.$ By the Pohozǎev identity (4.3) and (4.4),

[TABLE]

On the other hand, from $\{u_{n}\}\in S^{d}$, for $d$ small, there exist $\tilde{u}_{n}\in S_{r}$ and $\tilde{v}_{n}\in H^{s}(\mathbb{R}^{N})$ such that $u_{n}=\tilde{u}_{n}+\tilde{v}_{n}$ with $\|\tilde{v}_{n}\|\leq d$. Thus $\|u_{n}\|_{\mathcal{D}^{s,2}}^{2}\leq\|\tilde{v}_{n}\|_{\mathcal{D}^{s,2}}^{2}+\|\tilde{u}_{n}\|_{\mathcal{D}^{s,2}}^{2}\leq 1+\sup_{v\in S_{r}}\|v\|_{\mathcal{D}^{s,2}}^{2}\triangleq B$ which implies that $\|u_{b}\|_{\mathcal{D}^{s,2}}^{2}+A\leq 2B$, where $B$ is independent of $b,n$ and $d$. So

[TABLE]

Meanwhile, from $\limsup_{n\rightarrow\infty}I_{b}(u_{n})\leq D_{b}$, we get

[TABLE]

Together with $A\geq a^{\frac{N-2s}{2s}}S_{s}^{\frac{N}{2s}}$, we have $\frac{s}{N}(aS_{s})^{\frac{N}{2s}}\leq D_{b}+b\left|\frac{1}{4}-\frac{1}{2_{s}^{*}}\right|B^{2}\rightarrow E,\mbox{ as }b\rightarrow 0,$ which is a contradiction with $E<\frac{s}{N}(aS_{s})^{\frac{N}{2s}}$. So, the claim is true. Since $u_{n}\in S^{d}$, then for $d$ small, $u_{b}\not\equiv 0$. Thus, for $b$ and $d$ small, there exists $u_{b}\in H_{r}^{s}(\mathbb{R}^{N})$ which is a nontrivial solution of (1.1). In the following, we investigate the asymptotic behavior of $u_{b}$ as $b\rightarrow 0$. Noting that $D_{b}\rightarrow E$ as $b\rightarrow 0$, the similar proof as that in Proposition 4.1, we obtain that there exist $u\not\equiv 0$ such that $u_{b}\rightarrow u$ strongly in $H_{r}^{s}(\mathbb{R}^{N})$ with $L^{\prime}(u)=0$ and $L(u)=E$. The proof is finished. ∎

Acknowledgements.   This work is supported by the National Natural Science Foundation of China (11271364).