Nonlinear Choquard equations involving nonlocal operators

TL;DR

This paper investigates nonlinear Choquard equations with nonlocal operators, establishing conditions for existence and nonexistence of solutions, and demonstrating the presence of infinitely many solutions under certain parameters.

Contribution

It introduces new existence results for ground state solutions and multiple solutions for a class of nonlinear Choquard equations involving nonlocal operators.

Findings

Existence of a ground state solution when (N+α)/N < p < (N+α)/(N-1)

Nonexistence of solutions for p ≤ (N+α)/(N+1) or p ≥ (N+α)/(N-1)

Multiple solutions exist under specific parameter ranges.

Abstract

In this paper, we study nonlinear Choquard equations \begin{equation}\label{eq 1a1-} (-\Delta+id)^{\frac{1}{2}}u=(I_\alpha*{|u|^p})|u|^{p-2}u\ \ {\rm in} \ \ \mathbb{R}^N, \ \ \ u\in H^{\frac{1}{2}}(\mathbb{R}^N), \end{equation} where $(-\Delta+id)^\frac{1}{2}$ is a nonlocal operator, $p>0$, $N\geq2$ and $I_\alpha$ is the Riesz potential with order $\alpha\in(0,N)$. We show that there is a ground state solution to the above problem if $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$ and no solution if $0<p\leq\frac{N+\alpha}{N+1}$ or $p\geq\frac{N+\alpha}{N-1}$. Furthermore, the existence of infinity many solutions to the above problem is discussed when $p$ satisfies that $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$.

Equations

$$(-\Delta +id{)}^{\frac{1}{2}}u=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u\mathrm{in}{\mathbb{R}}^N,u\in H^{\frac{1}{2}}({\mathbb{R}}^N),$$

$$I_{\alpha }(x)=\frac{\Gamma (\frac{N-\alpha }{2})}{2^{\alpha }{\pi }^{\frac{N}{2}}\Gamma (\frac{\alpha }{2})|x{|}^{N-\alpha }},$$

$$H^{\frac{1}{2}}({\mathbb{R}}^N)=\{u\in L^2({\mathbb{R}}^N):(-\Delta +id{)}^{\frac{1}{4}}u\in L^2({\mathbb{R}}^N)\}$$

$$\Vert u{\Vert }_{H^{\frac{1}{2}}({\mathbb{R}}^N)}=\Vert (-\Delta +id{)}^{\frac{1}{4}}u{\Vert }_{L^2({\mathbb{R}}^N)}.$$

$$\left\{\begin{array}[]{lll}-\Delta u+u=(I_{2}\ast|u|^{2})u&{\rm in}\quad\mathbb{R}^{3},\\[5.69054pt] \lim_{|x|\to+\infty}u(x)=0\end{array}\right.$$

$$\left\{\begin{array}[]{lll}-\Delta u+u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u&{\rm in}\quad\mathbb{R}^{N},\\[5.69054pt] \lim_{|x|\to+\infty}u(x)=0,\end{array}\right.$$

$${\int }_{{\mathbb{R}}^N}|I_{\alpha }\ast v{|}^{\frac{Ns}{N-\alpha s}}dx\le C({\int }_{{\mathbb{R}}^N}|v{|}^{s}dx{)}^{\frac{N}{N-\alpha s}},$$

$$M_p=\inf \left\{{\int }_{{\mathbb{R}}^N}{\left|(-\Delta +id{)}^{\frac{1}{4}}u\right|}^{2}dx:{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |u{|}^p)(x)|u(x){|}^{p}dx=1\right\}.$$

$$\underset{m\to \infty }{\lim }{\int }_{\Omega }||w_m{|}^q-|w_m-w{|}^q-|w{|}^q{|}^{\frac{s}{q}}dx=0.$$

$$\underset{m\to \infty }{\lim }\left[{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |w_m{|}^p)(x)|w_m(x){|}^{p}dx-{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |w_m-w{|}^p)(x)|(w_m-w)(x){|}^{p}dx\right]$$

$$={\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |w{|}^p)(x)|w(x){|}^{p}dx.$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |w_m{|}^p)(x)|w_m(x){|}^{p}dx-{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |w_m-w{|}^p)(x)|(w_m-w)(x){|}^{p}dx$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast (|w_m{|}^p-|w_m-w{|}^p))(x)|(w_m-w)(x){|}^{p}dx$$

$$M_p=\inf \left\{{\int }_{{\mathbb{R}}^N}{\left|(-\Delta +id{)}^{\frac{1}{4}}u\right|}^{2}dx:{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |u{|}^p)(x)|u(x){|}^{p}dx=1\right\}.$$

$$\underset{z\in {\mathbb{R}}^N}{\sup }{\int }_{B_r(z)}|v_m(x){|}^{q}dx\to 0$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |v_m{|}^p)(x)|v_m(x){|}^{p}dx\to 0$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |v_m{|}^p)(x)|v_m(x){|}^{p}dx={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|v_m(x){|}^p|v_m(z){|}^p}{|x-z{|}^{N-\alpha }}dxdz$$

$${\left({\int }_{B_r(z)}|v_m(x){|}^{\frac{2Np}{N+\alpha }}dx\right)}^{\frac{N+\alpha }{2Np}}$$

$$\lambda =\frac{\frac{2Np}{N+\alpha }-q}{\frac{2N}{N-1}-q}\cdot \frac{\frac{2N}{N-1}}{\frac{2Np}{N+\alpha }}.$$

$${\int }_{B_r(z)}|v_m(x){|}^{\frac{2Np}{N+\alpha }}dx\le c_2^{\frac{2Np}{N+\alpha }}{\Vert v_m\Vert }_{L^q(B_r(z))}^{(1-\lambda )\frac{2Np}{N+\alpha }}\left({\int }_{{\mathbb{R}}^N}|(-\Delta +id{)}^{\frac{1}{4}}{v}_m{|}^{2}dx\right).$$

$${\int }_{B_r(z)}|v_m(x){|}^{\frac{2Np}{N+\alpha }}dx$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |v_m{|}^p(x))|v_m(x){|}^{p}dx$$

$${\int }_{{\mathbb{R}}^N}(|\nabla v_m(x){|}^2+|v_m(x){|}^2)dx\to M_p$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |v_m{|}^p)(x)|v_m(x){|}^{p}dx=1.$$

$$\delta =\underset{m\to \infty }{\lim }\underset{\tilde{x}\in {\mathbb{R}}^N}{\sup }{\int }_{B_1(\tilde{x})}|v_m(x){|}^{\frac{2Np}{N+\alpha }}dx>0.$$

$${\int }_{B_1(x_m)}|v_m(x){|}^{\frac{2Np}{N+\alpha }}dx>\frac{\delta }{2}.$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |w_m{|}^p)(x)|w_m(x){|}^{p}dx=1,{\int }_{{\mathbb{R}}^N}|(-\Delta +id{)}^{\frac{1}{4}}{w}_m{|}^{2}dx\to M_p,$$

$${\int }_{B_1(0)}|w_m(x){|}^{\frac{2Np}{N+\alpha }}dx>\frac{\delta }{2}.$$

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |w_m{|}^p)(x)|w_m(x){|}^{p}dx$$

$$M_p=\underset{m\to \infty }{\lim }\Vert w_m{\Vert }_{H^{\frac{1}{2}}({\mathbb{R}}^N)}^2=\Vert w{\Vert }_{H^{\frac{1}{2}}({\mathbb{R}}^N)}^2+\underset{m\to \infty }{\lim }\Vert w_m-w{\Vert }_{H^{\frac{1}{2}}({\mathbb{R}}^N)}^2$$

Full text

Nonlinear Choquard equations involving

nonlocal operators

Wanwan Wang

Department of Mathematics, Jiangxi Normal University,

Nanchang, Jiangxi 330022, PR China

††E-mail address: wwwang2014@yeah.net (W. Wang).††MSC2010: 35J60, 35J65, 35B06.††Keywords: Choquard equation, Nonlocal operator, Ground state solution, Pohozǎev identity.

Abstract. In this paper, we study nonlinear Choquard equations

[TABLE]

where $(-\Delta+id)^{\frac{1}{2}}$ is a nonlocal operator, $p>0$, $N\geq 2$ and $I_{\alpha}$ is the Riesz potential with order $\alpha\in(0,N)$. We show that there is a ground state solution to problem (1) if $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$ and no solution to problem (1) if $0<p\leq\frac{N+\alpha}{N+1}$ or $p\geq\frac{N+\alpha}{N-1}$. Furthermore, the existence of infinity many solutions to problem (1) is discussed when $p$ satisfies that $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$.

1. Introduction

Our purpose of this paper is to consider the solutions of nonlinear Choquard equations

[TABLE]

where $p>0$, $N\geq 2$, $I_{\alpha}:\mathbb{R}^{N}\backslash\{0\}\to\mathbb{R}$ is the Riesz potential with order $\alpha\in(0,N)$ given by

[TABLE]

here $\Gamma$ is gamma function, see [12]. The nonlocal operator $(-\Delta+id)^{\frac{1}{2}}$ can be characterized as $\mathcal{F}((-\Delta+id)^{\frac{1}{2}}\phi)(\xi)=(1+|\xi|^{2})^{\frac{1}{2}}\mathcal{F}(\phi)(\xi)$, here $\mathcal{F}$ is the Fourier transform. The Hilbert space $H^{\frac{1}{2}}(\mathbb{R}^{N})$ is defined as

[TABLE]

with the norm

[TABLE]

As early as in 1954, in a pioneering work of Pekar [11] where described the quantum mechanics of a polaron, the nonlinear Choquard equation

[TABLE]

is appeared. For general case, Moroz-Van Schaftingen in [9] studied the problem

[TABLE]

they obtained the results of existence, qualitative properties and decay asymptotic. In this paper, we consider the related nonlocal problem (1.1). To state our results, we first introduce Hardy-Littlewood-Sobolev inequality which states that if $s\in(1,\frac{N}{\alpha})$, then for every $v\in L^{s}(\mathbb{R}^{N})$, $I_{\alpha}*v\in L^{\frac{Ns}{N-\alpha s}}(\mathbb{R}^{N})$ and

[TABLE]

where $C>0$ depends on $\alpha$, $N$ and $s$. We know that the fractional Sobolev embedding $H^{\frac{1}{2}}(\mathbb{R}^{N})\subset L^{t}(\mathbb{R}^{N})$ for $t\in[2,2^{\sharp}]$, where $2^{\sharp}:=\frac{2N}{N-1}$, also note that $H^{\frac{1}{2}}(\mathbb{R}^{N})\subset L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^{N})$ if and only if $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$. Now we state our main theorem.

Theorem 1.1**.**

Assume that $N\geq 2$ and $\alpha\in(0,N)$.

$(i)$* If $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$, then there exists a $C^{2}$ positive ground state solution to problem (1.1).*

$(ii)$* If $0<p\leq\frac{N+\alpha}{N+1}$ or $p\geq\frac{N+\alpha}{N-1}$, then there is no nontrivial solution to problem (1.1).*

To prove the existence of solutions in Theorem 1.1 when $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$, we apply the critical points theory to the associated minimizing problem

[TABLE]

We note that the minimization of $M_{p}$ is a nontrivial solution of problem (1.1). Here we use the concentration compactness argument and a nonlocal version of Brezis-Lieb lemma to prove that $M_{p}$ can be achieved. Then we establish Pohozǎev identity to obtain the nonexistence results in Theorem 1.1.

Theorem 1.2**.**

Let $N\geq 2$, $\alpha\in(0,N)$ and $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$. Then there exists infinitely many distinct solutions to problem (1.1).

2. Preliminaries

In this section, we introduce some lemmas.

Lemma 2.1**.**

[15]** Let $\Omega$ be a domain in $\mathbb{R}^{N}$, $s>1$ and $\{w_{m}\}_{m\in\mathbb{N}}$ be a bounded sequence in $L^{s}(\Omega)$. If $w_{m}\to w$ almost everywhere on $\Omega$ as $m\to\infty$, then for every $q\in[1,s]$, we have that

[TABLE]

Lemma 2.2**.**

Let $\alpha\in(0,N)$, $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$ and $\{w_{m}\}_{m\in\mathbb{N}}$ be a bounded sequence in $L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^{N})$. Assume that

(i) ${w_{m}}$ weakly converges to ${w}$ in $L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^{N})$;

*(ii) ${w_{m}}\to w$ almost everywhere on $\mathbb{R}^{N}$.

Then*

[TABLE]

Proof. We observe that

[TABLE]

By the Hölder inequality, we have that

[TABLE]

Using Lemma 2.1 with $q=p$ and $s=\frac{2Np}{N+\alpha}$, we know that $|w_{m}|^{p}-|w_{m}-w|^{p}\to|w|^{p}$, strongly in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^{N})$ as $m\to\infty$. By the Hardy-Littlewood-Sobolev inequality, this implies that $I_{\alpha}*(|w_{m}|^{p}-|w_{m}-w|^{p})\to I_{\alpha}*|w|^{p}$ in $L^{\frac{2N}{N-\alpha}}(\mathbb{R}^{N})$ as $m\to\infty$. Since $|w_{m}-w|^{p}\rightharpoonup 0$ in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^{N})$ as $m\to\infty$, then $\int_{\mathbb{R}^{N}}(I_{\alpha}*(|w_{m}|^{p}-|(w_{m}-w)|^{p}))(x)|(w_{m}-w)(x)|^{p}\,dx\to 0$ as $m\to\infty$. This ends the proof. $\Box$

3. Ground state solution

In this section, we study the existence of ground state solutions of problem (1.1). To this end, let us consider the following minimizing problem

[TABLE]

Proposition 3.1**.**

The minimizing problem $M_{p}$ is achieved by a function $v\in{H^{\frac{1}{2}}(\mathbb{R}^{N})}$, which is a solution of problem (1.1) up to a translation.

To prove this result, we first introduce some lemma as follows.

Lemma 3.1**.**

Let $2\leq q<2^{\sharp}=\frac{2N}{N-1}$ and $r>0$. Suppose that $\{v_{m}\}_{m\in\mathbb{N}}$ is a bounded sequence in $H^{\frac{1}{2}}(\mathbb{R}^{N})$ and

[TABLE]

as $m\to\infty$. Then

[TABLE]

as $m\to\infty$ for $\frac{(N+\alpha)q}{2N}<p<\frac{N+\alpha}{N-1}$.

Proof. Let $l=t=\frac{2N}{N+\alpha}$, by Hardy-Littlewood-Sobolev inequality, we have that

[TABLE]

and

[TABLE]

where $c_{1},c_{2}>0$ and

[TABLE]

Choosing some suitable $q$ and $p$ such that $\lambda=\frac{2}{\frac{2Np}{N+\alpha}}$, we obtain that

[TABLE]

Now, covering $\mathbb{R}^{N}$ by balls of radius $r$, in such a way that each point of $\mathbb{R}^{N}$ is contained in at most $N+1$ balls, we have that

[TABLE]

Then

[TABLE]

The proof is complete. $\Box$

We now prove proposition 3.1.

Proof of Proposition 3.1  Let $\{v_{m}\}_{m\in\mathbb{N}}\subset H^{\frac{1}{2}}(\mathbb{R}^{N})$ be a minimizing sequence of $M_{p}$, which satisfies that

[TABLE]

and

[TABLE]

By Lemma 3.1, there exists $\delta>0$ such that

[TABLE]

So we may find $x_{m}\in\mathbb{R}^{N}$ such that

[TABLE]

Let $w_{m}(x)=v_{m}(x-x_{m})$, we have that

[TABLE]

which yields, up to a subsequence, that $w_{m}\rightharpoonup w$ in $H^{\frac{1}{2}}(\mathbb{R}^{N})$ and $w_{m}\to w$ almost everywhere on $\mathbb{R}^{N}$. Then

[TABLE]

Since $H_{loc}^{\frac{1}{2}}(\mathbb{R}^{N})\subset L_{loc}^{\frac{2Np}{N+\alpha}}(\mathbb{R}^{N})$ is compact, so we have that$\int_{B_{1}(0)}|w(x)|^{\frac{2Np}{N+\alpha}}\,dx\geq\frac{\delta}{2}>0$,we can claim that $w\neq 0$ almost everywhere on $\mathbb{R}^{N}$. Then $\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|w(x)|^{p}\,dx\neq 0$ almost everywhere on $\mathbb{R}^{N}$.

Using Lemma 2.1, we obtain that

[TABLE]

and

[TABLE]

Then $\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|w(x)|^{p}\,dx=1$. As a consequent, we get that $M_{p}=\|w\|_{H^{\frac{1}{2}}(\mathbb{R}^{N})}^{2}$.

The proof is completed. $\Box$

4. Regularity

To consider the regularity of solutions to problem (1.1), we transform (1.1) to the following extension problem

[TABLE]

as in the work of [14].

Proposition 4.1**.**

Suppose that $w\in H^{1}(\mathbb{R}_{+}^{N+1})$ is a weak solution of (4.1). Then $w\in L^{q}_{loc}(\mathbb{R}_{+}^{N+1})$. Moreover, $w\in C^{2,\alpha}(\mathbb{R}_{+}^{N+1})$.

Proof. Let $\varphi\in C^{\infty}(\mathbb{R}^{N+1})$, $0\leq\varphi\leq 1$ and for small fixed $R$,

[TABLE]

Multiplying (4.1) by $\varphi^{2}|w|^{2\beta_{0}}w$, where $\beta_{0}>0$ ,and integrating by parts, we have that

[TABLE]

Since

[TABLE]

So we have that

[TABLE]

Note that

[TABLE]

so we deduce from (4.2) and (4.3) that

[TABLE]

By the Sobolev theorem, we have that

[TABLE]

By the Hardy-Littlewood-Sobolev inequality, we have that

[TABLE]

where $\frac{1}{t_{1}}+\frac{1}{t_{2}}+\frac{N-\alpha}{N}=2$, we let $pt_{1}=2^{\sharp}=\frac{2N}{N-1}$ and $\frac{1}{t_{2}}=1+\frac{\alpha}{N}-\frac{1}{t_{1}}$. Note that

[TABLE]

we have that

[TABLE]

Let $\beta_{0}=\frac{N+\alpha}{N-1}-p>0$,  $(2\beta_{1}+p)t_{2}=2^{\sharp}(\beta_{0}+1)$,  we have that $\beta_{1}=\frac{2^{\sharp}(\beta_{0}+1)-pt_{2}}{2t_{2}}>\beta_{0}\leq\frac{2^{\sharp}-pt_{2}}{2t_{2}}$, so $\beta_{1}>\beta_{0}$. Let $(2\beta_{i}+p)t_{2}=2^{\sharp}(\beta_{i-1}+1)$. Repeating the procedure and using (4.2), we find $w(x,y)\in L_{loc}^{q}(\mathbb{R}_{+}^{N+1})$, where $q\in[2,\infty)$ and

[TABLE]

Now, we use the following auxiliary function

[TABLE]

Since $(-\triangle f+f)_{y}=0$  in $\mathbb{R}_{+}^{N+1}$, we have that $-\triangle f+f$ is independent of $y$. Hence, if $y=0$, $f\equiv 0$, we have $-\triangle f+f=f_{y}y=w_{y}$. Thus $w$ is a solution of the Dirichlet problem

[TABLE]

Then, by the estimates of Calderon-Zymund, we have $f\in W^{2,p}(\mathbb{R}_{+}^{N+1})$, $w\in W^{1,p}(\mathbb{R}_{+}^{N+1})$, thus, the Schauder estimates give $w\in C^{2,\alpha}(\mathbb{R}_{+}^{N+1})$. This proof is complete. $\Box$

5. Nonexistence

In this section, we prove a Pohožaev type identity for problem (4.1), which implies the non-existence results in Theorem 1.1.

Lemma 5.1**.**

Let $w\in H^{1}(\mathbb{R}_{+}^{N+1})\cap L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^{N})$ be a solution of (4.1) and $\nabla w\in H_{loc}^{1}(\mathbb{R}^{N})$. Then, there holds

[TABLE]

Proof. We take $\psi\in C_{c}^{1}(\mathbb{R}^{N}\times[0,\infty))$ such that $\psi=1$ on $B$, where set $B=\{z=(x,y)\in\mathbb{R}^{N}\times[0,\infty):|z|\leq 1\}$ Let $w\in H^{1}(\mathbb{R}_{+}^{N+1})$ be a bounded solutions of (4.1). By Proposition 4.1 we know that $w\in C^{2}(\mathbb{R}_{+}^{N+1})\cap C^{1}(\bar{\mathbb{R}_{+}^{N+1}})$. Let $\omega_{R}=B_{R}(0)\cap\mathbb{R}_{+}^{N+1}$, where $B_{R}(0)\subset\mathbb{R}_{+}^{N+1}$ is a ball centered at the originn with the radius $R$. By (4.1), we have

[TABLE]

Multiplying (4.1) by $(\nabla w,z)$, where $z=(x,y)\in\mathbb{R}_{+}^{N+1}$, and integrating on $\omega_{R}$, we deduce that

[TABLE]

So we have that

[TABLE]

By the same arguments in the proof of Proposition 3.1 in [9], we obtain that

[TABLE]

as $R\to\infty$. Since

[TABLE]

[TABLE]

we next claim that

[TABLE]

Here we only show that

[TABLE]

the other can be treated in the same way. To this end, by contradiction, we assume that

[TABLE]

then there exists $R_{0}$ such that, for all $R_{1}\geq R_{0}$,

[TABLE]

It yields a contradiction when $R_{1}>0$ large. So the claim holds. $\Box$

We now complete the proof of Theorem 1.1$(ii)$.

Proof of Theorem 1.1$(ii)$. Let $w\in H^{1}(\mathbb{R}_{+}^{N+1})$ be a solution of (4.1), we obtain the identity

[TABLE]

Hence, combine with equation (4.1), we have

[TABLE]

If $0<p\leq\frac{N+\alpha}{N+1}$ or $p\geq\frac{N+\alpha}{N-1}$, then $u=0$. $\Box$

6. Berestycki-Lions type solutions

The aim of this section is to establish the infinitely many bounded solutions in Theorem 1.2. We will apply the genus theory to an even functional, which is constrained in a manifold and obtain the infinitely many critical points of the functional. To this end, let us recall the following critical point theorems in [1].

Let $H$ be a real Hilbert space whose norm and inner product will be denoted respectively by $\|\cdot\|$ and $(\cdot,\cdot)$. Consider the manifold

[TABLE]

the tangent space of $\mathcal{M}$ at a given point $g\in\mathcal{M}$ is given by

[TABLE]

Let $J$ be a $C^{1}$ functional defined on $H$. Then the trace $J|_{\mathcal{M}}$ of $J$ on $\mathcal{M}$ is of class $C^{1}$ and for any $w\in\mathcal{M}$,

[TABLE]

Let $\sum(\mathcal{M})$ denote the set of compact and symmetric subsets of $\mathcal{M}$ . The genus $\gamma(A)$ of a set $A\in\sum(\mathcal{M})$ is defined as the least integer $n\geq 1$ such that there exists an odd continuous mapping $\varphi:A\rightarrow S^{n-1}$. For $k\geq 1$, we denote $\tau_{k}=\{A\in\sum(\mathcal{M}):\gamma(A)\geq k\}$.

We say that a functional $J$ defined on a manifold $\mathcal{M}$ satisfies the positive Palais-Smale condition (in short,$(PS)^{+}$) if for $0<c_{1}<c_{2}$, for every sequence $\{w_{m}\}\subset\mathcal{M}$ such that $\|J|_{\mathcal{M}}^{\prime}(w_{m})\|\rightarrow 0$ and $c_{1}\leq J(w_{m})\leq c_{2}$, there exists a convergent subsequence $\{w_{m_{i}}\}$ of $\{w_{m}\}$.

Proposition 6.1**.**

Let $J:H\rightarrow R$ be an even functional of class $C^{1}$. Suppose that J is bounded from above on $\mathcal{M}$ and $J|_{\mathcal{M}}$ satisfies the (PS) condition. Let

[TABLE]

Then for any $k\geq 1$, $b_{k}$ is a critical value of $J|_{\mathcal{M}}$ and $b_{1}\geq b_{2}\geq\cdots\geq b_{k}\geq\cdots$. If J only satisfies the $(PS)^{+}$ condition, then $b_{k}$ is a critical value of provided $b_{k}>0$.

To be convinent for the analysis, we denote

[TABLE]

Proposition 6.2**.**

Under the hypotheses of Proposition 6.1, suppose that $b_{k}=b_{k+1}=\cdots=b_{k+r-1}\equiv b$. Then $\gamma(K_{b})\geq r$. In particular, if $r\geq 2$, there exist infinitely many distinct critical points of $J|_{\mathcal{M}}$ corresponding to the critical value $b$.

By Proposition 6.1 and Proposition 6.2, under the conditions of Proposition 4.1, there always exist infinitely many distinct critical points of $J$ on $\mathcal{M}$. Let $E=H^{\frac{1}{2}}(\mathbb{R}^{N})$ and $E_{r}=\left\{v\in H^{\frac{1}{2}}(\mathbb{R}^{N}):\ v(x)=v(|x|)\right\}.$ Denote by $\mathcal{M}=\left\{v\in E_{r}:\ \|v\|_{E_{r}}=1\right\}$ the unit ball in $E_{r}$. Define the functional

[TABLE]

We next verify that $J$ satisfies the conditions of Proposition 6.1. In fact, by Hardy-Littlewood-Sobolev inequality, we have that

[TABLE]

where $l=t=\frac{2N}{N+\alpha}$. Then, by Min-Max method argument, we have that

[TABLE]

combining with the fact that $\|v\|_{E_{r}}=1$ and $H^{\frac{1}{2}}(\mathbb{R}^{N})\hookrightarrow L^{p}(\mathbb{R}^{N})$ for $2\leq p\leq\frac{2N}{N-1}$, we have that $\int_{B_{r}(z)}|v(x)|^{q}\,dx<+\infty$ and

[TABLE]

Thus, $J$ is bounded from above on $\mathcal{M}$.

Lemma 6.1**.**

Let $2<q<2^{\sharp}:=\frac{2N}{N-1}$ for $N\geq 2$, $\{\tilde{v}_{m}\}_{m}$ be a bounded sequence in $E_{r}$, suppose that $\tilde{v}_{m}\rightharpoonup 0$ in $H^{\frac{1}{2}}(\mathbb{R}^{N})$ as $m\to\infty$. Then

[TABLE]

as $m\to\infty$.

Proof. We know that $\int_{\mathbb{R}^{N}}(I_{\alpha}*|\tilde{v}_{m}|^{p})(x)|\tilde{v}_{m}|^{p}\,dx$ strongly converges, up to a subsequence. Denote by $\{v_{m}\}$ the restriction of functions $\{\tilde{v}_{m}\}$ to $\mathbb{R}^{N}$. Then $\{v_{m}\}\subset H^{\frac{1}{2}}(\mathbb{R}^{N})$ is uniformly bounded and $v_{m}$ is radially symmetric in x. Assuming $v_{m}\rightharpoonup 0$ in $H^{\frac{1}{2}}(\mathbb{R}^{N})$ as $m\to\infty$. By the results in [14], we have that

[TABLE]

as $m\to\infty$. Combining with Lemma 3.1, it implies that

[TABLE]

as $m\to\infty$,  and then

[TABLE]

as $m\to\infty$ by the fact that $\tilde{v}_{m}(x)=v_{m}(x)$.

Lemma 6.2**.**

$J|_{\mathcal{M}}$* satisfies the $(PS)^{+}$ condition.*

Proof. Let $\{v_{m}\}\subset\mathcal{M}$ be a $(PS)^{+}$ sequence for $J$, that is,

[TABLE]

as $m\to\infty$. Since $v_{m}$ is bounded in $E_{r}$, we may assume $v_{m}\rightharpoonup v$ in $E_{r}$. Then

[TABLE]

as $m\to\infty$ and for any $\varphi\in E_{r}$,

[TABLE]

By lemma 6.1, we have that

[TABLE]

together with the weak convergence of $\{v_{m}\}$, yield

[TABLE]

and

[TABLE]

Denote that $\xi^{m}=(I^{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p-1}$, $\xi=(I^{\alpha}*|v|^{p})(x)|v(x)|^{p-1}$, we have that $\xi^{m}\rightharpoonup\xi$ in $E_{r}$ and

[TABLE]

be the fact that

[TABLE]

Since $v_{m}\rightharpoonup v$ in $E_{r}$, we have that $E_{r}$ is compactly embedded in $L_{q}(\mathbb{R}^{N})$, where $2\leq q<2^{\sharp}$ and ${\frac{2^{\sharp}}{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}}<2^{\sharp}$ for $p<\frac{N+\alpha}{N-1}$ and $(p-1)\frac{2^{\sharp}}{p-1}=2^{\sharp}$, then

[TABLE]

[TABLE]

and

[TABLE]

we know that $E_{r}$ is embedded in $L_{q}(\mathbb{R}^{N})$, where $2\leq q\leq 2^{\sharp}$, so

[TABLE]

thus, we have that

[TABLE]

where ${\frac{2^{\sharp}}{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}}<2^{\sharp}$,  as $p<\frac{N+\alpha}{N-1}$,  $(p-2)\frac{2^{\sharp}}{p-2}=2^{\sharp}$, $p\frac{2^{\sharp}}{p}=2^{\sharp}$. So we have that

[TABLE]

[TABLE]

So we have that

[TABLE]

and

[TABLE]

Notice that $J(v_{m})\geq\alpha>0$, then we have that $J(v_{m})\geq\alpha$. So $v\neq 0$ and $\|v\|_{E_{r}}=1$. Combining $v_{m}\rightharpoonup v$ with $\|v_{m}\|_{E_{r}}\rightarrow\|v\|_{E_{r}}$, we obtain that $v_{m}$ strongly converges to $v$ in $E_{r}$.

Next we show that $b_{k}>0$ for each $k\geq 1$. Indeed, for $k\geq 1$, we denote

[TABLE]

Since $\pi_{k-1}$ is homeomorphic to $S^{k-1}$ by an odd homeomorphism, it follows that $\gamma(\pi_{k-1})=k$. The following result is due to Berestycki and Lions.

Lemma 6.3**.**

For all $k\geq 1$, there exists a constant $R=R(k)>0$ and an odd continuous mapping $\tau:\pi_{k-1}\to H_{0}^{1}(B_{R}(0))$ such that

(i) $\pi_{(}l)$ is a radial function for all $l\in\pi_{k-1}$ and $0\nsubseteq\tau(\pi_{k-1})$;

(ii) there exist $p,C>0$ such that $\rho\leq\|\nabla u\|_{L^{2}(B_{R(0)})}\leq C$ for $u\in\tau(\pi_{k-1})$;

(iii) for $u\in\tau(\pi_{k-1})$, $\int_{\mathbb{R}^{N}}|u(x)|^{q}\,dx\geq 1$.

Lemma 6.4**.**

When $N\geq 3$, there holds $b_{k}>0$ for each $k\geq 1$.

Proof. Let $\hat{\pi}_{k-1}=\tau(\pi_{k-1})$ and define for each $u\in H_{0}^{1}(B_{R}(0))$ an extension $\tilde{u}\in H^{1}(\mathbb{R}^{N})$ of u such that $\tilde{u}=u$ on $B_{R}(0)$ and $\tilde{u}=0$ on $\mathbb{R}^{N}\backslash B_{R}(0)$. Denote $\tilde{u}_{\sigma}(x)=\tilde{u}(\frac{x}{\sigma})$. For $u\in\hat{\pi}_{k-1}$, we define $\chi(u)=(1-y)(I_{\alpha}*|\tilde{u}_{\sigma}|^{p}(x))|\tilde{u}_{\sigma}(x)|^{p}$ and $\chi(u)=0$ if $y\not\in R\backslash(0,1)$, where $\sigma=\sigma(u)>0$ is determined by requiring $\chi(u)\in M$, that is,

[TABLE]

We deduce that

[TABLE]

and

[TABLE]

and

[TABLE]

Then

[TABLE]

Since the right hand side is increasing in $\sigma>0$, we find a unique $\sigma>0$ so that $\chi\in M$. By (ii) of lemma 6.3, we obtain

[TABLE]

Hence, there exists $\bar{\sigma}>0$ independent of $u\in\hat{\pi}_{k-1}$, such that $\sigma(u)\leq\bar{\sigma}$. By Poincare’s inequality, there holds

[TABLE]

which implies that $\sigma(u)$ has a lower bound $\bar{\sigma}>0$ independent of u.

Now we prove $b_{k}>0$ for each $k\geq 1$. In fact, we observe that

[TABLE]

Let $A_{k}=(1-y)\chi(\hat{\pi}_{k-1})$ for $0\leq y\leq 1$. Since $\chi$ is an odd continuous mapping, $A_{k}\in\Sigma(M)$. Furthermore, $(1-y)\chi\cdot\tau:\pi_{k-1}\rightarrow A_{k}$ is odd and continuous, so we have that $\gamma(A_{k})\geq\gamma(\pi_{k-1})=k$. Hence, $A_{k}\in\Gamma_{k}$. Set $\varepsilon_{k}=\underline{\sigma}^{N}$, we have that

[TABLE]

Therefore, for each $k\geq 1$,

[TABLE]

Proof of Theorem 1.2. By lemma 6.2, $J|_{\mathcal{M}}$ satisfies the $(PS)^{+}$ condition, and Lemma 6.4 yields $b_{k}>0$ for each $k>1$. Then the conclusion follows Proposition 6.1 and Proposition 6.2. $\Box$

Acknowledgements: The author would like to express the warmest gratitude to Prof. Jianfu Yang, for proposing the problem and for its active participation. This work is supported by the Jiangxi Provincial Natural Science Foundation (20161ACB20007).