section[0pt] *[8pt].

# Optimal regularity & Liouville property for stable solutions to semilinear
elliptic equations in $\mathbb{R}^{n}$ with $n\geq 10$ 00footnotetext: 2020
Mathematics Subject Classification: 35J61Key words and phases: semilienar
elliptic equation, stable solution, BMO regularity, Morry regularityThe second
author is partially funded by the Chinese Academy of Science and NSFC grant
No. 11688101.The first and third authors are supported by the National Natural
Science Foundation of China (No. 11871088 & No.12025102).∗ Corresponding
author.

Fa Peng, Yi Ru-Ya Zhang∗ and Yuan Zhou

Abstract. Let $0\leq f\in C^{0,1}({\mathbb{R}})$. Given a domain
$\Omega\subset\mathbb{R}^{n}$, we prove that any stable solution to the
equation $-\Delta u=f(u)$ in $\Omega$ satisfies

  * •

a BMO interior regularity when $n=10$,

  * •

an Morrey $M^{p_{n},4+2/(p_{n}-2)}$ interior regularity when $n\geq 11$, where

$p_{n}=\frac{2(n-2\sqrt{n-1}-2)}{n-2\sqrt{n-1}-4}.$

This result is optimal as hinted by e.g. [3, 7, 13], and answers an open
question raised by Cabré, Figalli, Ros-Oton and Serra [8]. As an application,
we show a sharp Liouville property: Any stable solution $u\in
C^{2}({{{\mathbb{R}}}^{n}})$ to $-\Delta u=f(u)$ in ${{{\mathbb{R}}}^{n}}$
satisfying the growth condition

$|u(x)|=\left\\{\begin{array}[]{ll}o\left(\log|x|\right)&{\rm as}\
|x|\to+\infty,\quad\quad{\rm when}\quad n=10;\\\
o\left(|x|^{-\frac{n}{2}+\sqrt{n-1}+2}\right)&{\rm as}\
|x|\to+\infty,\quad\quad{\rm when}\quad n\geq 11\end{array}\right.$

must be a constant. This extends the well-known Liouville property for radial
stable solutions obtained by Villegas [24].

##  1 Introduction

Let $\Omega$ be a bounded domain of ${\mathbb{R}}^{n}$ with $n\geq 2$. Given
any local Lipscitz function $f\colon{\mathbb{R}}\to\mathbb{R}$ (for short
$f\in C^{0,1}({\mathbb{R}})$), we consider the semilinear elliptic equation

$-\Delta u=f(u)\quad{\rm in}\quad\Omega,$ (1.1)

which is the Euler-Lagrange equation for the energy functional

${\mathcal{E}}(u):=\int_{\Omega}\left(\frac{1}{2}|Du|^{2}-F(u)\right)\,dx,$
(1.2)

where $F(t)=\int_{0}^{t}f(s)\,ds$ for $t\in{\mathbb{R}}$. A function $u\in
W^{1,2}(\Omega)$ is called as a weak solution to the equation (1.1) if
$f(u)\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ and

$\int_{\Omega}Du\cdot D\xi\,dx=\int_{\Omega}f(u)\xi\,dx\quad\forall\xi\in
C_{c}^{\infty}(\Omega),$

that is, $u$ is a critical point of the energy functional ${\mathcal{E}}$. We
say that a weak solution $u$ is _stable_ in $\Omega$ if $f^{\prime}_{-}(u)\in
L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ and

$\int_{\Omega}f^{\prime}_{-}(u)\xi^{2}\,dx\leq\int_{\Omega}|D\xi|^{2}\,dx\quad\forall\xi\in
C^{\infty}_{c}(\Omega),$ (1.3)

that is, the second variation of the energy functional $\mathcal{E}$ is
nonnegative. Here and below,

$f^{\prime}_{-}(t)=\liminf_{h\to 0}\frac{f(t+h)-f(t)}{h}\quad\forall
t\in{\mathbb{R}},$

and note that $f^{\prime}_{-}(t)=f^{\prime}(t)$ whenever $f\in
C^{1}({\mathbb{R}})$.

The study of stable solutions to semilinear elliptic equations can be traced
back to the seminal paper [4] by Crandall and Rabinowitz in 1975. The
regularity of stable solutions provides an important way to understand the
regularity of extremal solution $u^{\star}$ to the Gelfand-type problem

$\displaystyle\left\\{\begin{aligned} &-\Delta u=\lambda^{\star}f(u)&{\rm in}\
&\Omega\\\ &\quad u>0&{\rm in}\ &\Omega\\\ &\quad u=0&{\rm on}\
&\partial\Omega\end{aligned}\right.$ (1.4)

for some positive constant $\lambda^{\star}>0$. We refer to [2, 10, 19] and
the reference therein for a comprehensive analysis of (1.4) and related
topics. Note that the extremal solution $u^{\star}$ can be can be approximated
by stable solutions $\\{u_{\lambda}\\}_{\lambda<\lambda^{\star}}$, see e.g.
[13].

In dimension $n\leq 9$, Brezis [2] asked an open problem: whether the extremal
solution $u^{\star}$ to equation (1.4) is bounded for some $f$ and $\Omega$.
Since $u^{\star}$ is approximated by stable soluitons
$\\{u_{\lambda}\\}_{\lambda<\lambda^{\star}}$, it suffices to estbalish some
apriori bound for stable solutions. In recent years, there were several strong
efforts to study regularity for stable solutions and hence for Brezis’ open
problem. In particular, a positive answer was given by Nedev [23] when $n\leq
3$ and by Cabré [5] when $n=4$ (see also [6] for an alternative proof).

Very recently, Cabré, Figalli, Ros-Oton and Serra [8] provide a complete
answer to Brezis’ open problem when $f\geq 0$ based on certain Morrey-type
estimate for $n\geq 3$. Throughout this paper, for $p\in[1,\infty)$ and
$\beta\in(0,n)$, we define the Morrey norm

$\quad\|w\|_{M^{p,\beta}(\Omega)}:=\sup_{y\in\Omega,r>0}\left(r^{\beta-n}\int_{\Omega\cap
B_{r}(y)}|w|^{p}\,dx\right)^{1/p}<\infty,$ (1.5)

where $B_{r}(y)$ denotes the ball with center $y$ and radius $r>0$. We simply
write $B_{r}$ when the center of the ball is at the origin. In addition,
following the convention, we denote by $C(a,b,\cdots)$ a positive constant
depending only on the parameters $a$, $b$, $\cdots$.

In dimension $n\geq 10$, in particular, Cabré, Figalli, Ros-Oton and Serra [8,
Theorem 1.9] established the following regularity of stable solutions to the
equation (1.1).

######  Theorem 1.1 ([8]).

Suppose that $f\in C^{0,1}({\mathbb{R}})$ is nonnegative. If $u\in
C^{2}(B_{1})$ is a stable solution to (1.1) in $B_{1}$, then

$\displaystyle\|u\|_{M^{p,2+\frac{4}{p-2}}{(B_{1/2})}}\leq
C(n,p)\|u\|_{L^{1}(B_{1})}\quad{\rm for\ every}\quad p<p_{n}.$ (1.6)

where

$\displaystyle p_{n}:=\left\\{\begin{aligned} &\infty\quad&{\rm if}\ n=10,\\\
&\frac{2(n-2\sqrt{n-1}-2)}{n-2\sqrt{n-1}-4}\quad&{\rm if}\ n\geq
11.\end{aligned}\right.$ (1.7)

Moreover, suppose additionally that $f$ is nondecreasing, and $\Omega$ be a
bound domain of class $C^{3}$. If $u\in C^{2}(\Omega)\cap
C^{0}(\overline{\Omega})$ is a stable solution to (1.1) in $\Omega$ with
boundary $u=0$ on $\partial\Omega$, then

$\|u\|_{M^{p,2+\frac{4}{p-2}}{(\Omega)}}\leq
C(n,p,\Omega)\|u\|_{L^{1}(\Omega)}\quad{\rm for\ every}\quad p<p_{n}.$ (1.8)

We remark that the exponent $n-2\sqrt{n-1}-4$ changes sign when $n=10$, which
was already appeared in e.g. [21].

However for the endpoint case $p=p_{n}$, Cabré, Figalli, Ros-Oton and Serra
[8, Section 1.3] pointed out that it is an open question whether (1.6) holds.

As hinted by earlier results [7] in the radial symmetric case, when $n=10$,
instead of $L^{\infty}=M^{\infty,2}$ a more suitable space to consider is
class of functions with bounded mean oscillations (BMO space) as remarked
therein. Indeed, $u(x)=-2\log|x|$ is a stable solution to the equation (1.1)
in $B_{1}$ with $f(u)=2(n-2)e^{u}$. Obviously, $u\in{\rm BMO}(B_{1})$ but
$u\notin L^{\infty}(B_{1})$. Here and below, the BMO norm is defined as

$\|u\|_{{\rm
BMO}(\Omega)}:=\sup_{y\in\Omega,r>0}\inf_{c\in{\mathbb{R}}}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to30.87062pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to30.87062pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to23.8676pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to23.8676pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{\Omega\cap
B_{r}(y)}\left|u(x)-c\right|\,dx,$

where, $\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to5.83331pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to5.83331pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to4.5833pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to3.97914pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{E}v\,dx$
denotes the integral average of $v$ on a measurable set $E$.

On the other hand, when $n\geq 11$, also hinted by the results in [7], in the
range $p\leq p_{n}$ is the best possible to get (1.6). Besides, it was proven
in [3] that the function $u(x)=|x|^{-\frac{2}{q_{n}-1}}-1$ is the extremal
solution to

$\displaystyle\begin{aligned} &-\Delta u=\lambda^{\star}(1+u)^{q_{n}}\quad{\rm
in}\ B_{1};\quad u=0\quad\quad{\rm on}\ \partial B_{1}\end{aligned}$ (1.9)

with $\lambda^{\star}=\frac{2}{q_{n}}$ and
$q_{n}:=\frac{n-2\sqrt{n-1}}{n-2\sqrt{n-1}-4}$. It is easy to see that $u\in
M^{p,2+\frac{4}{p-2}}(B_{1/2})$ if and only if $p\leq p_{n}$. Recall that by
[13, Section 3.2.2], such extremal solution can be approximated by stable
solutions.

The first main purpose of this paper is to establish the following regularity
at the end-point $p_{n}$ for stable solutions to the equation (1.1) when
$n\geq 10$, and then answers the above open question by Cabré, Figalli, Ros-
Oton and Serra [8].

######  Theorem 1.2.

Suppose $f\in C^{0,1}({\mathbb{R}})$ is nonnegative. For any stable solution
$u\in C^{2}(B_{1})$ to (1.1) in $B_{1}$, when $n=10$ we have

$\displaystyle\|u\|_{{\rm BMO}(B_{1/2})}\leq C(n)\|u\|_{L^{1}(B_{1})},$ (1.10)

and when $n\geq 11$ we have

$\displaystyle\|u\|_{M^{p_{n},2+\frac{4}{p_{n}-2}}{(B_{1/2})}}\leq
C(n)\|u\|_{L^{1}(B_{1})}.$ (1.11)

Moreover, suppose additionally that $f$ is nondecreasing, and $\Omega$ is a
bounded smooth convex domain. For any positive stable solution $u\in
C^{2}(\overline{\Omega})$ to (1.1) with boundary $u=0$ on $\partial\Omega$,
when $n=10$ we have

$\displaystyle\|u\|_{{\rm BMO}(\Omega)}\leq C(n,\Omega)\|u\|_{L^{1}(\Omega)},$
(1.12)

and when $n\geq 11$ we have

$\displaystyle\|u\|_{M^{p_{n},2+\frac{4}{p_{n}-2}}{(\Omega)}}\leq
C(n,\Omega)\|u\|_{L^{1}(\Omega)}.$ (1.13)

As a direct consequence of the above a priori estimates, we have the following
result for stable solution in $W^{1,2}$.

######  Corollary 1.3.

Suppose that $\Omega\subset\mathbb{R}^{n}$ is a bounded smooth convex domain
and that $f\in C^{0,1}({\mathbb{R}})$ is nonnegative, nondecreasing, convex,
and satisfies $f(t)/t\to+\infty$ as $t\to+\infty$. For any stable solution
$u\in W^{1,2}_{0}(\Omega)$ to (1.1) with boundary $u=0$ on $\partial\Omega$,
we have (1.12) when $n=10$, and (1.13) when $n\geq 11$.

######  Remark 1.4.

(i) While writing this paper, via personal communication we learn that Figalli
and Mayboroda have independently proved (1.10) in Theorem 1.2 with $n=10$ via
a similar argument.

(ii) In Theorem 1.2 and Corollary 1.3 we only consider bounded smooth convex
domains so as to avoid technical discussions on the boundary estimate. We
believe that after suitable modifications, it is possible to relax this
assumption to bounded domains of $C^{3}$ class, as in [8].

As an application of Theorem 1.2, we prove the following Liouville property
for stable solutions to the equation

$-\Delta u=f(u)\quad{\rm in}\quad{\mathbb{R}}^{n}$ (1.14)

for $f\in C^{0,1}({{{\mathbb{R}}}^{n}})$.

######  Theorem 1.5.

Let $n\geq 10$ and $0\leq f\in
C^{0,1}_{{\mathop{\mathrm{missing}}{\,loc\,}}}({\mathbb{R}})$. Suppose that
$u\in C^{2}({\mathbb{R}}^{n})$ is a nonconstant stable solution to (1.14) in
${\mathbb{R}}^{n}$.

If $u$ is nonconstant, then

$\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to27.12424pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to27.12424pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to22.33397pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to22.33397pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{4R}\setminus
B_{R}}|u(x)|\,dx\geq\left\\{\begin{array}[]{ll}c\log R&\forall R\geq
R_{0},\quad{\rm if}\quad n=10,\\\ cR^{-\frac{n}{2}+2+\sqrt{n-1}}&\forall R\geq
R_{0},\quad{\rm if}\quad n\geq 11\end{array}\right.$ (1.15)

for some $R_{0}\geq 2$ and $c>0$.

In particular, if $u$ satisfies the growth condition

$|u(x)|=\left\\{\begin{array}[]{ll}o\left(\log|x|\right)&{\rm as}\
|x|\to+\infty,\quad\quad{\rm when}\quad n=10;\\\
o\left(|x|^{-\frac{n}{2}+2+\sqrt{n-1}}\right)&{\rm as}\
|x|\to+\infty,\quad\quad{\rm when}\quad n\neq 10,\end{array}\right.$ (1.16)

then $u$ must be a constant.

This problem has attracted a lot of attention in the literature. First of all,
for radial stable solutions, Villegas [24] in 2007 obtained the following
sharp Liouville property based on the monotone property by Cabré-Capella [11]
(see also [24, 13]).

######  Theorem 1.6 ([24]).

Let $n\geq 2$ and $f\in C^{1}({\mathbb{R}})$. Suppose that $u\in
C^{2}({{{\mathbb{R}}}^{n}})$ is a radial stable solution to (1.14).

If $u$ is not constant, then

$|u(x)|\geq\left\\{\begin{array}[]{lll}M\log|x|&\mbox{whenever}\ |x|\geq
r_{0},&{\rm when}\quad n=10,\\\
M|x|^{-\frac{n}{2}+\sqrt{n-1}+2}&\mbox{whenever}\ |x|\geq r_{0},&{\rm
when}\quad n\neq 10\end{array}\right.$ (1.17)

for some $M>0$ and $r_{0}\geq 10$.

In particular, if $u$ satisfies the growth condition (1.16), then $u$ must be
a constant.

Note that for radial stable solutions $u(x)$, the condition (1.15) is
equivalent to (1.17). Indeed, by [24], $u(r)=u(re_{1})$ is always monotone,
and hence

$\min\\{|u(4r)|,|u(r)|\\}\leq\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to24.81961pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to24.81961pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to20.02934pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to20.02934pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{4r}\setminus
B_{r}}|u(x)|\,dx\leq\max\\{|u(4r)|,|u(r)|\\}\quad\forall r>0,$

which implies the equivalence between (1.15) and (1.17).

Let $\beta_{n}=-\frac{n}{2}+2+\sqrt{n-1}$. Then $\beta_{n}<0$ when $n\geq 11$
and $\beta_{n}>0$ when $n\leq 9$. The sharpness of Theorem 1.6 (and also
Theorem 1.5) is demonstrated in the following sense by Villegas [24] (with a
slight modification at $n=10$).

  1. (i)

When $n\neq 10$, the radial smooth function
$(1+|x|^{2})^{\frac{\beta_{n}}{2}}$ is a stable solution to the equation
$-\Delta u=f_{\beta_{n}}(u)$ in ${\mathbb{R}}^{n}$, where when $n\geq 11$,

$\displaystyle f_{\beta_{n}}(s):=\left\\{\begin{array}[]{ll}0,&\quad{\rm
if}\quad s\leq 0,\\\
\beta_{n}(\beta_{n}-2)s^{1-\frac{4}{\beta_{n}}}-\beta_{n}(\beta_{n}+n-2)s^{1-\frac{2}{\beta_{n}}},&\quad{\rm
if}\quad s>0\end{array}\right.$

and when $n\leq 9$,

$\displaystyle
f_{\beta_{n}}(s):=\left\\{\begin{array}[]{ll}\beta_{n}(\beta_{n}-2)s^{1-\frac{4}{\beta_{n}}}-\beta_{n}(\beta_{n}+n-2)s^{1-\frac{2}{\beta_{n}}},&\quad{\rm
if}\quad s\geq 1,\\\ -(\beta_{n}-2)(n+2)(s-1)-n\beta_{n},&\quad{\rm if}\quad
s<1.\end{array}\right.$

See [24, Example 3.1] for details. Note that, when $n\geq 11$, by
$\beta_{n}<0$ and $\beta_{n}+n-2>0$, we have $f_{\beta_{n}}\geq 0$ in
${\mathbb{R}}$; while when $n\leq 9$, $f_{\beta_{n}}$ changes sign in
${\mathbb{R}}$.

  2. (ii)

When $n=10$, the radial smooth function $-\frac{1}{2}\log(1+|x|^{2})$ is a
stable solution to the equation $-\Delta u=f(u)$ in ${{{\mathbb{R}}}^{n}}$,
where $f(s)=(n-2)e^{2s}+2e^{4s}\geq 0$ in ${\mathbb{R}}$. This is a slight
modification of the [24, Example 3.1] with $n=10$. See the appendix for
details.

For general (nonradial) stable solutions $u\in C^{2}({{{\mathbb{R}}}^{n}})$ to
$-\Delta u=f(u)$ in ${{{\mathbb{R}}}^{n}}$, it is then natural to ask if
certain Liouville properties similar to Theorem 1.6 hold. Namely, when $f$
satisfies certain regularity assumption,

  * •

if $u$ satisfies (1.16), then is it necessary that $u$ is a constant?

  * •

if $u$ is nonconstant, is it possible to give some sharp lower bound for $|u|$
toward $\infty$?

Suppose that $0\leq f\in C^{1}({\mathbb{R}})$ and $u\in
C^{2}({{{\mathbb{R}}}^{n}})$ is a stable solution to (1.14). When $n\leq 4$,
Dupaigne-Farina [14] proved that if $|u|$ is bounded, then $u$ must be a
constant. Recently, with the aid of Cabré et al [8], Dupaigne-Farina [15]
showed that if $n\leq 9$ and $u(x)\geq-C[1+\log|x|]^{{\gamma}}$ for some
${\gamma}\geq 1$ and $C>0$, or if $n=10$ and $u\geq-C$ for some constant
$C>0$, then $u$ must be a constant. When $n\geq 10$, our result Theorem 1.5
finally answers the two questions above.

###  1.1 Ideas of the proofs

We sketch the ideas to prove Theorem 1.2 and Theorem 1.5. All of them heavily
rely on the the following decay estimate on the Dirichlet energy.

######  Lemma 1.7.

Let $n\geq 10$ and $f\in C^{0,1}({\mathbb{R}})$. For any
$y\in{{{\mathbb{R}}}^{n}}$ and $t>0$, if $u\in C^{2}(B_{2t}(y))$ is a stable
solution to the equation (1.1) in $B_{2t}(y)$, one has

$\displaystyle\left(\frac{r}{t}\right)^{-2(1+\sqrt{n-1})}\int_{B_{r}(y)}|Du|^{2}\,dx\leq
C(n)\int_{B_{t}(y)\setminus B_{t/2}(y)}|Du|^{2}\,dx,\quad\forall
r\leq\frac{t}{2}.$ (1.18)

See Section 2 for the proof of Lemma 1.7; the key point is that we take a
suitable test function in a celebrated lemma of Cabré et al [8] (see Lemma 2.1
below).

We also recall the following lemma, which was essentially established in [8],
see Lemma A.2 and Proposition 2.5 with its proof therein. For the convenience
of the reader, we give a sketch of the proof at the beginning of Section 3.

######  Lemma 1.8.

Let $0\leq f\in C^{0,1}({\mathbb{R}})$. For any stable solution $u\in
C^{2}(B_{2t}(y))$ to the equation (1.1) in $B_{2t}(y)$, one has

$\displaystyle\left(\int_{B_{t/2}(y)}|Du|^{2}\,dx\right)^{1/2}\leq
C(n)t^{-n/2}\int_{B_{t}(y)}|Du|\,dx$ (1.19)

and

$\displaystyle\int_{B_{t/2}(y)}|Du|\,dx\leq C(n)t^{-1}\int_{B_{t}(y)}|u|\,dx.$
(1.20)

Applying Lemma 1.7, Lemma 1.8 and some known boundary estimate, we are able to
prove Theorem 1.2 and Corollary 1.3. This is clarified in Section 3.

In order to prove Theorem 1.5, an auxiliary and crucial proposition is shown
in Section 4, which is especially applied in the case $n=10$.

######  Proposition 1.9.

Let $n\geq 3$. Suppose that $u\in
W^{1,1}_{\mathop{\mathrm{missing}}{\,loc\,}}({{{\mathbb{R}}}^{n}})$ is
superharmonic, that is, $-\Delta u\geq 0$ in ${{{\mathbb{R}}}^{n}}$ in
distributional sense. For any $0<r<R<\infty$ we have

$\displaystyle\int_{B_{R}\setminus B_{r}}|Du||x|^{-n+1}\,dx\leq
C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to30.81961pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to30.81961pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to26.02934pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to26.02934pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{r/2}\setminus
B_{r/4}}|u|\,dz+C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to29.12424pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to29.12424pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to24.33397pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to24.33397pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{4R}\setminus
B_{2R}}|u|\,dz.$ (1.21)

The main idea of showing Proposition 1.9 goes as follows. First, it is known
that

$\mbox{$Du_{\delta}(x)=D\Delta^{-1}[\Delta(u_{\delta}\eta)](x)$ for $x\in
B_{R}\setminus B_{r}$},$

where $u_{\delta}$ is a standard smooth mollification of $u$ and $\eta$ is a
suitable cut-off function. Next, thanks to the key fact $-\Delta
u_{\delta}\geq 0$, via some subtle kernel estimate and integration by parts,
we are able to prove (1.21) for $u_{\delta}$, and then a standard
approximation gives (1.21) as desired.

Theorem 1.5 is eventually proved in Section 5. The case $n\geq 11$ is
relatively simpler. In fact, by Lemma 1.7 and Lemma 1.8, one can build up the
following

$\displaystyle r^{-(1+\sqrt{n-1})}\left(\int_{B_{r}}|Du|^{2}\,dx\right)^{1/2}$
$\displaystyle\leq
C(n)R^{\frac{n}{2}-2-\sqrt{n-1}}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to33.12424pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to33.12424pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to28.33397pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to28.33397pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{3R}\setminus
B_{{3R}/4}}|u|\,dx\quad\forall 0<r<<R<\infty$

for stable solutions, which allows us to conclude Theorem 1.5 for $n\geq 11$.

As for the case when $n=10$, we first employ Lemma 1.7 and repeat Lemma 1.8 to
get

$\displaystyle r^{-(1+\sqrt{n-1})}\left(\int_{B_{r}}|Du|^{2}\,dx\right)^{1/2}$
$\displaystyle\leq C(n)\frac{1}{\log R}\int_{B_{R^{2}}\setminus
B_{4}}|Du||x|^{-n+1}\,dx\quad\mbox{ $\forall 0<r<<R<\infty$,}$

which, with the aid of Proposition 1.9, is then bounded from above by

$C(n)\frac{1}{\log R}\left(\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to22.98813pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to22.98813pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to18.19786pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to18.19786pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{2}\setminus
B_{1}}|u(z)|\,dz+\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to30.32425pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to30.32425pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to25.53398pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to25.53398pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{2R^{2}}\setminus
B_{R^{2}}}|u(z)|\,dz\right).$

From this we conclude Theorem 1.5 when $n=10$.

##  2 Proof of Lemma 1.7

Towards Lemma 1.7 we recall the following apriori bound by Cabré et al [8,
Lemma 2.1], which is obtained by taking test function $(x\cdot Du)\eta$ in the
stability condition (1.3).

######  Lemma 2.1.

Let $u\in C^{2}(B_{1})$ be a stable solution to equation (1.1) in $B_{1}$,
with $f\in C^{0,1}({\mathbb{R}})$. Then for all cut-off function $\eta\in
C^{0,1}_{c}(B_{1})$,

$\displaystyle\int_{B_{1}}|x\cdot Du|^{2}|D\eta|^{2}\,dx$
$\displaystyle\geq(n-2)\int_{B_{1}}|Du|^{2}\eta^{2}\,dx+2\int_{B_{1}}|Du|^{2}(x\cdot
D\eta)\eta\,dx-4\int_{B_{1}}(x\cdot Du)(Du\cdot D\eta)\eta\,dx.$ (2.1)

For convenience, for any $0<r<t<\infty$ and $y\in{{{\mathbb{R}}}^{n}}$, write
the annual $A_{r,t}(y):=B_{t}(y)\setminus\overline{B_{r}(y)}$; for simple
write $A_{r,t}=A_{r,t}(0)$.

###### Proof of Lemma 1.7.

It suffices to prove the following

$\displaystyle\left(\frac{r}{t}\right)^{-2(1+\sqrt{n-1})}\int_{B_{r}(y)}|Du|^{2}\,dx\leq
C(n)\int_{A_{r,t}(y)}|Du|^{2}\,dx,\quad\forall r\leq\frac{t}{2}.$ (2.2)

Indeed, applying (2.2) to $\frac{t}{2}$ and $t$, one has

$\displaystyle\left(\frac{1}{2}\right)^{-2(1+\sqrt{n-1})}\int_{B_{\frac{t}{2}}(y)}|Du|^{2}\,dx\leq
C(n)\int_{A_{\frac{t}{2},t}(y)}|Du|^{2}\,dx.$ (2.3)

If $\frac{t}{4}\leq r<\frac{t}{2}$, by $B_{r}(y)\subset B_{\frac{t}{2}}(y)$
and $\frac{1}{4}\leq\frac{r}{t}\leq 1$, (2.3) gives

$\displaystyle\left(\frac{r}{t}\right)^{-2(1+\sqrt{n-1})}\int_{B_{r}(y)}|Du|^{2}\,dx\leq
C(n)\int_{A_{\frac{t}{2},t}(y)}|Du|^{2}\,dx.$ (2.4)

If $0<r<\frac{t}{4}$, applying (2.2) to $r$ and $\frac{t}{2}$,and noting
$A_{r,\frac{t}{2}}\subset B_{\frac{t}{2}}$ one gets

$\displaystyle\left(\frac{r}{t/2}\right)^{-2(1+\sqrt{n-1})}\int_{B_{r}(y)}|Du|^{2}\,dx\leq
C(n)\int_{A_{r,\frac{t}{2}}(y)}|Du|^{2}\,dx\leq
C(n)\int_{B_{\frac{t}{2}}(y)}|Du|^{2}\,dx,$

which together with (2.3) yields

$\displaystyle\left(\frac{r}{t}\right)^{-2(1+\sqrt{n-1})}\int_{B_{r}(y)}|Du|^{2}\,dx\leq
C(n)\int_{A_{\frac{t}{2},t}(y)}|Du|^{2}\,dx.$

From this and (2.4) we conclude (1.18).

To prove (2.2), without loss of generality we may assume that $t=1$ and $y=0$.
Indeed, if $u(x)$ is a stable solution to $-\Delta u=f(u)$ in $B_{2t}(y)$,
then $v(x)=u(tx+y)$ is the stable solution to $-\Delta v=t^{2}f(v)$ in
$B_{2}$. Note that up to a change of variable $u$ satisfies (2.2) if and only
if $v$ satisfies (2.2) with $t=1$ and $y=0$.

Write $a=2(1+\sqrt{n-1})$. Let $r\in(0,\frac{1}{2}]$ be fixed and set

$\displaystyle\eta=\left\\{\begin{aligned} &r^{-\frac{a}{2}}&\quad{\rm if}\
0\leq|x|\leq r\\\ &|x|^{-\frac{a}{2}}\phi&\quad{\rm if}\ r<|x|\leq
1,\end{aligned}\right.$ (2.5)

where $\phi\in C^{\infty}_{c}(B_{1})$ satisfies

$\phi=1\quad{\rm in}\ B_{3/4}\quad\mbox{and}\quad|D\phi|\leq
5\chi_{B_{1}\setminus B_{3/4}}.$ (2.6)

Clearly, $\eta\in C^{0,1}_{c}(B_{1})$. Since $\eta=r^{-\frac{a}{2}}$ in
$B_{r}$ and hence $D\eta=0$ in $B_{r}$, submitting $\eta$ in inequality (2.1)
one has

$\displaystyle\int_{A_{r,1}}|x\cdot Du|^{2}|D\eta|^{2}\,dx$
$\displaystyle\geq(n-2)r^{-a}\int_{B_{r}}|Du|^{2}\,dx+(n-2)\int_{A_{r,1}}|Du|^{2}\eta^{2}\,dx$
$\displaystyle\quad+2\int_{A_{r,1}}|Du|^{2}(x\cdot
D\eta)\eta\,dx-4\int_{A_{r,1}}(x\cdot Du)(Du\cdot D\eta)\eta\,dx.$ (2.7)

Noting that

$D\eta=-\frac{a}{2}|x|^{-\frac{a}{2}-2}x\phi+|x|^{-\frac{a}{2}}D\phi\quad{\rm
in}\ A_{r,1},$

one has

$\displaystyle 2\int_{A_{r,1}}|Du|^{2}(x\cdot
D\eta)\eta\,dx-4\int_{A_{r,1}}(x\cdot Du)(Du\cdot D\eta)\eta\,dx$
$\displaystyle=-a\int_{A_{r,1}}|Du|^{2}|x|^{-a}\phi^{2}\,dx+2\int_{A_{r,1}}|Du|^{2}(x\cdot
D\phi)\phi|x|^{-a}\,dx$ $\displaystyle\quad+2a\int_{A_{r,1}}(x\cdot
Du)^{2}|x|^{-a-2}\phi^{2}\,dx-4\int_{A_{r,1}}(x\cdot Du)(Du\cdot
D\phi)\phi|x|^{-a}\,dx.$ (2.8)

Moreover, by

$|D\eta|^{2}=\frac{a^{2}}{4}|x|^{-a-2}\phi^{2}-2a|x|^{-a-2}(x\cdot
D\phi)\phi+|x|^{-a}|D\phi|^{2},$

one can write

$\displaystyle\int_{A_{r,1}}(Du\cdot x)^{2}|D\eta|^{2}\,dx$
$\displaystyle=\frac{a^{2}}{4}\int_{A_{r,1}}(Du\cdot
x)^{2}|x|^{-a-2}\phi^{2}\,dx+\int_{A_{r,1}}(Du\cdot
x)^{2}|x|^{-a}|D\phi|^{2}\,dx$ $\displaystyle\quad-a\int_{A_{r,1}}(Du\cdot
x)^{2}|x|^{-a-2}(x\cdot D\phi)\phi\,dx.$ (2.9)

Using (2) for the left hand side of (2) and (2) for the last two terms in the
right hand side of (2), and then moving all terms including $D\phi$ to the
left hand side and all other terms to the right hand side, we have

$\displaystyle\int_{A_{r,1}}|x\cdot
Du|^{2}|D\phi|^{2}|x|^{-a}\,dx-2\int_{A_{r,1}}|Du|^{2}(x\cdot
D\phi)\phi|x|^{-a}\,dx$ $\displaystyle\quad+4\int_{A_{r,1}}(x\cdot Du)(Du\cdot
D\phi)\phi|x|^{-a}\,dx-a\int_{A_{r,1}}|x|^{-a-2}(x\cdot Du)^{2}\phi(x\cdot
D\phi)\,dx$
$\displaystyle\geq(n-2)r^{-a}\int_{B_{r}}|Du|^{2}\,dx+(n-2)\int_{A_{r,1}}|Du|^{2}|x|^{-a}\phi^{2}\,dx$
$\displaystyle\quad-a\int_{A_{r,1}}|Du|^{2}|x|^{-a}\phi^{2}\,dx+2a\int_{A_{r,1}}(x\cdot
Du)^{2}|x|^{-a-2}\phi^{2}\,dx$
$\displaystyle\quad-\frac{a^{2}}{4}\int_{A_{r,1}}(Du\cdot
x)^{2}|x|^{-a-2}\phi^{2}\,dx$
$\displaystyle=(n-2)r^{-a}\int_{B_{r}}|Du|^{2}\,dx$
$\displaystyle\quad+\int_{A_{r,1}}\left\\{(n-2-a)|Du|^{2}+\left(2a-\frac{a^{2}}{4}\right)(Du\cdot
x)^{2}|x|^{-2}\right\\}|x|^{-a}\phi^{2}\,dx.$ (2.10)

Note that by $|D\phi|=0$ in $B_{3/4}$ and $|D\phi|\leq 5$ in $B_{1}$ as in
(2.6) and $a>2$,

$\displaystyle\int_{A_{r,1}}|x\cdot
Du|^{2}|D\phi|^{2}|x|^{-a}\,dx-2\int_{A_{r,1}}|Du|^{2}(x\cdot
D\phi)\phi|x|^{-a}\,dx$ $\displaystyle\quad+4\int_{A_{r,1}}(x\cdot Du)(Du\cdot
D\phi)\phi|x|^{-a}\,dx-a\int_{A_{r,1}}|x|^{-a-2}(x\cdot Du)^{2}\phi(x\cdot
D\phi)\,dx$ $\displaystyle\leq C(n)\int_{A_{\frac{3}{4},1}}|Du|^{2}\,dx.$
(2.11)

Additionally, note that $n\geq 10$ implies $a=2(1+\sqrt{n-1})\geq 8$, and
hence

$2a-\frac{a^{2}}{4}=\frac{a}{4}(8-a)\leq 0.$

By $|x|^{-1}|x\cdot Du|\leq|Du|$ in $A_{r,1}$ we have

$(n-2-a)|Du|^{2}+\left(2a-\frac{a^{2}}{4}\right)(Du\cdot
x)^{2}|x|^{-2}\geq\left(n-2+a-\frac{a^{2}}{4}\right)|Du|^{2}.$

Since

$n-2+a-\frac{a^{2}}{4}=-(\frac{a}{2}-[1-\sqrt{n-1}])(\frac{a}{2}-[1+\sqrt{n-1}])=0,$

we have

$\displaystyle(n-2-a)|Du|^{2}+\left(2a-\frac{a^{2}}{4}\right)(Du\cdot
x)^{2}|x|^{-2}\geq 0\quad\mbox{in $A_{r,1}$},$ (2.12)

which means that the last term in the right hand side of (2) is nonnegative.
From this, together with (2) and (2) we conclude (2.2). The proof is complete.
∎

######  Remark 2.2.

Recall that in [8], Cabré et al used the test funciton
$\eta=|x|^{-\frac{a}{2}}\xi$ with $\xi\in C^{\infty}_{c}(B_{1})$, which was
not enough to get (2.2).

##  3 Proofs of Theorem 1.2 and Corollary 1.3

In this section we prove Theorem 1.2 and Corollary 1.3. First, we show a
sketch of the proof of Lemma 1.8.

###### Proof of Lemma 1.8.

Up to considering $v(x)=u(tx+y)$ as did in the proof of Lemma 1.4, we may
assume that $t=1$ and $y=0$. The inequality (1.20) is given by [8, Lemma A.2].
The inequality (1.19) reads as $\|Du\|_{L^{2}(B_{{1/2}})}\leq
C(n)\|Du\|_{L^{1}(B_{1})}$, and will follow from the proof of [8, Proposition
2.5]. Indeed, in [8, Proposition 2.5], Cabre et al proved that

$\|Du\|_{L^{2}(B_{{1/2}})}\leq C(n)\|u\|_{L^{1}(B_{1})}.$ (3.1)

In their proof, first they obtained a bound of $\|Du\|_{L^{2}(B_{{1/2}})}$ via
$\|Du\|_{L^{1}(B_{1/2})}$ and also some other small terms. Next, they used
$\|Du\|_{L^{1}(B_{{1/2}})}\leq C(n)\|u\|_{L^{1}(B_{1})}$. Finally, via an
iteration argument, they got (3.1). If we directly apply the iteration
argument without using $\|Du\|_{L^{1}(B_{{1/2}})}\leq
C(n)\|u\|_{L^{1}(B_{1})}$, one gets $\|Du\|_{L^{2}(B_{{1/2}})}\leq
C(n)\|Du\|_{L^{1}(B_{1})}$. ∎

Recall that $u_{E}=\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to5.83331pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to5.83331pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to4.5833pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to3.97914pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{E}u\,dx$
denotes the integral average of $u$ on a measurable set $E$. The interior
regularity (1.10)& (1.11) in Theorem 1.2 is a consequence of Lemma 1.7 and
(1.19), together with some standard embedding argument.

###### Proofs of (1.10) and (1.11) in Theorem 1.2.

Let $u\in C^{2}(B_{2})$ be stable solution to equation (1.1). Write
$\beta=n-2-2\sqrt{n-1}$. For any $y\in B_{1/2}$ if $r>1/8$, by Lemma 1.8 we
have

$r^{\beta-n}\int_{B_{r}(y)\cap B_{1/2}}|Du|^{2}\,dx\leq
C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to11.66075pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to11.66075pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to10.0434pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to10.0434pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{1/2}}|Du|^{2}\,dx\leq
C(n)\|u\|^{2}_{L^{1}(B_{1})}$

and if $0<r<1/8$, by Lemma 1.7 and Lemma 1.8 again we have

$r^{\beta-n}\int_{B_{r}(y)\cap B_{1/2}}|Du|^{2}\,dx\leq
r^{\beta}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.70406pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.70406pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to12.4788pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to12.4788pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{r}(y)}|Du|^{2}\,dx\leq
C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to20.78831pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to20.78831pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to16.56306pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to16.56306pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{1/4}(y)}|Du|^{2}\,dx\leq
C(n)\|u\|^{2}_{L^{1}(B_{1})}.$

This means that $Du\in M^{2,\beta}(B_{1/2})$ with
$\|Du\|_{M^{2,\beta}(B_{1/2})}\leq C(n)\|u\|_{L^{1}(B_{1})}$.

If $n=10$, then $\beta=2$ and ${2\beta}/{(\beta-2)}=\infty$. Thanks to
Sobolev-Poincaré inequality one can easily check that $Du\in
M^{2,\beta}(B_{1/2})$ implies $u\in BMO(B_{1/2})$ with a norm bound
$\|u\|_{BMO(B_{1/2})}\leq C(n)\|Du\|_{M^{2,\beta}(B_{1/2})}$. If $n\geq 11$,
then $p_{n}={2\beta}/{(\beta-2)}<\infty$ and $\beta=4+2/(p_{n}-2)$. By the
embedding result in [1] and also [9, Section 4], $Du\in M^{2,\beta}(B_{1/2})$
implies $u\in M^{2\beta/(\beta-2),\beta}(B_{1/2})$ with its norm bound
$\|u\|_{M^{p_{n},\beta}(B_{1/2})}\leq C(n)\|Du\|_{M^{2,\beta}(B_{1/2})}$. This
proves (1.10) and (1.11). ∎

To prove the global regularity (1.12) and (1.13) in Theorem 1.2, we need the
following a priori $L^{\infty}$-bound in a neighborhood of $\partial\Omega$
for $C^{2}$ solution when $\Omega$ is a bounded smooth convex domain, see [5,
Proposition 3.2] and [12, 18, 20]. For $\rho>0$ we write

$\Omega_{\rho}:=\\{x\in\Omega:{\rm dist}(x,\partial\Omega)<\rho\\}.$

######  Lemma 3.1.

Suppose that that $f\in C^{0,1}({\mathbb{R}})$ is nonnegative and $\Omega$ is
a smooth convex domain in ${\mathbb{R}}^{n}$. There exist positive constants
$\rho$ and ${\gamma}$ depending only on the domain $\Omega$ such that for any
positive solution $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ to (1.1)
one has

$\|u\|_{L^{\infty}(\Omega_{\rho})}\leq\frac{1}{{\gamma}}\|u\|_{L^{1}(\Omega)}.$
(3.2)

Note that, as $f\geq 0$, the maximal principle shows that any solution $u\in
C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ to (1.1) with $0$ boundary is
always nonnegative; and the strong maximal principle further shows that $u$ is
always positive in the domain $\Omega$.

###### Proofs of (1.12) and (1.13) in Theorem 1.2.

Let $\beta=n-2-2\sqrt{n-1}$ and let $\rho,{\gamma}$ be as in Lemma 3.1. We
first consider the case $n\geq 11$. For any $y\in\overline{\Omega}$ and $r>0$,
write

$\displaystyle r^{\beta-n}\int_{\Omega\cap B_{r}(y)}|u|^{p_{n}}\,dx$
$\displaystyle=r^{\beta-n}\int_{\Omega_{\rho}\cap
B_{r}(y)}|u|^{p_{n}}\,dx+r^{\beta-n}\int_{(\Omega\setminus\Omega_{\rho})\cap
B_{r}(y)}|u|^{p_{n}}\,dx$ $\displaystyle:=\Phi_{1}(y,r)+\Phi_{2}(y,r).$

To see (1.12), obviously, we only need to prove $\Phi_{1}(y,r)\leq
C(n,\Omega)\|u\|_{L^{1}(\Omega)}^{p_{n}}$ and $\Phi_{2}(y,r)\leq
C(n,\rho,\Omega)\|u\|^{p_{n}}_{L^{1}(\Omega)}$ for any $y\in\Omega$ and $r>0$.

Note that

$r^{\beta-n}|\Omega_{\rho}\cap B_{r}(y)|\leq C(n)r^{\beta-n}\leq
C(n)\quad\mbox{when $r<1$ and $\leq|\Omega_{\rho}|$ when $r>1$,}$

by $2<\beta<n$ and Lemma 3.1, we have

$\displaystyle\Phi_{1}(y,r)\leq r^{\beta-n}|\Omega_{\rho}\cap
B_{r}(y)|\|u\|_{L^{\infty}(\Omega_{\rho})}^{p_{n}}\leq
C(n,\Omega)\|u\|_{L^{1}(\Omega)}^{p_{n}}.$

Next, to get $\Phi_{2}(y,r)\leq C(n,\rho,\Omega)\|u\|^{p_{n}}_{L^{1}(\Omega)}$
for any $y\in\Omega$ and $r>0$, we only need to consider
$y\in\Omega\setminus\Omega_{\rho}$ and $0<r<\rho/8$. Indeed, for
$y\in\Omega_{\rho}$, if
$r<{\mathop{\mathrm{missing}}{\,dist\,}}(y,\Omega\setminus\Omega_{\rho})$,
then $\Phi_{2}(y,r)=0$, and if $r\geq\Omega\setminus\Omega_{\rho}$, then
$\Phi_{2}(y,r)\leq C(n)\Phi_{2}(\bar{y},2r)$, where $\bar{y}$ is the closest
point in $\Omega\setminus\Omega_{\rho}$ and $B(y,r)\subset B(\bar{y},2r)$.
Moreover for any $y\in\Omega\setminus\Omega_{\rho}$ and $r\geq\rho/8$,

$\Phi_{2}(y,r)\leq\rho^{\beta-n}\int_{\Omega\setminus\Omega_{\rho}}|u|^{p_{n}}\,dx\leq\sum_{i=1}^{N}\rho^{\beta-n}\int_{\Omega\setminus\Omega_{\rho}\cap
B_{\rho/9}(x_{i})}|u|^{p_{n}}\,dx=\sum_{i=1}^{N}\Phi(x_{i},\rho/9),$

where $\\{B(x_{i},\rho/9)\\}_{i=1}^{N}$ is a cover of the compact set
$\Omega\setminus\Omega_{\rho}$,
$\\{x_{i}\\}_{i=1}^{N}\subset\Omega\setminus\Omega_{\rho}$ and $N$ depending
on $\Omega$ and $\rho$.

On the other hand, for any $y\in{\Omega\backslash\Omega_{\rho}}$ and
$0<r<\rho/8$, since $u$ is stale solution in $B_{\rho}(y)\subset\Omega$, by
(1.11) with a scaling argument we have $u\in M^{p_{n},\beta}(B_{\rho/8}(y))$
with $\|u\|_{M^{p_{n},\beta}(B_{\rho/8}(y))}\leq
C(n,\rho)\|u\|_{L^{1}(B_{\rho/2}(y))}$, in particular

$\Phi_{2}(y,r)\leq r^{\beta}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.70406pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.70406pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to12.4788pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to12.4788pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{r}(y)}|u|^{p_{n}}\,dx\leq
C(n,\rho)\|u\|^{p_{n}}_{L^{1}(\Omega)}$

as desired. This proves (1.13).

In the case $n=10$, for any $y\in\Omega$, if $r>\frac{1}{9}\rho$, we have

$r^{-n}\int_{\Omega\cap B_{r}(y)}|u|\,dx\leq C(n,\rho)\|u\|_{L^{1}(\Omega)}.$

Below we assume that $0<r<\frac{1}{9}\rho$. If
$y\in\Omega\setminus\Omega_{8\rho/9}$, we have
$\rho<\frac{9}{8}{\mathop{\mathrm{missing}}{\,dist\,}}(y,\partial\Omega)$.
Since $0<r<\frac{1}{8}{\mathop{\mathrm{missing}}{\,dist\,}}(y,\partial\Omega)$
and $u$ is a stale solution in
$B_{{\mathop{\mathrm{missing}}{\,dist\,}}(y,\partial\Omega)}(y)\subset\Omega$,
by (1.10) with a scaling we have

$\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to16.70406pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.70406pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to12.4788pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to12.4788pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{r}(y)}|u-u_{B_{r}(y)}|\,dx\leq
C(n,\rho)\|u\|_{L^{1}(B_{{\mathop{\mathrm{missing}}{\,dist\,}}(y,\partial\Omega)}(y))}\leq
C(n,\rho)\|u\|_{L^{1}(\Omega)}.$

For $y\in\Omega_{8\rho/9}$, noting
$0<r<\frac{1}{9}\rho\leq{\mathop{\mathrm{missing}}{\,dist\,}}(y,\partial\Omega_{\rho})$,
one has $\Omega\cap B_{r}(y)\subset\Omega\setminus\Omega_{\rho}$. Thus

$r^{-n}\int_{\Omega\cap B_{r}(y)}|u|\,dx=r^{-n}\int_{\Omega_{\rho}\cap
B_{r}(y)}\left|u\right|\,dx\leq C(n,\rho)\|u\|_{L^{1}(\Omega)}.$

Combining these estimates, we obtain (1.12). ∎

We finally prove Corollary 1.3.

###### Proof of Corollary 1.3.

Let $u\in W^{1,2}_{0}(\Omega)$ be a stable solution to (1.1) with $0$
boundary. By [13, Corollary 3.2.1](see also the proof in [8, Theorem 4.1] and
[14, Theorem 5]), there is a nonnegative, nondecreasing sequence $(f_{k})$ of
convex functions in $C^{1}({\mathbb{R}})$ such that $f_{k}\to f$ pointwise in
$[0,\infty)$ and a nondecreasing sequence $(u_{k})$ in
$C^{2}(\overline{\Omega})\cap W^{1,2}_{0}(\Omega)$ such that $u_{k}$ is a weak
stable solution to

$\displaystyle-\Delta u_{k}=f_{k}(u_{k})\quad{\rm in}\quad\Omega;\
u_{k}=0\quad{\rm on}\quad\partial\Omega.$ (3.3)

and

$u_{k}\to u\quad{\rm in}\quad W^{1,2}(\Omega)\quad{\rm as}\quad k\to+\infty.$

If $n=10$, applying (1.12) to $u_{k}$ one has

$\displaystyle\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to30.87062pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to30.87062pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to23.8676pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to23.8676pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{\Omega\cap
B_{r}(y)}\left|u_{k}(x)-\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to30.87062pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to30.87062pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to23.8676pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to23.8676pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{\Omega\cap
B_{r}(y)}u_{k}\,dz\right|\,dx\leq\|u_{k}\|_{{\rm BMO}(\Omega)}\leq
C(n,\Omega)\int_{\Omega}|u_{k}|\,dx,\quad\forall r>0,\ \forall
y\in\overline{\Omega}.$

By $u_{k}\to u$ in $W^{1,2}(\Omega)$ as $k\to+\infty$ we conclude that
$\|u\|_{BMO(\Omega)}\leq C(n)\|u\|_{L^{1}(\Omega)}$ as desired.

If $n\geq 11$, applying (1.13) to $u_{k}$ we have

$\displaystyle r^{\beta-n}\int_{\Omega\cap B_{r}(y)}|u_{k}|^{p_{n}}\,dx\leq
C(n,\Omega,\rho)(\|u_{k}\|_{L^{1}(\Omega)})^{p_{n}},\quad\forall
y\in\overline{\Omega},\quad\forall r>0,$ (3.4)

where $\beta=\frac{2p_{n}}{p_{n}-2}\in(0,n)$. By $u_{k}\to u$ in
$W^{1,2}(\Omega)$ as $k\to+\infty$ we deduce that $u_{k}\in L^{p_{n}}(\Omega)$
uniformly in $k\geq 0$, and hence, $u_{k}\to u$ weakly in $L^{p_{n}}(\Omega)$.
Thus, letting $k\to+\infty$ in (3.4) we conclude
$\|u\|_{M^{p_{n},\beta}(\Omega)}\leq C(n)\|u\|_{L^{1}(\Omega)}$ as desired. ∎

##  4 Proof of Proposition 1.9

Let $0<r<R<\infty$. Let $\eta\in C^{\infty}_{c}(A_{\frac{r}{4},4R})$ satisfy
that

$\displaystyle 0\leq\eta\leq 1\quad\mbox{in}\ A_{\frac{r}{4},4R}\
\quad\mbox{and}\quad\eta=1\ \mbox{in}\ A_{\frac{r}{2},2R},$ (4.1)
$\displaystyle|D\eta|^{2}+|D^{2}\eta|\leq\frac{C}{r^{2}}\ \mbox{ in }\
A_{\frac{r}{4},\frac{r}{2}}\ \mbox{and}\
|D\eta|^{2}+|D^{2}\eta|\leq\frac{C}{R^{2}}\ \mbox{in $A_{2R,4R}$,}$ (4.2)

where $C>0$ is a universal constant.

Let $u_{\delta}=u\ast\phi_{\delta}$ for $\delta>0$, where $\phi_{\delta}$ is
the standard smooth mollifier and supported in $B(0,\delta)$. Recall that
$u\in W^{1,\,1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\mathbb{R}^{n})$ and
$u_{\delta}\to u$ in
$W^{1,\,1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\mathbb{R}^{n})$. Since
$-\Delta u\geq 0$ is a locally finite measure, we have $-\Delta
u_{\delta}=(-\Delta u)\ast\phi_{\delta}\geq 0$ everywhere. By
$u_{\delta}\eta\in C^{\infty}_{c}({{{\mathbb{R}}}^{n}})$, one has

$u_{\delta}\eta(x)=\Delta^{-1}[\Delta(u_{\delta}\eta)](x)=c(n)\int_{{{{\mathbb{R}}}^{n}}}\frac{1}{|x-y|^{n-2}}\Delta(u_{\delta}\eta)(y)\,dy\quad\forall
x\in{{{\mathbb{R}}}^{n}}$

and hence

$D(u_{\delta}\eta)(x)=D\Delta^{-1}[\Delta(u_{\delta}\eta)](x)=c(n)(2-n)\int_{{{{\mathbb{R}}}^{n}}}\frac{x-y}{|x-y|^{n}}\Delta(u_{\delta}\eta)(y)\,dy\quad\forall
x\in{{{\mathbb{R}}}^{n}}.$

Noting

$\Delta(u_{\delta}\eta)(y)=\Delta
u_{\delta}(y)\eta(y)+u_{\delta}(y)\eta(y)+Du_{\delta}(y)\cdot D\eta(y),$

for $0<\delta<<r/8$ we write

$\displaystyle\int_{A_{r,R}}{|Du_{\delta}|}{|x|^{-n+1}}\,dx=$
$\displaystyle\int_{A(r,\,R)}|D(u_{\delta}\eta)|{|x|^{-n+1}}\,dx$
$\displaystyle=$
$\displaystyle\int_{A_{r,R}}\left|\int_{\mathbb{R}^{n}}\frac{x-y}{|x-y|^{n}}\Delta(u_{\delta}\eta)(y)\,dy\right||x|^{-n+1}\,dx$
$\displaystyle\leq$ $\displaystyle
C(n)\int_{A_{r,R}}\left|\int_{\mathbb{R}^{n}}\frac{x-y}{|x-y|^{n}}\Delta
u_{\delta}(y)\eta(y)\,dy\right|{|x|^{-n+1}}\,dx$
$\displaystyle+C(n)\int_{A_{r,R}}\left|\int_{\mathbb{R}^{n}}\frac{x-y}{|x-y|^{n}}u_{\delta}(y)\Delta\eta(y)\,dy\right|{|x|^{-n+1}}\,dx$
$\displaystyle+C(n)\int_{A_{r,R}}\left|\int_{\mathbb{R}^{n}}\frac{x-y}{|x-y|^{n}}Du_{\delta}(y)\cdot
D\eta(y)\,dy\right|{|x|^{-n+1}}\,dx$ $\displaystyle=:$ $\displaystyle
I_{1}+I_{2}+I_{3}.$

In order to control $I_{1}$ from above, first by $-\Delta u_{\delta}\geq 0$
and (4.1) one has

$I_{1}\leq\int_{{{{\mathbb{R}}}^{n}}}\int_{{{{\mathbb{R}}}^{n}}}|x-y|^{-n+1}|x|^{-n+1}\,dx(-\Delta
u_{\delta})(y)\eta(y)\,dy.$

Employing the triangle inequality, for $y\in{{{\mathbb{R}}}^{n}}$, we further
get

$\displaystyle\int_{{{{\mathbb{R}}}^{n}}}|x-y|^{-n+1}|x|^{-n+1}\,dx$
$\displaystyle\leq
2^{n-1}\int_{\\{|x|>2|y|\\}}|x|^{-2n+2}\,dx+2^{n-1}\int_{\\{|x|<\frac{1}{2}|y|\\}}|x|^{-n+1}|y|^{-n+1}\,dx$
$\displaystyle\quad+\int_{\\{\frac{1}{2}|y|\leq|x|\leq
2|y|\\}}|x-y|^{-n+1}|y|^{-n+1}\,dx$ $\displaystyle\leq
C(n)|y|^{-n+2}+C(n)|y|^{-n+2}+\int_{\\{|y-x|\leq
3|y|\\}}|x-y|^{-n+1}|y|^{-n+1}\,dx$ $\displaystyle\leq C(n)|y|^{-n+2}.$ (4.3)

This together with $-\Delta u_{\delta}\geq 0$ again gives

$I_{1}\leq C(n)\int_{{{{\mathbb{R}}}^{n}}}(-\Delta
u_{\delta})|y|^{-n+2}\eta(y)\,dy.$

Via integration by parts and using $\eta\in
C^{\infty}_{c}(A_{\frac{r}{4},4R})$, we have

$\int_{{{{\mathbb{R}}}^{n}}}(-\Delta
u_{\delta})|y|^{-n+2}\eta(y)\,dy=\int_{A_{\frac{r}{4},4R}}u_{\delta}[-\Delta|y|^{-n+2}\eta(y)+D|y|^{-n+2}\cdot
D\eta(y)-|y|^{-n+2}\Delta\eta(y)]\,dy.$

Observing that $\Delta|y|^{n-2}=0$ in $A_{\frac{r}{4},4R}$, and using
(4.1)&(4.2) we arrive at

$\displaystyle I_{1}$ $\displaystyle\leq
C(n)\int_{A_{\frac{r}{4},4R}}u_{\delta}(y)[(2-n)|y|^{-n}y\cdot
D\eta(y)-|y|^{-n+2}\Delta\eta(y)]\,dy$ $\displaystyle\leq
C(n)\int_{A_{\frac{r}{4},4R}}|u_{\delta}(y)|[r^{-n}\chi_{A_{\frac{r}{4},\frac{r}{2}}}+R^{-n}\chi_{A_{2R,4R}}]\,dy$
$\displaystyle\leq C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to10.36111pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to10.36111pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{\frac{r}{4},\frac{r}{2}}}|u_{\delta}|\,dz+\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.49722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.49722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{2R,4R}}|u_{\delta}|\,dz.$

For $I_{2}$, by (4) and (4.1)

$\displaystyle I_{2}$
$\displaystyle\leq\int_{{{{\mathbb{R}}}^{n}}}\int_{A(r,\,R)}|x-y|^{-n+1}|x|^{-n+1}\,dx|u_{\delta}|(y)|\Delta\eta(y)|\,dy$
$\displaystyle\leq
C(n)\int_{{{{\mathbb{R}}}^{n}}}|y|^{-n+2}|u_{\delta}|(y)|\Delta\eta(y)|\,dy$
$\displaystyle\leq
C(n)\int_{{{{\mathbb{R}}}^{n}}}|u_{\delta}(y)|[r^{-n}\chi_{A_{\frac{r}{4},\frac{r}{2}}}+R^{-n}\chi_{A_{2R,4R}}]\,dy$
$\displaystyle\leq C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to10.36111pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to10.36111pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{\frac{r}{4},\frac{r}{2}}}|u_{\delta}|\,dz+C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.49722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.49722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{2R,4R}}|u_{\delta}|\,dz.$

Now let us estimate $I_{3}$. First via integration by parts one gets

$\displaystyle\int_{\mathbb{R}^{n}}|x-y|^{-n}(x-y)Du_{\delta}(y)\cdot
D_{\delta}\eta(y)\,dy$
$\displaystyle=\int_{\mathbb{R}^{n}}|x-y|^{-n}(x-y)u_{\delta}(y)\Delta\eta(y)\,dy+\int_{\mathbb{R}^{n}}u_{\delta}(y)D[|x-y|^{-n}(x-y)]D\eta(y)\,dy.$

Since

$|D[|x-y|^{-n}(x-y)]|\leq C(n)|x-y|^{-n},$

we obtain

$\displaystyle\left|\int_{\mathbb{R}^{n}}|x-y|^{-n}(x-y)Du_{\delta}(y)\cdot
D\eta(y)\,dy\right|$ $\displaystyle\leq
C(n)\left|\int_{\mathbb{R}^{n}}|x-y|^{-n+1}u_{\delta}(y)\Delta\eta(y)\,dy\right|+C(n)\int_{\mathbb{R}^{n}}|x-y|^{-n}|u_{\delta}(y)||D\eta(y)|\,dy.$

As a consequence,

$\displaystyle I_{3}\leq$ $\displaystyle
C(n)I_{2}+C(n)\int_{\mathbb{R}^{n}}\int_{A_{r,R}}|x-y|^{-n}|x|^{-n+1}\,dx|u_{\delta}(y)||D\eta(y)|\,dy=:C(n)I_{2}+C(n)\widetilde{I_{3}}.$

In order to estimate $\widetilde{I_{3}}$, first we note that (4.1) gives

$\widetilde{I_{3}}\leq
C(n)\int_{{{{\mathbb{R}}}^{n}}}\int_{A_{r,R}}|x-y|^{-n}|x|^{-n+1}\,dx|u_{\delta}(y)|[r^{-1}\chi_{A_{\frac{r}{4},\frac{r}{2}}}+R^{-1}\chi_{A_{2R,4R}}]\,dy$

For any $x\in A_{r,R}$, if $y\in A_{\frac{r}{4},\frac{r}{2}}$ we have
$|x-y|\geq|x|/2$ and hence

$\int_{A_{r,R}}|x-y|^{-n}|x|^{-n+1}\,dx\leq
C(n)\int_{A_{r,R}}|x|^{-2n+1}\,dx\leq C(n)r^{-n+1};$

if $y\in A_{2R,4R}$, then $|x-y|\geq R$ and hence

$\int_{A_{r,R}}|x-y|^{-n}|x|^{-n+1}\,dx\leq
C(n)R^{-n}\int_{A_{r,R}}|x|^{-n+1}\,dx\leq C(n)R^{-n+1}.$

Thus it follows that

$\widetilde{I_{3}}\leq
C(n)\int_{{{{\mathbb{R}}}^{n}}}|u_{\delta}(y)|[r^{-n}\chi_{A_{\frac{r}{4},\frac{r}{2}}}+R^{-n}\chi_{A_{2R,4R}}]\,dy\leq
C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to10.36111pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to10.36111pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{\frac{r}{4},\frac{r}{2}}}|u_{\delta}|\,dz+C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.49722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.49722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{2R,4R}}|u_{\delta}|\,dz.$

To conclude,

$\int_{A_{r,R}}|Du_{\delta}||x|^{-n+1}\,dx\leq
C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to10.36111pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to10.36111pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{\frac{r}{4},\frac{r}{2}}}|u_{\delta}|\,dz+C(n)\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.49722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.49722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{2R,4R}}|u_{\delta}|\,dz.$

By letting $\delta\to 0$ and noting $u_{\delta}\to u$ in
$W^{1,\,1}_{\mathop{\mathrm{missing}}{\,loc\,}}$, we conclude (1.21).

##  5 Proof of Theorem 1.5

Since $u$ satisfies (1.16), then $u$ does not satisfy (1.15). We only need to
show that if $u$ is nonconstant, then (1.15) holds. Equivalently, it suffices
to show that if $u$ does not satisfy (1.15), then $u$ is a constant. Namely,
there exists a sequence $\\{R_{j}\\}_{j\in{\mathbb{N}}}$ towards $\infty$ so
that

$\displaystyle\frac{1}{\log R_{j}}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to17.49911pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to17.49911pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.99911pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.99911pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{R_{j},4R_{j}}}|u(z)|\,dz\to
0\quad\mbox{ as $j\to\infty$\quad when $n=10$ }$ (5.1)

and

$\displaystyle
R_{j}^{\frac{n}{2}-2-\sqrt{n-1}}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to17.49911pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to17.49911pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.99911pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.99911pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{R_{j},4R_{j}}}|u(x)|\,dx\to
0\quad\mbox{as $j\to\infty$\quad when $n\geq 11$.}$ (5.2)

On the other hand, given any $0<r<\infty$, applying (1.18) for any $R>4r$ we
have

$\displaystyle
r^{-(1+\sqrt{n-1})}\left(\int_{B_{r}}|Du|^{2}\,dx\right)^{1/2}\leq
C(n)R^{-(1+\sqrt{n-1})}\left(\int_{A_{R,2R}}|Du|^{2}\,dx\right)^{1/2}.$

Observe that the annual $A_{1,2}$ can be covered by
$\\{B_{\frac{1}{8}}(y_{i})\\}_{i=1}^{N}$ with $y_{1},\cdots,y_{N}\in A_{1,2}$
and $N\leq C(n)$.

$\chi_{A_{1,2}}\leq\sum_{i=1}^{N}\chi_{B_{\frac{1}{8}}(y_{i})}\leq\sum_{i=1}^{N}\chi_{B_{\frac{1}{4}}(y_{i})}\leq
C(n)\chi_{A_{\frac{3}{4},3}}.$

Below we consider the case $n\geq 11$ and the case $n=10$ separately.

Case $n\geq 11$. For each $i$, applying (1.19) and (1.20), one attains

$\left(\int_{B_{\frac{R}{8}}(Ry_{i})}|Du|^{2}\,dx\right)^{1/2}\leq
C(n)R^{-\frac{n+2}{2}}\int_{B_{\frac{R}{4}}(Ry_{i})}|u|\,dx\leq
C(n)R^{\frac{n-2}{2}}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.49722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.49722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{\frac{3R}{4},3R}}|u|\,dx.$

Thus by summing over all these balls,

$\displaystyle\int_{A_{R,2R}}|Du|^{2}\,dx$ $\displaystyle\leq
C(n)R^{n-2}\left(\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.49722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.49722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{\frac{3R}{4},3R}}|u|\,dx\right)^{2},$

and we eventually obtain

$\displaystyle r^{-(1+\sqrt{n-1})}\left(\int_{B_{r}}|Du|^{2}\,dx\right)^{1/2}$
$\displaystyle\leq
C(n)R^{\frac{n}{2}-2-\sqrt{n-1}}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to16.49722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to16.49722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.99722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{\frac{3R}{4},3R}}|u|\,dx.$

Taking $R=\frac{4R_{j}}{3}$, applying (5.2) and letting $j\to\infty$, one
concludes

$\int_{B_{r}}|Du|^{2}\,dx=0.$

By the arbitrariness of $r>0$, we obtain
$\|Du\|_{L^{2}({{{\mathbb{R}}}^{n}})}=0$, which implies that $u$ is a
constant.

Case $n=10$. For each $i$, applying (1.19), one attains

$\left(\int_{B_{\frac{R}{8}}(Ry_{i})}|Du|^{2}\,dx\right)^{1/2}\leq
C(n)R^{-\frac{n}{2}}\int_{B_{\frac{R}{4}}(Ry_{i})}|Du|\,dx\leq
C(n)R^{\frac{n-2}{2}}\int_{A_{\frac{R}{2},4R}}|Du||x|^{-n+1}\,dx.$

Thus

$\displaystyle\int_{A_{R,2R}}|Du|^{2}\,dx$ $\displaystyle\leq
C(n)R^{n-2}\left(\int_{A_{\frac{R}{2},4R}}|Du||x|^{-n+1}\,dx\right)^{2}.$
(5.3)

We therefore obtain

$\displaystyle r^{-(1+\sqrt{n-1})}\left(\int_{B_{r}}|Du|^{2}\,dx\right)^{1/2}$
$\displaystyle\leq
C(n)R^{-\frac{n}{2}-2-\sqrt{n-1}}\int_{A_{\frac{R}{2},4R}}|Du||x|^{-n+1}\,dx$
$\displaystyle=C(n)\int_{A_{\frac{R}{2},4R}}|Du||x|^{-n+1}\,dx,$

where in the last identity we use $\frac{n}{2}-2-\sqrt{n-1}=5-2-3=0$.

For $R>2^{5}+r>4$, let $m$ be the largest integer so that $m\leq\log_{2}R-3$.
Applying the (5.3) to $2^{j}R$ with $j=1,\cdots m$, one has

$\displaystyle r^{-(1+\sqrt{n-1})}\left(\int_{B_{r}}|Du|^{2}\,dx\right)^{1/2}$
$\displaystyle\leq
C(n)\frac{1}{m}\sum_{j=1}^{m}\int_{A_{\frac{2^{j}R}{2},4(2^{j}R)}}|Du||x|^{-n+1}\,dx$
$\displaystyle\leq C(n)\frac{1}{m}\int_{A_{R,2^{m+2}R}}|Du||x|^{-n+1}\,dx$
$\displaystyle\leq C(n)\frac{1}{\log
R}\int_{A_{4,\frac{R^{2}}{2}}}|Du||x|^{-n+1}\,dx.$

By (1.21), one has

$r^{-(1+\sqrt{n-1})}\left(\int_{B_{r}}|Du|^{2}\,dx\right)^{1/2}\leq
C(n)\frac{1}{\log R}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to10.36111pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to10.36111pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to8.86111pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{1,2}}|u(z)|\,dz+C(n)\frac{1}{\log
R^{2}}\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to17.69722pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to17.69722pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to16.19722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to16.19722pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{A_{{R^{2}},2R^{2}}}|u(z)|\,dz.$

Taking $R=\sqrt{R_{j}}$ and letting $j\to\infty$, by (5.1) one concludes

$\int_{B_{r}}|Du|^{2}\,dx=0.$

Then the arbitrariness of $r>0$ implies
$\|Du\|_{L^{2}({{{\mathbb{R}}}^{n}})}=0$, which further implies that $u$ is a
constant.

## Appendix A radial stable solution when $n=10$

Suppose $n=10$ in this appendix. Villegas [24] proved that
$\frac{1}{2}\log(1+|x|^{2})$ is a stable solution to equation $-\Delta
u=-(n-2)e^{-2u}-2e^{-4u}$ in ${\mathbb{R}}^{n}.$ Note that
$-(n-2)e^{-2s}-2e^{-4s}\leq 0$ in ${\mathbb{R}}$.

Below, we show that $u=-\frac{1}{2}\log(1+|x|^{2})$ is a stable solution to
the equation

$-\Delta u=f(u)\quad{\rm in}\quad{\mathbb{R}}^{n},$

where $f(s)=(n-2)e^{2s}+2e^{4s}\geq 0$ in ${\mathbb{R}}$.

First we show that $u$ is a solution. Indeed, for any
$x\in{{{\mathbb{R}}}^{n}}$ a direct calculation gives

$\displaystyle-\Delta u(x)$
$\displaystyle=\left((1+|x|^{2})^{-1}x_{i}\right)_{x_{i}}=\frac{n}{1+|x|^{2}}+2\frac{|x|^{2}}{(1+|x|^{2})^{2}}=(n-2)\frac{1}{1+|x|^{2}}+2\frac{1}{(1+|x|^{2})^{2}}.$

By $e^{2u(x)}=(1+|x|^{2})^{-1}$, we have

$\displaystyle-\Delta u(x)$ $\displaystyle=(n-2)e^{2u(x)}+2e^{4u(x)}=f(u(x)).$

Next, we show that $u$ is stable. Note that
$f^{\prime}(s)=2(n-2)e^{2s}+8e^{4s}$ for $s\in{\mathbb{R}}$. Given any $x\neq
0$, writing $r=|x|$ and noting $e^{2u(x)}=(1+|x|^{2})^{-1}$, we have

$\displaystyle
f^{\prime}(u(x))=2(n-2)e^{2u(x)}+8e^{4u(x)}=\frac{2(n-2)}{1+r^{2}}+\frac{8}{(1+r^{2})^{2}}$

By $n=10$ we have

$\displaystyle f^{\prime}(u(x))$
$\displaystyle=\frac{16r^{2}(1+r^{2})+8r^{2}}{r^{2}(1+r^{2})^{2}}=\frac{16r^{4}+24r^{2}}{r^{2}(1+r^{2})^{2}}<\frac{16(1+r^{2})^{2}}{{r^{2}(1+r^{2})^{2}}}=\frac{(n-2)^{2}}{4|x|^{2}}.$

By this, and the Hardy inequality, we have

$\int_{{\mathbb{R}}^{n}}f^{\prime}(u)\xi^{2}\,dx\leq\frac{(n-2)^{2}}{4}\int_{{\mathbb{R}}^{n}}\frac{\xi^{2}}{|x|^{2}}\,dx\leq\int_{{\mathbb{R}}^{n}}|D\xi|^{2}\,dx\quad\forall\xi\in
C^{\infty}_{c}({\mathbb{R}}^{n}).$

Thus $u$ is a stable solution to $-\Delta u=f(u)$ in ${{{\mathbb{R}}}^{n}}$.

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Fa Peng

School of Mathematical Science, Beihang University, Changping District Shahe
Higher Education Park South Third Street No. 9, Beijing 102206, P. R. China

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Yi Ru-Ya Zhang

Academy of Mathematics and Systems Science, the Chinese Academy of Sciences,
Beijing 100190, P. R. China

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Yuan Zhou

School of Mathematical Science, Beijing Normal University, Haidian District
Xinjiekou Waidajie No.19, Beijing 10875, P. R. China

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