#  Nonlinear Choquard equations involving
nonlocal operators

Wanwan Wang

Department of Mathematics, Jiangxi Normal University,

Nanchang, Jiangxi 330022, PR China

††E-mail address<EMAIL_ADDRESS>(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
>
> $(-\Delta+id)^{\frac{1}{2}}u=(I_{\alpha}*{|u|^{p}})|u|^{p-2}u\ \ {\rm in}\ \
> \mathbb{R}^{N},\ \ \ u\in H^{\frac{1}{2}}(\mathbb{R}^{N}),$ (1)
>
> 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

$(-\Delta+id)^{\frac{1}{2}}u=(I_{\alpha}*{|u|^{p}})|u|^{p-2}u\ \ {\rm in}\ \
\mathbb{R}^{N},\ \ \ u\in H^{\frac{1}{2}}(\mathbb{R}^{N}),$ (1.1)

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

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

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

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

with the norm

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

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

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

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

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

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

$\int_{\mathbb{R}^{N}}|I_{\alpha}*v|^{\frac{Ns}{N-\alpha s}}dx\leq
C(\int_{\mathbb{R}^{N}}|v|^{s}dx)^{\frac{N}{N-\alpha s}},$

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

$M_{p}=\inf\left\\{\int_{\mathbb{R}^{N}}\left|(-\Delta+id)^{\frac{1}{4}}u\right|^{2}dx:\int_{\mathbb{R}^{N}}(I_{\alpha}*|u|^{p})(x)|u(x)|^{p}\,dx=1\right\\}.$
(1.2)

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

$\lim_{m\to\infty}{\int_{\Omega}|{|w_{m}|^{q}-|w_{m}-w|^{q}-|w|^{q}}|^{\frac{s}{q}}}\,dx=0.$

######  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

$\displaystyle\lim_{m\to\infty}\left[\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}|^{p})(x)|w_{m}(x)|^{p}\,dx-\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}-w|^{p})(x)|(w_{m}-w)(x)|^{p}\,dx\right]$
$\displaystyle=\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|w(x)|^{p}\,dx.$

Proof. We observe that

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}|^{p})(x)|w_{m}(x)|^{p}\,dx-\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}-w|^{p})(x)|(w_{m}-w)(x)|^{p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*(|w_{m}|^{p}-|w_{m}-w|^{p}))(x)(|w_{m}(x)|^{p}-|(w_{m}-w)(x)|^{p})\,dx$
$\displaystyle+2\int_{\mathbb{R}^{N}}(I_{\alpha}*(|w_{m}|^{p}-|w_{m}-w|^{p}))(x)|(w_{m}-w)(x)|^{p}\,dx.$

By the Hölder inequality, we have that

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*(|w_{m}|^{p}-|w_{m}-w|^{p}))(x)|(w_{m}-w)(x)|^{p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*(|w_{m}|^{p}-|w_{m}-w|^{p}-|w|^{p}))(x)|(w_{m}-w)(x)|^{p}\,dx$
$\displaystyle+\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|(w_{m}-w)(x)|^{p}\,dx$
$\displaystyle\leq$
$\displaystyle\left({\int_{\mathbb{R}^{N}}|\left(I_{\alpha}*(|w_{m}|^{p}-|w_{m}-w|^{p}-|w|^{p})\right)|^{\frac{2N}{N-\alpha}}}(x)\,dx\right)^{\frac{N-\alpha}{2N}}$
$\displaystyle\left(\int_{\mathbb{R}^{N}}(|(w_{m}-w)(x)|^{p})^{\frac{2N}{N+\alpha}}\,dx\right)^{\frac{N+\alpha}{2N}}+\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|(w_{m}-w)(x)|^{p}\,dx.$

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

$M_{p}=\inf\left\\{\int_{\mathbb{R}^{N}}\left|(-\Delta+id)^{\frac{1}{4}}u\right|^{2}dx:\int_{\mathbb{R}^{N}}(I_{\alpha}*|u|^{p})(x)|u(x)|^{p}\,dx=1\right\\}.$
(3.1)

######  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

$\sup_{z\in{\mathbb{R}^{N}}}\int_{B_{r}(z)}|v_{m}(x)|^{q}\,dx\to 0$

as $m\to\infty$. Then

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p}\,dx\to 0$

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

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p}\,dx\
=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|v_{m}(x)|^{p}|v_{m}(z)|^{p}}{|x-z|^{N-\alpha}}\,dxdz$
$\displaystyle\leq$ $\displaystyle
c_{1}\|{|v_{m}|^{p}}\|_{L^{l}(\mathbb{R}^{N})}\|{|v_{m}|^{p}}\|_{L^{t}(\mathbb{R}^{N})}\
=c_{1}\left(\int_{\mathbb{R}^{N}}|v_{m}(x)|^{\frac{2Np}{N+\alpha}}\,dx\right)^{\frac{N+\alpha}{N}}$

and

$\displaystyle\left(\int_{B_{r}(z)}|v_{m}(x)|^{\frac{2Np}{N+\alpha}}\,dx\right)^{\frac{N+\alpha}{2Np}}$
$\displaystyle\leq$
$\displaystyle\left[\left(\int_{B_{r}(z)}|v_{m}(x)|^{q}\,dx\right)^{\frac{1}{q}}\right]^{1-\lambda}\left[\left(\int_{B_{r}(z)}|v_{m}(x)|^{2^{\sharp}}\,dx\right)^{\frac{1}{2^{\sharp}}}\right]^{\lambda}$
$\displaystyle\leq$ $\displaystyle
c_{2}{\|v_{m}\|}^{1-\lambda}_{L^{q}(B_{r}(z))}\left(\int_{\mathbb{R}^{N}}|(-\Delta+id)^{\frac{1}{4}}v_{m}|^{2}\,dx\right)^{\frac{\lambda}{2}},$

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

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

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

$\displaystyle\int_{B_{r}(z)}|v_{m}(x)|^{\frac{2Np}{N+\alpha}}\,dx\leq
c_{2}^{\frac{2Np}{N+\alpha}}{\|v_{m}\|}^{(1-\lambda)\frac{2Np}{N+\alpha}}_{L^{q}(B_{r}(z))}\left(\int_{\mathbb{R}^{N}}|(-\Delta+id)^{\frac{1}{4}}v_{m}|^{2}\,dx\right).$

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

$\displaystyle\int_{B_{r}(z)}|v_{m}(x)|^{\frac{2Np}{N+\alpha}}\,dx$
$\displaystyle\leq$
$\displaystyle(N+1)c_{2}^{\frac{2Np}{N+\alpha}}\sup_{y\in{\mathbb{R}^{N}}}[\int_{B_{r}(z)}{|v_{m}(x)|}^{q}\,dx]^{\frac{(1-\lambda)\frac{2Np}{N+\alpha}}{q}}\left(\int_{\mathbb{R}^{N}}|(-\Delta+id)^{\frac{1}{4}}v_{m}|^{2}\,dx\right).$

Then

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p}(x))|v_{m}(x)|^{p}\,dx$
$\displaystyle\leq$ $\displaystyle
c_{3}\left((\sup_{z\in{\mathbb{R}^{N}}}\int_{B_{r}(z)}|v_{m}(x)|^{q}\,dx)^{\frac{1}{q}(\frac{2Np}{N+\alpha}-2)}(\int_{\mathbb{R}^{N}}|(-\Delta+id)^{\frac{1}{4}}v_{m}|^{2}\,dx)\right)^{\frac{N+\alpha}{N}}.$

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

$\int_{\mathbb{R}^{N}}(|\nabla v_{m}(x)|^{2}+|v_{m}(x)|^{2})\,dx\to M_{p}$

and

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p}\,dx=1.$

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

$\delta=\lim_{m\to\infty}\sup_{\tilde{x}\in\mathbb{R}^{N}}\int_{B_{1}(\tilde{x})}|v_{m}(x)|^{\frac{2Np}{N+\alpha}}\,dx>0.$

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

$\int_{B_{1}(x_{m})}|v_{m}(x)|^{\frac{2Np}{N+\alpha}}\,dx>\frac{\delta}{2}.$

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

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}|^{p})(x)|w_{m}(x)|^{p}\,dx=1,\ \ \
\int_{\mathbb{R}^{N}}|(-\Delta+id)^{\frac{1}{4}}w_{m}|^{2}\,dx\to M_{p},$

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

$\int_{B_{1}(0)}|w_{m}(x)|^{\frac{2Np}{N+\alpha}}\,dx>\frac{\delta}{2}.$

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

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}|^{p})(x)|w_{m}(x)|^{p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|w(x)|^{p}\,dx+\lim_{m\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}-w|^{p})(x)|(w_{m}-w)(x)|^{p}\,dx$

and

$\displaystyle
M_{p}=\lim_{m\to\infty}{\|w_{m}\|_{H^{\frac{1}{2}}(\mathbb{R}^{N})}^{2}}={\|w\|_{H^{\frac{1}{2}}(\mathbb{R}^{N})}^{2}}+\lim_{m\to\infty}{\|w_{m}-w\|_{H^{\frac{1}{2}}(\mathbb{R}^{N})}^{2}}$
$\displaystyle\geq$ $\displaystyle
M_{p}\left(\lim_{m\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*|w_{m}-w|^{p})(x)|(w_{m}-w)(x)|^{p}\,dx\right)^{\frac{2}{p+1}}$
$\displaystyle\ \
+M_{p}\left(\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|w(x)|^{p}\,dx\right)^{\frac{2}{p+1}}$
$\displaystyle=$ $\displaystyle
M_{p}\left(\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|w(x)|^{p}\,dx\right)^{\frac{2}{p+1}}+M_{p}\left(1-\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x)|w(x)|^{p}\,dx\right)^{\frac{2}{p+1}}$

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

$\left\\{\begin{array}[]{lll}-\Delta w(x,y)+w(x,y)=0,\quad\
(x,y)\in\mathbb{R}_{+}^{N+1},\\\\[5.69054pt] \frac{\partial
w}{\partial\nu}(x,0)=(I_{\alpha}*|w|^{p})(x,0)|w(x,0)|^{p-2}w(x,0)\ \ \ \
x\in\mathbb{R}^{N},\end{array}\right.$ (4.1)

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 $R$,

$\varphi(x,y)=\left\\{\begin{array}[]{lll}\displaystyle 1\quad\quad(x,y)\in
B_{R}(0),\\\\[5.69054pt] \phantom{}\displaystyle 0\quad\quad(x,y)\not\in
B_{2R}(0),\end{array}\right.\qquad|\nabla\varphi|\leq\frac{C}{R}.$

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

$\int_{\mathbb{R}_{+}^{N+1}}(\nabla
w\nabla(\varphi^{2}|w|^{2\beta_{0}}w)+\varphi^{2}|w|^{2\beta_{0}+2})\,dxdy=\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})(x,0)\varphi^{2}(x,0)|w(x,0)|^{2\beta_{0}+p}\,dx.$
(4.2)

Since

$\displaystyle\int_{\mathbb{R}_{+}^{N+1}}\nabla
w\nabla(\varphi^{2}|w|^{2\beta_{0}}w)\,dxdy$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}^{N+1}}\left((2\beta_{0}+1))\varphi^{2}|w|^{2\beta_{0}}|\nabla
w|^{2}+2\varphi|w|^{2\beta_{0}}w\nabla\varphi\nabla w\right)\,dxdy.$

So we have that

$\displaystyle\int_{\mathbb{R}_{+}^{N+1}}(2\beta_{0}+1))\varphi^{2}|w|^{2\beta_{0}}|\nabla
w|^{2}\,dxdy$
$\displaystyle=\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})\varphi^{2}|w|^{2\beta_{0}+p}\,dx-\int_{\mathbb{R}_{+}^{N+1}}\left(\varphi^{2}|w|^{2\beta_{0}+2}+2\varphi|w|^{2\beta_{0}}w\nabla\varphi\nabla
w\right)\,dxdy$
$\displaystyle\leq\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})\varphi^{2}|w|^{2\beta_{0}+p}\,dx+\int_{\mathbb{R}_{+}^{N+1}}2\varphi|w|^{2\beta_{0}+1}w\nabla\varphi\nabla
w\,dxdy$
$\displaystyle\leq\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})\varphi^{2}|w|^{2\beta_{0}+p}\,dx+\left(\int_{\mathbb{R}_{+}^{N+1}}2\varepsilon\varphi^{2}|w|^{2\beta_{0}}w|\nabla\varphi|^{2}+2C_{\varepsilon}w^{2(\beta_{0}+1)}|\nabla\varphi|^{2}\right)\,dxdy.$

Note that

$\displaystyle\int_{\mathbb{R}_{+}^{N+1}}|\nabla(\varphi|w|^{\beta_{0}}w)|^{2}\,dxdy$
(4.3) $\displaystyle\leq$ $\displaystyle
C(\beta_{0}+1)\int_{\mathbb{R}_{+}^{N+1}}\left((2\beta_{0}+1)\varphi^{2}|w|^{2\beta_{0}}|\nabla
w|^{2}+2|w|^{2\beta_{0}+2}|\nabla\varphi|^{2}\right)\,dxdy,$

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

$\displaystyle\int_{\mathbb{R}_{+}^{N+1}}(|\nabla(\varphi|w|^{\beta_{0}}w)|^{2}+\varphi^{2}|w|^{2\beta_{0}+2})\,dxdy$
$\displaystyle\leq
C(\beta_{0}+1)\int_{\mathbb{R}_{+}^{N+1}}\left((2\beta_{0}+1)\varphi^{2}|w|^{2\beta_{0}}|\nabla
w|^{2}+2|w|^{2\beta_{0}+2|\nabla\varphi|^{2}}+\varphi^{2}|w|^{2\beta_{0}+2}\right)\,dxdy$
$\displaystyle\leq
C(\beta_{0}+1)\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})\varphi^{2}|w|^{2\beta_{0}+p}\,dx+C(\beta_{0}+1)\int_{\mathbb{R}_{+}^{N+1}}w^{2(\beta_{0}+1)}|\nabla\varphi|^{2}\,dxdy.$

By the Sobolev theorem, we have that

$\displaystyle
C(R)\left(\int_{\mathbb{R}_{+}^{N+1}}(|\varphi||w|^{\beta_{0}+1})^{\frac{2N}{N-1}}\,dx\right)^{\frac{N-1}{N}}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}_{+}^{N+1}}(|\varphi||w|^{\beta_{0}+1})^{\frac{2N}{N-2}}\,dx\right)^{\frac{N-2}{N}}$
$\displaystyle\leq
C\int_{\mathbb{R}_{+}^{N+1}}\left(|\nabla(\varphi|w|^{\beta_{0}}w)|^{2}+\varphi^{2}|w|^{2\beta_{0}+2})\,dxdy\right)$
$\displaystyle\leq
C(\beta_{0}+1)\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})\varphi^{2}|w|^{2\beta_{0}+p}\,dx+C(\beta_{0}+1)\int_{\mathbb{R}_{+}^{N+1}}w^{2(\beta_{0}+1)}|\nabla\varphi|^{2}\,dxdy.$

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

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|w|^{p})\varphi^{2}|w|^{2\beta_{0}+p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{w^{p}(z)w^{2\beta_{0}+p}(x)\varphi^{2}(x)}{|x-y|^{N-\alpha}}\,dzdx\leq\||w|^{p}\|_{L^{t_{1}}}\||w|^{2\beta_{0}+p}\varphi^{2}\|_{L_{t_{2}}},$

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

$(2\beta_{0}+p)t_{2}\leq\frac{2N}{N-1},$

we have that

$\displaystyle\beta_{0}\leq\frac{\frac{2N}{N-1}-pt_{2}}{2t_{2}}=\frac{N}{N-1}(1+\frac{\alpha}{N}-\frac{p(N-1)}{2N})-\frac{p}{2}=\frac{N+\alpha}{N-1}-p$

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

$\displaystyle\|w(x,y)\|_{L^{q}(B_{R}(0))}\leq
C(q,R)\|w(x,y)\|_{L^{2}(B_{2R}(0))}.$ (4.4)

Now, we use the following auxiliary function

$f(x,y)=\int_{0}^{y}w(x,t)\,dt\ \ for\ (x,y)\in\mathbb{R}_{+}^{N+1}.$

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

$\left\\{\begin{array}[]{lll}\displaystyle-\triangle
f(x,y)+f(x,y)=(I_{\alpha}*|w|^{p})(x,0)|w(x,0)|^{p-2}w(x,0)&{\rm for}\quad
x\in\mathbb{R}^{N},\ y>0\\\\[5.69054pt] \phantom{}\displaystyle f(x,0)=0&{\rm
for}\quad x\in\mathbb{R}^{N}.\end{array}\right.$ (4.5)

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

$\frac{N-1}{2}\int_{\mathbb{R}_{+}^{N+1}}|\nabla
w|^{2}\,dxdy+\frac{N+1}{2}\int_{\mathbb{R}_{+}^{N+1}}w^{2}\,dxdy=\frac{N+\alpha}{2p}\int_{\mathbb{R}^{N}\times\\{0\\}}(I_{\alpha}*|w|^{p})|w|^{p}\,dx.$
(5.1)

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

$\int_{\omega_{R}}(|\nabla
w|^{2}+w^{2})\,dxdy=\int_{\partial{w_{R}}}w\frac{\partial
w}{\partial\nu}\,dS.$ (5.2)

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

$\displaystyle 0=\int_{\omega_{R}}(-\Delta w+w)(\nabla w,z)\,dxdy$
$\displaystyle=$ $\displaystyle\int_{\omega_{R}}\nabla(\frac{1}{2}|\nabla
w|)z\,dxdy+\int_{\omega_{R}}|\nabla
w|^{2}\,dxdy+\frac{1}{2}\int_{\omega_{R}}(\nabla
w^{2},z)\,dxdy-\int_{\partial{w_{R}}}(\nabla w,z)\frac{\partial
w}{\partial\nu}\,dS$ $\displaystyle=$
$\displaystyle-\frac{N-1}{2}\int_{\omega_{R}}|\nabla
w|^{2}\,dxdy+\frac{1}{2}\int_{\partial{w_{R}}}|\nabla
w|^{2}(z,\nu)\,dS-\frac{N+1}{2}\int_{\omega_{R}}w^{2}\,dxdy$
$\displaystyle+\frac{1}{2}\int_{\partial{w_{R}}}w^{2}(z,\nu)\,dS-\int_{\partial{w_{R}}}(\nabla
w,z)\frac{\partial w}{\partial\nu}\,dS.$

So we have that

$\displaystyle\int_{\omega_{R}}\frac{N-1}{2}\int_{\omega_{R}}|\nabla
w|^{2}\,dxdy+\frac{N+1}{2}\int_{\omega_{R}}w^{2}\,dxdy$ (5.3) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\partial{w_{R}}\cap\\{y>0\\}}|\nabla
w|^{2}(z,\nu)\,dS+\frac{1}{2}\int_{\partial{w_{R}}\cap\\{y>0\\}}w^{2}(z,\nu)\,dS$
$\displaystyle-$ $\displaystyle\int_{\partial{w_{R}}\cap\\{y>0\\}}(\nabla
w,z)\frac{\partial
w}{\partial\nu}\,dS-\int_{B_{R}(0)\times\\{0\\}}(\nabla_{x}w,x)\frac{\partial
w}{\partial\nu}\,dx$

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

$\int_{B_{R}(0)\times\\{0\\}}(I_{\alpha}*|w|^{p})|w|^{p-2}w(\nabla_{x}w,x)\,dx\rightarrow-\frac{N+\alpha}{2p}\int_{B_{R}(0)\times\\{0\\}}(I_{\alpha}*|w|^{p})|w|^{p}\,dx$

as $R\to\infty$. Since

$\left|\int_{\partial{w_{R}}\cap\\{y>0\\}}|\nabla w|^{2}(z,\nu)\,dS\right|\leq
CR\int_{\partial{w_{R}}\cap\\{y>0\\}}|\nabla w|^{2}\,dS,$
$\left|\int_{\partial{w_{R}}\cap\\{y>0\\}}w^{2}(z,\nu)\,dS\right|\leq
CR\int_{\partial{w_{R}}\cap\\{y>0\\}}w^{2}\,dS,$

we next claim that

$R\int_{\partial{w_{R}}\cap\\{y>0\\}}|\nabla w|^{2}\,dS\to 0,\ \ \
R\int_{\partial{w_{R}}\cap\\{y>0\\}}w^{2}\,dS\to 0\quad{\rm as}\quad
R\to+\infty.$

Here we only show that

$R\int_{\partial{w_{R}}\cap\\{y>0\\}}w^{2}\,dS\to 0\quad{\rm as}\quad
R\to+\infty,$

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

$\lim_{R\rightarrow\infty}\inf\int_{\partial{w_{R}}\cap\\{y>0\\}}w^{2}\,dS=c_{0}>0,$

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

$\displaystyle
C\geq\int_{\mathbb{R}_{+}^{N+1}}w^{2}\,dx\geq\int_{R_{0}}^{R_{1}}\int_{\partial{w_{R}}\cap\\{y>0\\}}w^{2}\,dxdR\geq\frac{c_{0}}{2}\int_{R_{0}}^{R_{1}}\frac{1}{R}\,dR=\frac{c_{0}}{2}\log\frac{R_{1}}{R_{0}}.$

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

$\int_{\mathbb{R}^{N}}|\nabla
w|^{2}+\int_{\mathbb{R}^{N}}|w|^{2}=\int_{\mathbb{R}^{N}}(I_{\alpha}\ast|w|^{p})|w|^{p}.$

Hence, combine with equation (4.1), we have

$(\frac{N-1}{2}-\frac{N+\alpha}{2p})\int_{\mathbb{R}^{N}}|\nabla
w|^{2}\,dx+(\frac{N+1}{2}-\frac{N+\alpha}{2p})\int_{\mathbb{R}^{N}}w^{2}\,dx=0.$

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

$\mathcal{M}:=\\{w\in H:\|w\|_{H}=1\\},$

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

$T_{w}\mathcal{M}=\\{w\in H:(g,w)=0\\}.$

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
$w\in\mathcal{M}$,

$\langle J|_{\mathcal{M}}^{\prime}(w),g\rangle=\langle J^{\prime}(w),g\rangle\
\ \ \ g\in T|_{w}\mathcal{M}.$

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

$b_{k}=\sup_{A\in\tau_{k}}\inf_{w\in A}J(w).$

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

$K_{p}=\left\\{x\in\mathcal{M}:J(w)=b,J|_{\mathcal{M}}^{\prime}(w)=0\right\\}.$

######  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

$J(v)=\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p}\,dx\ \ \ for\
v\in E_{r}.$

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

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p}\,dx=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|v(z)|^{p}|v(x)|^{p}}{|x-z|^{N-\alpha}}\,dxdz$
$\displaystyle\leq
c\||v|^{p}\|_{l}\|v|^{p}\|_{t}=c\left(\int_{\mathbb{R}^{N}}|v(x)|^{\frac{2Np}{N+\alpha}}\,dx\right)^{\frac{N+\alpha}{N}},$

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

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p}\,dx$
$\displaystyle\leq$ $\displaystyle
c_{6}\left(\sup_{z\in{\mathbb{R}^{N}}}(\int_{B_{r}(z)}|v(x)|^{q}\,dx)^{\frac{1}{q}(\frac{2Np}{N+\alpha}-2)}\int_{\mathbb{R}^{N}}|(-\Delta+id)^{\frac{1}{4}}u|^{2}\,dx\right)^{\frac{N+\alpha}{N}},$

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

$\displaystyle\sup_{z\in{\mathbb{R}^{N}}}\int_{B_{r}(z)}|v(x)|^{q}\,dx<+\infty.$

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

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|\tilde{v}_{m}|^{p})(x)|\tilde{v}_{m}(x)|^{p}\,dx\to
0$

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

$\sup_{z\in{\mathbb{R}^{N}}}\int_{B_{r}(z)}|v_{m}(x)|^{2}\,dx\to 0$

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

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p}\,dx\to 0$

as $m\to\infty$, and then

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|\tilde{v}_{m}|^{p})(x)|\tilde{v}_{m}(x)|^{p}\,dx\to
0$

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,

$0<\alpha\leq J(v_{m})\leq C,\quad J|_{\mathcal{M}}^{\prime}(v_{m})\to 0$

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

$<J^{\prime}(v_{m})-<J^{\prime}(v_{m}),v_{m}>v_{m},\varphi>\to 0$

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

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p-2}v_{m}(x)\varphi(x)\,dx-(\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p}\,dx)\cdot$
$\displaystyle\int_{\mathbb{R}^{N}}(\nabla
v_{m}(x)\nabla\varphi(x)+v_{m}(x)\varphi(x))\,dx=o(1).$

By lemma 6.1, we have that

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p}\,dx\to\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p}\,dx\quad{\rm
as}\quad m\to\infty,$

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

$\int_{\mathbb{R}^{N}}(\nabla
v_{m}(x)\nabla\varphi(x)+v_{m}(x)\varphi(x))\,dx\to\int_{\mathbb{R}^{N}}(\nabla
v(x)\nabla\varphi(x)+v(x)\varphi(x))\,dx\quad{\rm as}\quad m\to\infty$

and

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p-2}v_{m}(x)\varphi(x)\,dx\to\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p-2}v(x,0)\varphi(x)\,dx.$

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

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p-1}\varphi(x)\,dx-\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p-1}\varphi(x)\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*(|v_{m}|^{p}-|v|^{p}))(x)|v_{m}(x)|^{p-1}\varphi(x)\,dx+\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)(|v_{m}(x)|^{p-1}-|v(x)|^{p-1})\varphi(x)\,dx$
$\displaystyle\leq$ $\displaystyle
C_{1}\int_{\mathbb{R}^{N}}(I_{\alpha}*(|v_{m}|^{p-1}+|v|^{p-1})|v_{m}-v|)(x)|v_{m}(x)|^{p-1}\varphi(x)\,dx$
$\displaystyle\ \
+C_{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*|v^{p}|)(x)((|v_{m}|^{p-2}+|v|^{p-2})|(v_{m}-v)|)(x)|\varphi(x)\,dx$
$\displaystyle\leq$ $\displaystyle
C_{1}\left(\int_{\mathbb{R}^{N}}(|v_{m}|^{p-1}+|v|^{p-1})^{\frac{1}{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}}|(v_{m}-v)(x)|^{\frac{1}{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}}\,dx\right)^{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}(|v_{m}(z)|^{p-1}\varphi(z))^{\frac{2^{\sharp}}{p}}\,dz\right)^{\frac{p}{2^{\sharp}}}$
$\displaystyle\ \
+C_{2}\left(\int_{\mathbb{R}^{N}}|v^{p}(z)|^{\frac{2^{\sharp}}{p}}\,dz\right)^{\frac{p}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}[((|v_{m}|^{p-2}+|v|^{p-2})|(v_{m}-v)|)(x)|\varphi(x)]^{\frac{1}{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}}\,dx\right)^{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}},$

be the fact that

$\displaystyle\left(\int_{\mathbb{R}^{N}}\left((|v_{m}|^{p-1}+|v|^{p-1})|(v_{m}-v)(x)|\right)^{\frac{1}{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}}\,dx\right)^{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{R}^{N}}|(v_{m}-v)(x)|^{\frac{2^{\sharp}}{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}}\,dx\right)^{\frac{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}(|v_{m}|^{p-1}+|v|^{p-1})^{\frac{2^{\sharp}}{p-1}}\,dx\right)^{\frac{p-1}{2^{\sharp}}}.$

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

$\left(\int_{\mathbb{R}^{N}}(|v_{m}|^{p-1}+|v|^{p-1})^{\frac{2^{\sharp}}{p-1}}\,dx\right)^{\frac{p-1}{2^{\sharp}}}<\infty,$
$\left(\int_{\mathbb{R}^{N}}|(v_{m}-v)(x)|^{\frac{2^{\sharp}}{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}}\,dx\right)^{\frac{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}{2^{\sharp}}}\rightarrow
0$

and

$\displaystyle\left(\int_{\mathbb{R}^{N}}(|v_{m}(z)|^{p-1}\varphi(z))^{\frac{2^{\sharp}}{p}}\,dz\right)^{\frac{p}{2^{\sharp}}}\leq\left(\int_{\mathbb{R}^{N}}|v_{m}(z)|^{2^{\sharp}}\,dz\right)^{\frac{p-1}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}\varphi(z)^{2^{\sharp}}\,dz\right)^{\frac{1}{2^{\sharp}}}$

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

$\left(\int_{\mathbb{R}^{N}}(|v_{m}(z)|^{p-1}\varphi(z))^{\frac{2^{\sharp}}{p}}\,dz\right)^{\frac{p}{2^{\sharp}}}<+\infty,$

thus, we have that

$\displaystyle\left(\int_{\mathbb{R}^{N}}|v^{p}(z)|^{\frac{2^{\sharp}}{p}}\,dz\right)^{\frac{p}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}[((|v_{m}|^{p-2}+|v|^{p-2})|(v_{m}-v)|)(x)|\varphi(x)]^{\frac{1}{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}}\,dx\right)^{1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{R}^{N}}((|v_{m}|^{p-2}+|v|^{p-2})(x)\varphi(x))^{\frac{2^{\sharp}}{p-1}}\,dx\right)^{\frac{p-1}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}|(v_{m}-v)|)(x)|^{\frac{2^{\sharp}}{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}}\,dx\right)^{\frac{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}{2^{\sharp}}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{R}^{N}}((v_{m}|^{p-2}+|v|^{p-2})(x))^{\frac{2^{\sharp}}{p-2}}\,dx\right)^{\frac{p-2}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}\varphi(x)^{2^{\sharp}}\,dx\right)^{\frac{1}{2^{\sharp}}}\cdot$
$\displaystyle\ \
\left(\int_{\mathbb{R}^{N}}|(v_{m}-v)|)(x)|^{\frac{2^{\sharp}}{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}}\,dx\right)^{\frac{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}{2^{\sharp}}}$

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

$\left(\int_{\mathbb{R}^{N}}|(v_{m}-v)|)(x)|^{\frac{2^{\sharp}}{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}}\,dx\right)^{\frac{(1+\frac{\alpha}{N}-\frac{p}{2^{\sharp}})2^{\sharp}-p+1}{2^{\sharp}}}\to
0,$
$\left(\int_{\mathbb{R}^{N}}((v_{m}|^{p-2}+|v|^{p-2})(x))^{\frac{2^{\sharp}}{p-2}}\,dx\right)^{\frac{p-2}{2^{\sharp}}}\left(\int_{\mathbb{R}^{N}}\varphi(x)^{2^{\sharp}}\,dx\right)^{\frac{1}{2^{\sharp}}}<+\infty.$

So we have that

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|v_{m}|^{p})(x)|v_{m}(x)|^{p-2}v_{m}(x)\varphi(x)\,dx\to\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p}(x))|v(x)|^{p-2}v(x)\varphi(x)\,dx$

and

$\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p-2}v(x)\varphi(x)\,dx-\int_{\mathbb{R}^{N}}(I_{\alpha}*|v|^{p})(x)|v(x)|^{p}\,dx\int_{\mathbb{R}^{N}}\left(\nabla
v\nabla\varphi+v\varphi\right)\,dx=0.$

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

$\pi_{k-1}=\\{l=(l_{1},l_{2},...l_{k})\in\mathbb{R}^{k}\ \
|\sum_{i=1}^{k}|l_{i}|=1\\}.$

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,

$\int_{\mathbb{R}^{N}}(|\nabla\chi(u))|^{2}+|\chi|^{2})\,dx=1.$

We deduce that

$\displaystyle
1=\int_{\mathbb{R}_{+}^{N+1}}(|\nabla\chi(u))|^{2}+|\chi|^{2})\,dxdy$
$\displaystyle=\int_{0}^{1}\,dy\int_{\mathbb{R}^{N}}|(\nabla_{x}[(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x)|\tilde{u}_{\sigma}(x)|^{p}],-(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x)|\tilde{u}_{\sigma}(x)|^{p})|^{2}\,dx$
$\displaystyle+\int_{0}^{1}\,dy\int_{\mathbb{R}^{N}}(1-y)^{2}|(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x)|\tilde{u}_{\sigma}(x)|^{p}|^{2}\,dx$
$\displaystyle=\int_{0}^{1}\,dy\int_{B_{\sigma
R}(0)}|(\nabla_{x}[(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x)|\tilde{u}_{\sigma}(x)|^{p}],-(I_{\alpha}*|\tilde{u}_{\sigma}(x)|^{p})|\tilde{u}_{\sigma}(x)|^{p})|^{2}\,dx$
$\displaystyle+\int_{0}^{1}\,dy\int_{B_{\sigma
R}(0)}(1-y)^{2}|(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x)|\tilde{u}_{\sigma}(x)|^{p}|^{2}\,dx$

and

$\displaystyle\nabla_{x}[(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x)|\tilde{u}_{\sigma}(x)|^{p}]$
$\displaystyle=$
$\displaystyle\nabla_{x}[\int_{\mathbb{R}^{N}}\frac{|u(z)|^{p}}{|\frac{x}{\sigma}-z|^{N-\alpha}}\,dz|u(\frac{x}{\sigma})|^{p}]$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|u(z)|^{p}(\alpha-N)|\frac{x}{\sigma}-z|^{\alpha-N-2}(\frac{x}{\sigma}-z)\frac{1}{\sigma}\,dz|u(\frac{x}{\sigma})|^{p}$
$\displaystyle+\int_{\mathbb{R}^{N}}\frac{|u(z)|^{p}}{|\frac{x}{\sigma}-z|^{N-\alpha}}\,dzp|u(\frac{x}{\sigma})|^{p-2}u(\frac{x}{\sigma})\frac{1}{\sigma}$

and

$\displaystyle\int_{B_{\sigma
R}(0)}|\int_{\mathbb{R}^{N}}|u(z)|^{p}(\alpha-N)|\frac{x}{\sigma}-z|^{\alpha-N-2}(\frac{x}{\sigma}-z)\frac{1}{\sigma}\,dz|u(\frac{x}{\sigma})|^{p}$
$\displaystyle+\int_{\mathbb{R}^{N}}\frac{|u(z,0)|^{p}}{|\frac{x}{\sigma}-z|^{N-\alpha}}\,dzp|u(\frac{x}{\sigma})|^{p-2}u(\frac{x}{\sigma})\frac{1}{\sigma}|^{2}\,dx$
$\displaystyle=$
$\displaystyle\int_{B_{R}(0)}|\int_{\mathbb{R}^{N}}|u(z)|^{p}(\alpha-N)|y-z|^{\alpha-N-2}(y-z)\frac{1}{\sigma}\,dz|u(y)|^{p}$
$\displaystyle+\int_{\mathbb{R}^{N}}\frac{|u(z,0)|^{p}}{|y-z|^{N-\alpha}}\,dzp|u(y)|^{p-2}u(y)\frac{1}{\sigma}|^{2}\sigma^{N}\,dy$
$\displaystyle=$
$\displaystyle\frac{1}{\sigma^{2}}\sigma^{N}\int_{B_{R}(0)}|\nabla_{x}[(I_{\alpha}*|u|^{p}(x))|u(x)|^{p}]|^{2}\,dx.$

Then

$\displaystyle\int_{\mathbb{R}_{+}^{N+1}}(|\nabla\chi(u))|^{2}+|\chi|^{2})\,dxdy=\sigma^{N-2}\int_{B_{R}(0)}|\nabla_{x}[(I_{\alpha}*|u|^{p}(x,0))|u(x,0)|^{p}]|^{2}\,dx$
$\displaystyle+\frac{4}{3}\sigma^{N}\int_{B_{R}(0)}|(I_{\alpha}*|u|^{p}(x,0))|u(x,0)|^{p}|^{2}\,dx.$

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

$1\geq\sigma^{N-2}\int_{B_{R}(0)}|\nabla_{x}u|^{2}\,dx\geq\sigma^{N-2}\rho^{2}.$

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

$1\leq C(\sigma^{N-2}+\sigma^{N}),$

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

$\displaystyle\int_{\mathbb{R}^{N}\times\\{0\\}}|\chi(u)|^{p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}\times\\{0\\}}|(1-y)(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x,0)|\tilde{u}_{\sigma}(x,0)|^{p}|^{p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}\times\\{0\\}}|(1-y)|^{p}|(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x,0)|\tilde{u}_{\sigma}(x,0)|^{p}|^{p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|(I_{\alpha}*|\tilde{u}_{\sigma}|^{p})(x,0)|\tilde{u}_{\sigma}(x,0)|^{p}|^{p}\,dx$
$\displaystyle=$ $\displaystyle\int_{\mathbb{B}_{\sigma
R}}|(I_{\alpha}*|u(\frac{x}{\sigma},0)|^{p})(x,0)|u(\frac{x}{\sigma},0)|^{p}|^{p}\,dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{B}_{R}}|(I_{\alpha}*|u(y,0)|^{p})(x,0)|u(y,0)|^{p}|^{p}\,dy$
$\displaystyle\geq$ $\displaystyle\sigma^{N}\cdot
1\geq\underline{\sigma}^{N}.$

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

$\int_{\mathbb{R}^{N}\times\\{0\\}}(I_{\alpha}*|u|^{p})|u|^{p-2}u\,dx\geq\varepsilon_{k},\
\ \ \ u\in A_{k}.$

Therefore, for each $k\geq 1$,

$b_{k}=\sup_{A\in\Gamma_{k}}\inf_{u\in A_{k}}J(u)\geq\inf_{u\in
A_{k}}J(u)\geq\varepsilon_{k}>0.$

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).

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# A note on the strong consistency of M-estimates in linear models††thanks:
This work is supported by the National Natural Science Foundation of China
(11171001, 11201001, 11426032), the Natural Science Foundation of Anhui
Province (1308085QA03, 1408085QA02), the Science Fund for Distinguished Young
Scholars of Anhui Province (1508085J06) and Introduction Projects of Anhui
University Academic and Technology Leaders.

Xinghui Wang, Shuhe Hu
Department of Statistics, Anhui University, Hefei 230601, P.R. China
Corresponding author. E-mail address<EMAIL_ADDRESS>(S.H. Hu);
<EMAIL_ADDRESS>(X.H. Wang).

Abstract We improve a known result on the strong consistency of M-estimates of
the regression parameters in a linear model for independent and identically
distributed random errors under some mild conditions.

Keywords linear model, strong consistency, M-estimate

MR(2010) Subject Classification 62F12

We consider the linear model

$\displaystyle Y_{i}=x_{i}^{\prime}\beta_{0}+e_{i},i=1,2,\cdots,n,$ (1)

where $x_{i}$ are $p\times 1$ known design vectors, $\beta_{0}$ is a $p\times
1$ unknown vector of regression coefficients and $\\{e_{i}\\}$ are error
variables. The M-estimate $\hat{\beta}_{n}$ of $\beta_{0}$ is defined by
minimizing

$\displaystyle\sum\limits_{i=1}^{n}\rho(Y_{i}-x_{i}^{\prime}\beta),$ (2)

where $\rho$ is a convex function. Important examples include Huber’s estimate
with $\rho(x)=(x^{2}I(|x|\leq c))/2+(c|x|-c^{2}/2)I(|x|>c)$, $c>0$, where
$I(A)$ is the indicator function of the set $A$, the $\mathscr{L}^{q}$
regression estimate with $\rho(x)=|x|^{q}$, $1\leq q\leq 2$, and regression
quantiles with $\rho(x)=\rho_{\alpha}(x)=\alpha x^{+}+(1-\alpha)(-x)^{+}$,
$0<\alpha<1$, where $x^{+}=\max(x,0)$. In particular, if $q=1$ or
$\alpha=1/2$, then the minimizer of (2) is called the least absolute deviation
estimate. There is a substantial amount of work concerning asymptotic
properties of M-estimates.

We assume that $\rho$ is a non-monotonic convex function on $\mathbb{R}$ with
right and left derivatives $\psi_{+}$ and $\psi_{-}$. Choose $\psi$ such that
$\psi_{-}(u)\leq\psi(u)\leq\psi_{+}(u)$ for all $u\in\mathbb{R}$. Write
$S_{n}=\sum_{i=1}^{n}x_{i}x_{i}^{\prime}$, $d_{n}=\max_{1\leq i\leq
n}x_{i}^{\prime}S_{n}^{-1}x_{i}$, and assume that $S_{n_{0}}>0$ for some
integer $n_{0}$ and that $n\geq n_{0}$.

Zhao (2002) established the following result.

Theorem A. Assume that $\\{e_{i},i\geq 1\\}$ is a sequence of independent and
identically distributed (i.i.d.) random variables. Suppose there exist
positive constants $\Delta$, $C_{0}$, $C_{1}$ and $\delta\in(0,1]$ such that
the following conditions are satisfied:

$\displaystyle\psi(u+h)-\psi(u)\leq
C_{0}~{}~{}\textrm{for}~{}~{}h\in(0,\Delta)~{}~{}\textrm{and}~{}~{}u\in\mathbb{R},$
(3)
$\displaystyle\textrm{E}\psi(e_{1})=0,~{}~{}~{}|\textrm{E}\psi(e_{1}+u)|\geq
C_{1}|u|~{}~{}\textrm{ for}~{}~{}|u|<\Delta,$ (4) $d_{n}=O(n^{-\delta}).$

Assume further that $\textrm{E}|\psi(e_{1})|^{1/\delta}<\infty$ if
$0<\delta<1$ and $\textrm{E}|\psi(e_{1})|^{q}<\infty$ for some $q>1$ if
$\delta=1$. Then $\hat{\beta}_{n}\rightarrow\beta_{0}$ a.s as
$n\rightarrow\infty$.

We will further discuss the strong consistency of $\hat{\beta}_{n}$ and obtain
the following result.

Theorem 1. In model (1), assume that $\\{e_{i},i\geq 1\\}$ is a sequence of
i.i.d. random variables. Suppose that conditions (3) and (4) are satisfied and

$\displaystyle d_{n}=O(n^{-1}).$

Then $\hat{\beta}_{n}\rightarrow\beta_{0}$ a.s as $n\rightarrow\infty$.

_Proof_ For the technical proof, see the Appendix.
_Remark._ In Theorem A, $\textrm{E}|\psi(e_{1})|^{q}<\infty$ for some $q>1$ is
needed for the case $\delta=1$. Theorem 1 is obtained without the condition
$\textrm{E}|\psi(e_{1})|^{q}<\infty$ for some $q>1$ in the case $\delta=1$.
Theorem 1 improves the result of Theorem A.

## Appendix

To prove the main results of the paper, we need the following lemmas.
Lemma 1 (c.f. Bennett, 1962). Assume that $\\{X_{n},n\geq 1\\}$ is a sequence
of independent random variables such that $\textrm{E}X_{n}=0$ and $|X_{n}|\leq
b$ for all $n\geq 1$ and some $b>0$. Denote
$B_{n}^{2}=\sum_{i=1}^{n}\textrm{E}X_{i}^{2}$. Then for all $\varepsilon>0$,

$\displaystyle
P\Big{(}\Big{|}\sum\limits_{i=1}^{n}X_{i}\Big{|}>\varepsilon\Big{)}\leq
2\exp\Big{\\{}-\frac{\varepsilon^{2}}{2b\varepsilon+2B_{n}^{2}}\Big{\\}}.$

Lemma 2 (c.f. Choi and Sung, 1987). Let $\\{X_{n},n\geq 1\\}$ be a sequence of
i.i.d. random variables with $\textrm{E}X_{1}=0$. Assume that $\\{a_{ni},1\leq
i\leq n,n\geq 1\\}$ is an array of constants. If $\max_{1\leq i\leq
n}|a_{ni}|=O(n^{-1})$, then

$\displaystyle\sum\limits_{i=1}^{n}a_{ni}X_{i}\rightarrow
0~{}~{}\textrm{as}~{}~{}n\rightarrow\infty.$

_Proof of Theorem 1._ Let $x_{ni}=S_{n}^{-1/2}x_{i}$,
$\beta_{n0}=S_{n}^{1/2}\beta_{0}$ and
$\hat{\beta}_{n}^{*}=S_{n}^{1/2}\hat{\beta}_{n}$. Then

$\displaystyle\sum\limits_{i=1}^{n}x_{ni}x_{ni}^{\prime}=I_{p},~{}~{}\sum\limits_{i=1}^{n}||x_{ni}||^{2}=p,~{}~{}d_{n}=\max\limits_{1\leq
i\leq n}||x_{ni}||^{2},$ (A.1)

where $I_{p}$ is the $p\times p$ identity matrix, and $||\cdot||$ is the
Euclidean norm on $\mathbb{R}^{p}$.

Model (1) can be rewritten as

$\displaystyle Y_{i}=x_{ni}^{\prime}\beta_{n0}+e_{i},~{}~{}i=1,2,\cdots,n$
(A.2)

and

$\displaystyle\sum\limits_{i=1}^{n}\rho(Y_{i}-x_{ni}^{\prime}\hat{\beta}_{n}^{*})=\min\limits_{\beta}\sum\limits_{i=1}^{n}\rho(Y_{i}-x_{ni}^{\prime}\beta).$
(A.3)

Without loss of generality, we assume that the true parameter $\beta_{0}=0$ in
model (1), i.e., $\beta_{n0}=0$ in (A.2). Denote the unit sphere
$U=\\{\beta:\beta\in\mathbb{R}^{p},||\beta||=1\\}$. Let $\varepsilon>0$ be any
given constant. Without loss of generality, it can be assumed that
$2C_{2}\varepsilon<\Delta$. Define

$\displaystyle
D_{n}(\beta)=\sum\limits_{i=1}^{n}\\{\rho(e_{i}-x_{ni}^{\prime}\beta)-\rho(e_{i})\\},~{}~{}\beta\in\mathbb{R}^{p}$

and

$\displaystyle D_{n}(\varepsilon n^{1/2}\gamma)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}\\{\rho(e_{i}-\varepsilon
n^{1/2}x_{ni}^{\prime}\gamma)-\rho(e_{i})\\}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma}\\{\psi(e_{i}+t)-\psi(e_{i})\\}dt+\sum\limits_{i=1}^{n}w_{ni}^{\prime}\gamma\psi(e_{i}),~{}~{}\gamma\in
U,$ $\displaystyle=:$ $\displaystyle I_{1n}(\gamma)+I_{2n}(\gamma),$

where $w_{ni}=-\varepsilon n^{1/2}x_{ni}$. Hence

$\displaystyle\inf\limits_{\gamma\in U}D_{n}(\varepsilon
n^{1/2}\gamma)\geq\inf\limits_{\gamma\in
U}I_{1n}(\gamma)+\inf\limits_{\gamma\in
U}I_{2n}(\gamma)\geq\inf\limits_{\gamma\in
U}I_{1n}(\gamma)-\sup\limits_{\gamma\in U}|I_{2n}(\gamma)|.$ (A.4)

We can divide $U$ into $N$ parts, $U_{1}$, $U_{2}$, $\cdots$, $U_{N}$, such
that the diameter of each part is less than $n^{-2}$ and
$N\leq(2n^{2}+1)^{p}$. Let $T_{j}$ be the smallest close convex set covering
$U_{j}$. For a fixed $T_{j}$, there are three cases as follows.

i) $w_{ni}^{\prime}\gamma\geq 0$ for each $\gamma\in T_{j}$, then there exists
a $\gamma_{ij}\in T_{j}$ such that
$w_{ni}^{\prime}\gamma_{ij}=\inf\\{w_{ni}^{\prime}\gamma:\gamma\in T_{j}\\}$.

ii) $w_{ni}^{\prime}\gamma\leq 0$ for each $\gamma\in T_{j}$, then there
exists a $\gamma_{ij}\in T_{j}$ such that
$w_{ni}^{\prime}\gamma_{ij}=\sup\\{w_{ni}^{\prime}\gamma:\gamma\in T_{j}\\}$.

iii) $w_{ni}^{\prime}\gamma>0$ for some $\gamma\in T_{j}$, and
$w_{ni}^{\prime}\gamma<0$ for some $\gamma\in T_{j}$, then there exists a
$\gamma_{ij}\in T_{j}$ such that $w_{ni}^{\prime}\gamma_{ij}=0$.

Write

$\displaystyle
G(t)=\textrm{E}\psi(e_{i}+t),~{}~{}\Psi_{i}(t)=\psi(e_{i}+t)-\psi(e_{i})-G(t),~{}~{}t\in\mathbb{R}.$

By the monotonicity of $\psi$,

$\displaystyle\inf\limits_{\gamma\in U}I_{1n}(\gamma)$ $\displaystyle\geq$
$\displaystyle\inf\limits_{1\leq j\leq N}\inf\limits_{\gamma\in
T_{j}}I_{1n}(\gamma)\geq\inf\limits_{1\leq j\leq
N}\sum\limits_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}\\{\psi(e_{i}+t)-\psi(e_{i})\\}dt$
(A.5) $\displaystyle\geq$ $\displaystyle\inf\limits_{1\leq j\leq
N}\sum\limits_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt\Big{\\{}1-\frac{|\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}\Psi_{i}(t)dt|}{\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt}\Big{\\}}.$

Let $\gamma\in U_{j}$ and $\gamma_{ij}\in T_{j}$. By (A.1) and the definition
of $U_{j}$ and $T_{j}$, for large enough $n$,

$\displaystyle\sum\limits_{i=1}^{n}(x_{ni}^{\prime}\gamma_{ij})^{2}-\sum\limits_{i=1}^{n}(x_{ni}^{\prime}\gamma)^{2}$
$\displaystyle\leq$
$\displaystyle|\sum\limits_{i=1}^{n}(x_{ni}^{\prime}\gamma_{ij})^{2}-\sum\limits_{i=1}^{n}(x_{ni}^{\prime}\gamma)^{2}|$
$\displaystyle=$
$\displaystyle\Big{|}\sum\limits_{i=1}^{n}(\gamma_{ij}-\gamma)^{\prime}x_{ni}x_{ni}^{\prime}(\gamma_{ij}+\gamma)\Big{|}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{i=1}^{n}\parallel\gamma_{ij}-\gamma\parallel\parallel
x_{ni}\parallel^{2}(\parallel\gamma_{ij}-\gamma\parallel+2\parallel\gamma\parallel)$
$\displaystyle\leq$
$\displaystyle\sum\limits_{i=1}^{n}n^{-2}(n^{-2}+2)\parallel
x_{ni}\parallel^{2}$ $\displaystyle\leq$ $\displaystyle 3pn^{-2}<1/2,$

combining (A.1), we obtain that for $1\leq j\leq N$,

$\displaystyle\sum\limits_{i=1}^{n}(x_{ni}^{\prime}\gamma)^{2}$
$\displaystyle\geq$
$\displaystyle\sum\limits_{i=1}^{n}(x_{ni}^{\prime}\gamma)^{2}-1/2=\gamma^{\prime}\sum\limits_{i=1}^{n}x_{ni}x_{ni}^{\prime}\gamma-1/2$
(A.6) $\displaystyle=$ $\displaystyle\parallel\gamma\parallel^{2}-1/2=1/2.$

By (A note on the strong consistency of M-estimates in linear models††thanks:
This work is supported by the National Natural Science Foundation of China
(11171001, 11201001, 11426032), the Natural Science Foundation of Anhui
Province (1308085QA03, 1408085QA02), the Science Fund for Distinguished Young
Scholars of Anhui Province (1508085J06) and Introduction Projects of Anhui
University Academic and Technology Leaders.) and the selection of
$\varepsilon$, for large enough $n$ and for $i=1,2,\cdots,n$ and
$j=1,2,\cdots,N$,

$\displaystyle|w_{ni}^{\prime}\gamma_{ij}|=|\varepsilon
n^{1/2}x_{ni}^{\prime}\gamma_{ij}|\leq
C_{2}\varepsilon\parallel\gamma_{ij}\parallel\leq
C_{2}\varepsilon(1+n^{-2})<2C_{2}\varepsilon<\Delta.$

It follows that by (4) and (A.6),

$\displaystyle\inf\limits_{1\leq j\leq
N}\sum\limits_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma ij}G(t)dt$
$\displaystyle\geq$ $\displaystyle\inf\limits_{1\leq j\leq
N}C_{1}\sum\limits_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}tdt$
$\displaystyle=$ $\displaystyle\inf\limits_{1\leq j\leq
N}\frac{C_{1}}{2}\sum\limits_{i=1}^{n}(w_{ni}^{\prime}\gamma_{ij})^{2}$
$\displaystyle\geq$ $\displaystyle\frac{C_{1}\varepsilon^{2}n}{4}.$

For $1\leq j\leq N$,, denote
$Y_{ni}^{(j)}=\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}\Psi_{i}(t)dt$. By (3) and
(A.6), one has that

$\displaystyle|Y_{ni}^{(j)}|\leq
2C_{0}|w_{ni}^{\prime}\gamma_{ij}|<4C_{0}C_{2}\varepsilon=:C_{3}.$

For $1\leq j\leq N$, we also obtain that by

$\displaystyle\sum\limits_{i=1}^{n}\textrm{Var}(Y_{ni}^{(j)})$
$\displaystyle\leq$
$\displaystyle\sum\limits_{i=1}^{n}\textrm{E}\\{\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}(\psi(e_{i}+t)-\psi(e_{i}))dt\\}^{2}$
$\displaystyle\leq$ $\displaystyle
C_{0}\sum\limits_{i=1}^{n}|w_{ni}^{\prime}\gamma_{ij}|\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt$
$\displaystyle\leq$
$\displaystyle\frac{C_{3}}{2}\sum\limits_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt.$

Define event $A_{n}$ by

$\displaystyle A_{n}:=\Big{\\{}\sup\limits_{1\leq j\leq
N}\frac{|\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}\Psi_{i}(t)dt|}{\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt}\geq\frac{1}{2}\Big{\\}}.$

Noting the fact that $N\leq(2n^{2}+1)^{p}$ and using Lemma 1, we have that

$\displaystyle P(A_{n})$ $\displaystyle\leq$
$\displaystyle\sum\limits_{j=1}^{N}P\Big{(}\Big{|}\sum_{i=1}^{n}Y_{ni}^{(j)}\Big{|}\geq\frac{1}{2}\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt\Big{)}$
$\displaystyle\leq$ $\displaystyle
2\sum\limits_{j=1}^{N}\exp\Big{\\{}-\frac{\frac{1}{4}(\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt)^{2}}{2C_{3}\times\frac{1}{2}\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt+2C_{3}\times\frac{1}{2}\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt}\Big{\\}}$
$\displaystyle=$ $\displaystyle
2\sum\limits_{j=1}^{N}2\exp\Big{\\{}-\frac{1}{8C_{3}}\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt\Big{\\}}$
$\displaystyle\leq$ $\displaystyle 2(2n^{2}+1)^{p}\exp(-C_{4}n),$

thus $\sum_{n=1}^{\infty}P(A_{n})<\infty$. By Borel-Cantelli lemma, it follows
that $P(A_{n},i.o.)=0$. Hence with probability one for large enough $n$,

$\displaystyle\sup\limits_{1\leq j\leq
N}\frac{\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}\Psi_{i}(t)dt}{\sum_{i=1}^{n}\int_{0}^{w_{ni}^{\prime}\gamma_{ij}}G(t)dt}<\frac{1}{2},$

which implies by (A.5) that with probability one for large enough $n$,

$\displaystyle\inf\limits_{\gamma\in
U}I_{1n}(\gamma)\geq\frac{C_{1}\varepsilon^{2}n}{8}.$ (A.7)

Denote

$\displaystyle x_{ni}=\left(\begin{array}[]{c}x_{ni1}\\\ x_{ni2}\\\ \vdots\\\
x_{nip}\end{array}\right),\gamma=\left(\begin{array}[]{c}\gamma_{1}\\\
\gamma_{2}\\\ \vdots\\\
\gamma_{p}\end{array}\right),~{}~{}a_{ni}=\left\\{\begin{array}[]{cc}\frac{x_{nik}}{n^{1/2}},1\leq
i\leq n\\\ 0,i\geq n\end{array}\right.\textrm{for fixed}~{}~{}k=1,2,\cdots,p.$

By (A.1) and $d_{n}=O(n^{-1})$, for fixed $k=1,2,\cdots,N$,
$|x_{nik}|\leq\parallel x_{ni}\parallel\leq
d_{n}^{1/2}\leq\sqrt{C_{2}}n^{-1/2}$ and $\sum_{i=1}^{n}x_{nik}^{2}=1$, thus

$\displaystyle|a_{ni}|\leq\sqrt{C_{2}}n^{-1}~{}~{}\textrm{for}~{}~{}n\geq 1.$

Using Lemma 2, it follows that

$\displaystyle
n^{-1/2}\sum\limits_{i=1}^{n}x_{nik}\psi(e_{i})=\sum\limits_{i=1}^{n}a_{ni}\psi(e_{i})\rightarrow
0~{}~{}a.s.$

Hence,

$\displaystyle n^{-1}\sup\limits_{\gamma\in U}|I_{2n}(\gamma)|$
$\displaystyle=$ $\displaystyle n^{-1}\sup\limits_{\gamma\in
U}|\sum\limits_{i=1}^{n}w_{ni}^{\prime}\gamma\psi(e_{i})|=n^{-1}\sup\limits_{\gamma\in
U}|\sum\limits_{i=1}^{n}\varepsilon n^{1/2}x_{ni}^{\prime}\gamma\psi(e_{i})|$
$\displaystyle=$ $\displaystyle\varepsilon n^{-1/2}\sup\limits_{\gamma\in
U}\Big{|}\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{p}x_{nik}\gamma_{k}\psi(e_{i})\Big{|}$
$\displaystyle=$ $\displaystyle\varepsilon n^{-1/2}\sup\limits_{\gamma\in
U}\Big{|}\sum\limits_{k=1}^{p}\Big{(}\sum\limits_{i=1}^{n}x_{nik}\psi(e_{i})\Big{)}\gamma_{k}\Big{|}$
$\displaystyle\leq$ $\displaystyle\varepsilon n^{-1/2}\sup\limits_{\gamma\in
U}\sqrt{\sum\limits_{k=1}^{p}\Big{(}\sum\limits_{i=1}^{n}x_{nik}\psi(e_{i})\Big{)}^{2}}\sqrt{\sum\limits_{k=1}^{p}\gamma_{k}^{2}}$
$\displaystyle=$
$\displaystyle\varepsilon\sqrt{\sum\limits_{k=1}^{p}\Big{(}n^{-1/2}\sum\limits_{i=1}^{n}x_{nik}\psi(e_{i})\Big{)}^{2}}\rightarrow
0~{}~{}a.s.,$

which implies that with probability one for large enough $n$,

$\displaystyle\sup\limits_{\gamma\in
U}|I_{2n}(\gamma)|\leq\frac{C_{2}\varepsilon^{2}n}{16}.$ (A.9)

Combining (A.4) with (A.7) and (A.9), we have that with probability one for
large enough $n$,

$\displaystyle\inf\limits_{\gamma\in U}D_{n}(\varepsilon
n^{1/2}\gamma)\geq\frac{C_{2}\varepsilon^{2}n}{16}.$

By the convexity of $D_{n}(\cdot)$, $D_{n}(0)=0$ and the definition (A.3) of
$\hat{\beta}_{n}^{*}$, it follow that

$\displaystyle\Big{\\{}\inf\limits_{\gamma\in U}D_{n}(\varepsilon
n^{1/2}\gamma)>0\Big{\\}}\subset\\{\parallel\hat{\beta}_{n}^{*}\parallel\leq\varepsilon
n^{1/2}\\}.$

Thus for any given $\varepsilon>0$, with probability one for large enough $n$,

$\displaystyle\parallel\hat{\beta}_{n}^{*}\parallel\leq\varepsilon n^{1/2},$

which implies that

$\displaystyle n^{-1/2}\parallel\hat{\beta}_{n}^{*}\parallel\rightarrow
0~{}~{}a.s.~{}~{}\textrm{as}~{}~{}n\rightarrow\infty.$

Denote $\zeta(A)$ for the smallest eigenvalue of a positive definite matrices
$A$. Based on the result that if $A$ and $B$ are two positive definite
matrices of order $p$, then

$\displaystyle\textrm{tr}(AB)\geq\mu(A)\zeta(B),$

where $\mu(A)$ is the largest eigenvalue of $A$ (see Chen & Zhao, 1995). Now
we take $M=n_{0}$ and $n>n_{0}$ so that $S_{n}\geq S_{M}>0$. By (A note on the
strong consistency of M-estimates in linear models††thanks: This work is
supported by the National Natural Science Foundation of China (11171001,
11201001, 11426032), the Natural Science Foundation of Anhui Province
(1308085QA03, 1408085QA02), the Science Fund for Distinguished Young Scholars
of Anhui Province (1508085J06) and Introduction Projects of Anhui University
Academic and Technology Leaders.), one has that

$\displaystyle\zeta(S_{M})(\zeta(S_{M}))^{-1}$ $\displaystyle\leq$
$\displaystyle\zeta(S_{M})\textrm{tr}(S_{n}^{-1})\leq\textrm{tr}(S_{M}^{1/2}S_{n}^{-1}S_{M}^{1/2})$
$\displaystyle=$
$\displaystyle\textrm{tr}(S_{n}^{-1}S_{M})=\sum\limits_{i=1}^{M}x_{i}^{\prime}S_{n}^{-1}x_{i}\leq
Md_{n}\leq MC_{2}n^{-1},$

and there exists a positive constant $C_{5}$ such that $1\leq
C_{5}n^{-1/2}\zeta(S_{n}^{1/2})$. It follows that

$\displaystyle\parallel\hat{\beta}_{n}\parallel\leq C_{5}n^{-1/2}\parallel
S_{n}^{1/2}\hat{\beta}_{n}\parallel=C_{5}n^{-1/2}\parallel\hat{\beta}_{n}^{*}\parallel\rightarrow
0~{}~{}a.s.~{}~{}\textrm{as}~{}~{}n\rightarrow\infty.$

The proof of the theorem is completed.

## References

  * [1] Bennett G (1962) Probability inequalities for sums of independent random variables. J Amer Statist Assoc 57: 33-45
  * [2] Chen XR, Zhao LC (1995) Strong consistency of $M$-estimates of multiple regression coefficients. Syst Sci Math Sci 8: 82-87
  * [3] Choi BD, Sung SH (1987) Almost sure convergence theorems of weighted sums of random variables. Stoch Anal Appl 5: 365-377
  * [4] Zhao LC (2002) Strong consistency of M-estimates in linear models. Sci China Ser A 45: 1420-1427
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