#  On semi-linear elliptic equation arising from Micro-Electromechanical
Systems with contacting elastic membrane

Huyuan Chen

Institute of Mathematical Sciences, New York University Shanghai,

Shanghai, PR China

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Ying Wang

Departamento de Ingeniería Matemática, Universidad de Chile,

Santiago, Chile

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Feng Zhou

Department of Mathematics, East China Normal University,

Shanghai, PR China

Email<EMAIL_ADDRESS>

> Abstract. This paper is concerned with the nonlinear elliptic problem
> $-\Delta u=\frac{\lambda}{(a-u)^{2}}$ on a bounded domain $\Omega$ of
> $\mathbb{R}^{N}$ with Dirichlet boundary conditions. This problem arises
> from Micro-Electromechanical Systems devices in the case that the elastic
> membrane contacts the ground plate on the boundary. We analyze the
> properties of minimal solutions to this equation when $\lambda>0$ and the
> function $a:\bar{\Omega}\to[0,1]$ satisfying $a(x)\geq\kappa{\rm
> dist}(x,\partial\Omega)^{\gamma}$ for some $\kappa>0$ and $\gamma\in(0,1)$.
> Our results show how the boundary decay of the membrane works on the
> solutions and pull-in voltage $\lambda$.

††AMS Subject Classifications: 36J08, 35B50, 35J15.††Key words: MEMS; Pull-in
voltage; Stable; Minimal solutions.

##  1\. Introduction

Micro-Electromechanical Systems (MEMS) are often used to combine electronics
with micro-size mechanical devices in the design of various types of
microscopic machinery. They are successfully utilized in components of many
commercial systems, including accelerometers for airbag deployment in
automobiles, ink jet printer heads, optical switches, chemical sensors, etc.
In MEMS devices, a key component is called the electrostatic actuation, which
is based on an electrostatic-controlled tunable, it is a simple idealized
electrostatic device. The upper part of this electrostatic device consists of
a thin and deformable elastic membrane that is fixed along its boundary and
which lies above a rigid grounded plate. This elastic membrane is modeled as a
dielectric with a small but finite thickness. The upper surface of the
membrane is coated with a negligibly thin metallic conducting film. When a
voltage $\lambda$ is applied to the conducting film, the thin dielectric
membrane deflects towards the bottom plate, and when $\lambda$ is increased
beyond a certain critical value $\lambda^{*}-$known as pull-in voltage$-$the
steady state of the elastic membrane is lost, and proceeds to touchdown or
snap through at a finite time creating the so-called pull-in instability.

A mathematical model of the physical phenomena, leading to a partial
differential equation for the dimensionless deflection of the membrane, was
derived and analyzed in [4, 5, 6, 7, 8, 12, 15] and reference therein. In the
damping-dominated limit, and using a narrow-gap asymptotic analysis, the
dimensionless deflection $u$ of the membrane on a bounded domain $\Omega$ in
$\mathbb{R}^{2}$ is found to satisfy the equation

$-\Delta u=\frac{\lambda}{(1-u)^{2}}\qquad{\rm in}\quad\Omega$ (1.1)

with the Dirichlet boundary condition. Here the term on the right hand side of
equation (1.1) is the Coulomb force. Later on, Ghoussoub and Guo in [4, 5]
studied the nonlinear elliptic problem

$-\Delta u=\frac{\lambda f(x)}{(1-u)^{2}}\qquad{\rm in}\quad\Omega$ (1.2)

with the Dirichlet boundary condition, which models a simple electrostatic
MEMS device consisting of a thin dielectric elastic membrane with boundary
supported at $0$ above a rigid ground plate located at $1$. Here $\Omega$ is a
bounded domain of $\mathbb{R}^{N}$ and the function $f\geq 0$ represents the
permittivity profile and $\lambda>0$ is a constant which is increasing with
respect to the applied voltage. We know that for any given suitable $f$, there
exists a critical value $\lambda^{*}$ (pull-in voltage) such that if
$\lambda\in(0,\lambda^{*})$, problem (1.2) is solvable, while for
$\lambda>\lambda^{*}$, no solution for (1.2) exists.

In an effort to achieve better MEMS design, the membrane can be
technologically fabricated into non-flat shape like the surface of a semi-
ball, which contacts the ground plate along the boundary. In this paper, we
study how the shape of membranes effects on the pull-in voltage. In what
follows, we denote that $\Omega$ is a $C^{2}$ bounded domain in
$\mathbb{R}^{N}$ with $N\geq 1$, $\rho(x)={\rm dist}(x,\partial\Omega)$ for
$x\in\Omega$, the function $a:\bar{\Omega}\to[0,1]$ is in the class of
$C^{\gamma}(\Omega)\cap C(\bar{\Omega})$ and satisfies

$a(x)\geq\kappa\rho(x)^{\gamma},\quad\forall\ x\in\Omega$ (1.3)

for some $\kappa>0$ and $\gamma\in(0,1)$. Our purpose of this paper is to
consider the minimal solutions to elliptic equation

$\left\\{\begin{array}[]{lll}-\Delta u=\frac{\lambda}{(a-u)^{2}}&{\rm
in}\quad\ \Omega,\\\\[5.69054pt] \phantom{-}0<u<a&{\rm in}\quad\
\Omega,\\\\[5.69054pt] \phantom{-\Delta}u=0&{\rm
on}\quad\partial\Omega,\end{array}\right.$ (1.4)

where parameter $\lambda>0$ characterizes the relative strength of the
electrostatic and mechanical forces in the system. Equation (1.4) models a
MEMS device that the static deformation of the surface of membrane when it is
applied voltage $\lambda$, where $a$ is initially undeflected state of the
elastic membrane that contacts the ground plate on the boundary. For this
equation, we have the following existence results.

######  Theorem 1.1.

Assume that $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$ satisfies (1.3) with
$\gamma\in(0,\frac{2}{3}]$ and $\kappa>0$, then there exists a finite pull-in
voltage $\lambda^{*}:=\lambda^{*}(\kappa,\gamma)>0$ such that

$(i)$ for $\lambda\in(0,\lambda^{*})$, problem (1.4) admits a minimal solution
$u_{\lambda}$ and the mapping: $\lambda\mapsto u_{\lambda}$ is strictly
increasing;

$(ii)$ for $\lambda>\lambda^{*}$, there is no solution for (1.4);

$(iii)$ assume more that there exists $c_{0}\geq\kappa$ such that

$a(x)\leq c_{0}\rho(x)^{\gamma},\quad x\in\Omega,$ (1.5)

then there exists $\lambda_{*}:=\lambda_{*}(\kappa,\gamma)\in(0,\lambda^{*}]$
such that for $\lambda\in(0,\lambda_{*})$, $u_{\lambda}\in H^{1}_{0}(\Omega)$
and

${\rm
for}\quad\gamma\not=\frac{1}{2},\qquad\frac{\lambda}{c_{1}}\rho(x)^{\min\\{1,2-2\gamma\\}}\leq
u_{\lambda}(x)\leq c_{1}\lambda\rho(x)^{\min\\{1,2-2\gamma\\}},\qquad\forall
x\in\Omega;$ ${\rm
for}\quad\gamma=\frac{1}{2},\qquad\frac{\lambda}{c_{1}}\rho(x)\ln\frac{1}{\rho(x)}\leq
u_{\lambda}(x)\leq c_{1}\lambda\rho(x)\ln\frac{1}{\rho(x)},\qquad\forall x\in
A_{\frac{1}{2}},$

where $c_{1}\geq 1$ and $A_{\frac{1}{2}}=\\{x\in\Omega:\
\rho(x)<\frac{1}{2}\\}$.

In particular, assume more that $\Omega=B_{1}(0)$ and

$a(x)=\kappa(1-|x|^{2})^{\gamma},\qquad\forall x\in B_{1}(0),$

then the mappings: $\gamma\mapsto\lambda_{*}(\kappa,\gamma)$,
$\gamma\mapsto\lambda^{*}(\kappa,\gamma)$ are decreasing and

$(2/3)^{5}\kappa^{3}\leq\lambda_{*}(\kappa,2/3)\leq\lambda_{*}(\kappa,\gamma)\leq\lambda^{*}(\kappa,\gamma)\leq\lambda^{*}(\kappa,0)\leq(4/3)^{2}N\kappa^{3}.$

We remark that the membrane contacts the ground plate on the boundary with
decay rate $\rho^{\gamma}$, $\gamma\in(0,\frac{2}{3}]$, there still has a
positive finite pull-in voltage $\lambda^{*}$, but the decay of $a$ plays an
important role in decay of minimal solution, the regularity of minimal
solution and the estimate of $\lambda^{*}$. Theorem 1.1 shows that the
membrane of the MEMS device could be designed as the surface of the unit semi-
ball, that is,

$\Omega=B_{1}(0)\quad{\rm and}\quad a(x)=(1-|x|^{2})^{\frac{1}{2}},$

which is equivalent to the case that $a(x)=\rho(x)^{\frac{1}{2}}$, so there
exists a positive finite pull-in voltage $\lambda^{*}$. However, the decay
rate of function $a$ determined completely nonexistence of pull-in voltage
when $\gamma>\frac{2}{3}$. Precisely, we have the following non-existence
result.

######  Theorem 1.2.

Assume that $a\in C(\bar{\Omega})$ is positive and satisfies (1.5) with
$\gamma\in(\frac{2}{3},1)$ and $c_{0}>0$. Then problem (1.4) admits no
nonnegative solution for any $\lambda>0.$

We notice that for $\gamma\leq\frac{2}{3}$ and fixed $\kappa$, the finite
pull-in voltage $\lambda^{*}$ depends on $\gamma$, however, when
$\gamma=\frac{2}{3}$, $\lambda^{*}>0$ and $\lambda^{*}=0$ for
$\gamma>\frac{2}{3}$. Therefore, there is a gap at $\gamma=\frac{2}{3}$. From
Theorem 1.2, we learn that the membrane of the MEMS device should be made
steeply enough near the boundary, otherwise the device does not work.

As normal, it is challenging to study the extremal solution, i.e., the
solution of (1.4) when $\lambda=\lambda^{*}$. Especially, the decay of
function $a$ makes this issue more subtle. From Theorem 1.1, we observe that
the mapping $\lambda\mapsto u_{\lambda}$ is increasing and uniformly bounded
by function $a$, then it is well-defined that

$u_{\lambda^{*}}:=\lim_{\lambda\to\lambda^{*}}u_{\lambda}\quad{\rm in}\ \
\bar{\Omega},$ (1.6)

where $u_{\lambda}$ is the minimal solution of (1.4) with
$\lambda\in(0,\lambda^{*})$. Our final purpose in this paper is to prove that
$u_{\lambda^{*}}$ is a solution of (1.4) in some weak sense and it is called
the extremal solution. The extremal solution always is found in a weak sense
and then it could be improved the regularity until to the classical sense when
the dimension $N$ is suitable.

######  Definition 1.1.

A function $u$ is a weak solution of (1.4) if $0\leq u\leq a$ and

$\int_{\Omega}u(-\Delta\xi)dx=\int_{\Omega}\frac{\lambda\xi}{(a-u)^{2}}dx,\quad\forall\xi\in
C_{c}^{2}(\Omega),$

where $C_{c}^{2}(\Omega)$ is the space of all $C^{2}$ functions with compact
support in $\Omega$.

A solution (or weak solution) $u$ of (1.4) is stable (resp. semi-stable) if

$\int_{\Omega}|\nabla\varphi|^{2}dx>\int_{\Omega}\frac{2\lambda\varphi^{2}}{(a-u)^{3}}dx,\quad({\rm
resp.}\ \geq)\quad\forall\varphi\in H^{1}_{0}(\Omega)\setminus\\{0\\}.$

We make use of the functions’ space $C^{2}_{c}(\Omega)$ in Definition 1.1
replacing $C_{0}^{2}(\Omega):=C^{2}(\Omega)\cap C_{0}(\Omega)$, used in the
case of $a\equiv 1$, to avoid the singularity at the boundary of
$\frac{1}{(a-u_{\lambda})^{2}}$.

######  Theorem 1.3.

Assume that $\lambda\in(0,\lambda^{*})$, the function $a\in
C^{\gamma}(\Omega)\cap C(\bar{\Omega})$ satisfies (1.3) and(1.5) with
$c_{0}\geq\kappa>0$, $\gamma\in(0,\frac{2}{3}]$, $u_{\lambda}$ is the minimal
solution of (1.4) and $u_{\lambda^{*}}$ is given by (1.6). Then

$(i)$ $u_{\lambda^{*}}$ is a weak solution of (1.4) and $u_{\lambda^{*}}\in
W^{1,\frac{N}{N-\beta}}_{0}(\Omega)$ for any $\beta\in(0,\gamma)$.

$(ii)$ $u_{\lambda}$ is a stable solution of (1.4) with
$\lambda\in(0,\lambda_{*})$.

$(iii)$ if $\gamma=\frac{2}{3}$, we have that $\lambda^{*}=\lambda_{*}$,
$u_{\lambda^{*}}$ is a semi-stable weak solution of (1.4).

Assume more that $1\leq N\leq 7$, $\Omega=B_{1}(0)$ and
$a(x)=\kappa(1-|x|^{2})^{\frac{2}{3}}$, $u_{\lambda^{*}}$ is a classical
solution of (1.4).

We note that $(i)$ the extremal solution $u_{\lambda^{*}}$ is a weak solution
but in a weaker sense comparing with the case $a\equiv 1$; $(ii)$ the main
difficulty to study the stability of $u_{\lambda}$ with
$\lambda\in(0,\lambda_{*})$ arises from the decay on the boundary of $a$ and
to overcome this difficult, we make use of the generalized Hardy’s inequality;
$(iii)$ in the case of $\gamma=\frac{2}{3}$, the regularity of the extremal
solution could be improved into classical sense for $1\leq N\leq 7$, the same
range of the dimension with the case $a\equiv 1$.

The paper is organized as follows. Section 2 is devoted to do the estimate for
$\mathbb{G}_{\Omega}[\rho^{\tau-2}]$ when $\tau\in(0,2)$ and then we prove
that there exists pull-in voltage $\lambda^{*}$ such that problem (1.4) admits
a minimal solution when $\lambda\in(0,\lambda^{*})$. We also analyze the
boundary decay of the minimal solution. In Section 3, we do the estimate for
$\lambda_{*}$ and $\lambda^{*}$ when $\Omega=B_{1}(0)$ and
$a(x)=\kappa(1-|x|)^{\gamma}$. In Section 4, we study the properties of
$u_{\lambda^{*}}$, including its regularity and stability.

##  2\. Existence

Denote by $G_{\Omega}$ the Green kernel of $-\Delta$ in $\Omega\times\Omega$
and by $\mathbb{G}_{\Omega}[\cdot]$ the Green operator defined as

$\mathbb{G}_{\Omega}[f](x)=\int_{\Omega}G_{\Omega}(x,y)f(y)dy,\quad\forall
f\in L^{1}(\Omega,\rho^{-1}dx).$

To do the existence of a minimal solution of problem (1.4), the following
estimates play an important role.

######  Lemma 2.1.

Let $\tau\in(0,2)$, $A_{\frac{1}{2}}=\\{x\in\Omega:\ \rho(x)<\frac{1}{2}\\}$.
For $x\in A_{\frac{1}{2}}$, we denote

$\varrho_{\tau}(x)=\left\\{\begin{array}[]{lll}\rho(x)^{\min\\{1,\tau\\}}&{\rm
if}\ \ \tau\in(0,1)\cup(1,2),\\\\[5.69054pt] \rho(x)\ln\frac{1}{\rho(x)}&{\rm
if}\ \ \tau=1\end{array}\right.$ (2.1)

and we make $C^{1}$ extension of $\varrho_{\tau}$ into $\Omega\setminus
A_{\frac{1}{2}}$ such that $\varrho_{\tau}>0$ in $\Omega\setminus
A_{\frac{1}{2}}$.

Then there exists $c_{\tau}>1$ such that

$\frac{1}{c}_{\tau}\varrho_{\tau}(x)\leq\mathbb{G}_{\Omega}[\rho^{\tau-2}](x)\leq
c_{\tau}\varrho_{\tau}(x),\quad\forall x\in\Omega.$

Proof. Since the domain $\Omega$ is $C^{2}$, then there exists $\delta_{1}>0$
such that the distance function $\rho(\cdot)$ is $C^{2}$ in
$A_{\delta_{1}}=\\{x\in\Omega:\ \rho(x)<\delta_{1}\\}.$ Let us define

$V_{\tau}(x)=\left\\{\begin{array}[]{lll}l(x),&x\in\Omega\setminus
A_{\delta_{1}},\\\\[5.69054pt] \rho(x)^{\tau},&x\in
A_{\delta_{1}},\end{array}\right.$ (2.2)

where $\tau$ is a parameter in $(0,2)$ and $l$ is a positive function such
that $V_{\tau}$ is $C^{2}$ in $\Omega$. Now we do the estimate of $-\Delta
V_{\tau}$ near the boundary of $\Omega$. Without loss of generality, we may
assume that $0\in\partial\Omega$ and $e_{N}=(0,\cdots,0,1)$ is the unit normal
vector at $0$ pointing inside. We observe that there exists
$\delta_{2}\in(0,\delta_{1})$ such that for any $t\in(0,\delta_{2})$,
$te_{N}\in A_{\delta_{1}}$ and
$V_{\tau}(te_{N})=\rho(te_{N})^{\tau}=t^{\tau}$. Then

$\frac{\partial^{2}V_{\tau}(te_{N})}{\partial
x_{N}^{2}}=\tau(\tau-1)t^{\tau-2}\quad{\rm
and}\quad|\frac{\partial^{2}V_{\tau}(te_{N})}{\partial x_{i}^{2}}|\leq
c_{2},\quad i=1,2,\cdots,N-1,$

where $c_{2}>0$ is independent of $t$. Thus,

$\tau(\tau-1)t^{\tau-2}-c_{2}\leq\Delta
V_{\tau}(te_{N})\leq\tau(\tau-1)t^{\tau-2}+c_{2}.$

Since $\Omega$ is a $C^{2}$ bounded domain, $\partial\Omega$ is compact and
then

$\tau(\tau-1)\rho(x)^{\tau-2}-c_{2}\leq\Delta
V_{\tau}(x)\leq\tau(\tau-1)\rho(x)^{\tau-2}+c_{2},\quad\forall x\in
A_{\delta_{2}}.$ (2.3)

For $\tau\in(0,1)$, we observe that $\tau(\tau-1)<0$ and by (2.3),

$\frac{1}{2}\tau(1-\tau)\rho(x)^{\tau-2}\leq-\Delta V_{\tau}(x)\leq
2\tau(1-\tau)\rho(x)^{\tau-2},\quad\forall x\in A_{\delta_{2}}.$

Combining with the fact that
$-\Delta\mathbb{G}_{\Omega}[\rho^{\tau-2}]=\rho^{\tau-2}$ in $\Omega$, we
obtain that

$\frac{1}{2\tau(1-\tau)}(-\Delta
V_{\tau})\leq-\Delta\mathbb{G}_{\Omega}[\rho^{\tau-2}]\leq\frac{2}{\tau(1-\tau)}(-\Delta
V_{\tau})\quad{\rm in}\ \ A_{\delta_{2}}.$

For $x\in\partial\Omega$, we have that $V_{\tau}(x)=0$ and $G_{\Omega}(x,y)=0$
for $y\in\Omega$ and then

$\mathbb{G}_{\Omega}[\rho^{\tau-2}](x)=\int_{\Omega}G_{\Omega}(x,y)\rho(y)^{\tau-2}dy=0.$

For $x\in\Omega$ satisfying $\rho(x)=\delta_{2}$, we have that
$V_{\tau}(x)={\delta_{2}}^{\tau}$ and then

$\frac{1}{c_{3}}V_{\tau}(x)\leq\mathbb{G}_{\Omega}[\rho^{\tau-2}](x)\leq
c_{3}V_{\tau}(x)\quad{\rm for}\ \rho(x)=\delta_{2},$ (2.4)

where $c_{3}>1$. By Comparison Principle, there exists $c_{4}>1$ depends on
$\tau$ such that

$\frac{1}{c_{4}}V_{\tau}\leq\mathbb{G}_{\Omega}[\rho^{\tau-2}]\leq
c_{4}V_{\tau}\quad{\rm in}\ \ A_{\delta_{2}}.$ (2.5)

Since $\mathbb{G}_{\Omega}[\rho^{\tau-2}]$ and $V_{\tau}$ is bounded in
$\Omega\setminus A_{\delta_{2}}$, then (2.5) holds in $\Omega$.

For $\tau\in(1,2)$, by the fact of (2.3), we observe that

$\frac{1}{2}\tau(\tau-1)\rho(x)^{\tau-2}\leq\Delta V_{\tau}(x)\leq
2\tau(\tau-1)\rho(x)^{\tau-2},\quad\forall x\in A_{\delta_{2}}.$

Letting $W_{\tau}=\mathbb{G}_{\Omega}[1]-V_{\tau}$, then there exists
$c_{5}>1$ depends on $\tau$ such that

$\frac{1}{c_{5}}\rho(x)^{\tau-2}\leq-\Delta W_{\tau}(x)=-1+\Delta
V_{\tau}(x)\leq c_{5}\rho(x)^{\tau-2},\quad\forall x\in A_{\delta_{2}}.$

By the fact that $-\Delta\mathbb{G}_{\Omega}[\rho^{\tau-2}]=\rho^{\tau-2}$ in
$\Omega$, we have

$\frac{1}{c_{6}}W_{\tau}(x)\leq\mathbb{G}_{\Omega}[\rho^{\tau-2}](x)\leq
c_{6}W_{\tau}(x),\ \ \ \ \forall x\in\partial A_{\delta_{2}}$ (2.6)

for some $c_{6}>1$ depends on $\tau$. Using the Comparison Principle, there
exists $c_{7},c_{8}>1$ depend on $\tau$ such that

$\frac{1}{c_{8}}\rho(x)\leq\frac{1}{c_{7}}W_{\tau}(x)\leq\mathbb{G}_{\Omega}[\rho^{\tau-2}](x)\leq
c_{7}W_{\tau}(x)\leq c_{8}\rho(x),\quad\ x\in A_{\delta_{2}},$

and then

$\frac{1}{c_{8}}\rho(x)\leq\mathbb{G}_{\Omega}[\rho^{\tau-2}](x)\leq
c_{8}\rho(x),\quad\ x\in\Omega.$

For $\tau=1$, we redefine

$V_{1}(x)=\left\\{\begin{array}[]{lll}l(x),&x\in\Omega\setminus
A_{\delta_{1}},\\\\[5.69054pt] \rho(x)\ln\frac{1}{\rho(x)},&x\in
A_{\delta_{1}},\end{array}\right.$

where $l$ is a positive function such that $V_{1}$ is $C^{2}$ in $\Omega$. By
direct computation, there exist $\delta_{3}\in(0,\delta_{1})$ and $c_{9}>0$
such that

$\rho(x)^{-1}-c_{9}\leq-\Delta V_{1}(x)\leq\rho(x)^{-1}+c_{9},\quad\forall
x\in A_{\delta_{3}}.$

Then it follows by Comparison Principle that

$\frac{1}{c_{10}}\rho\ln\frac{1}{\rho}\leq\mathbb{G}_{\Omega}[\rho^{-1}]\leq
c_{10}\rho\ln\frac{1}{\rho}\qquad{\rm in}\quad A_{\delta_{3}}$

for some $c_{10}>1$. The proof is complete. $\Box$

By Lemma 2.1, we have the following results.

######  Corollary 2.1.

For $\gamma\in(\frac{2}{3},1)$, we have that

$\lim_{x\in\Omega,x\to\partial\Omega}\mathbb{G}_{\Omega}[\rho^{-2\gamma}](x)\rho(x)^{-\gamma}=+\infty.$
(2.7)

Proof. Let $\tau=2-2\gamma$, by $\gamma\in(\frac{2}{3},1)$, we have that
$\tau<\gamma<1$. Using Lemma 2.1, we obtain that

$\mathbb{G}_{\Omega}[\rho^{-2\gamma}](x)\geq c_{11}\rho(x)^{\tau},\quad\forall
x\in\Omega$

for some $c_{11}>0$, which implies (2.7). $\Box$

Now we are ready to show the existence of pull-in voltage $\lambda^{*}$ to
problem (1.4) such that (1.4) admits a solution for
$\lambda\in(0,\lambda^{*})$ and no solution for $\lambda>\lambda^{*}$.

######  Proposition 2.1.

Assume that $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$ satisfies (1.3) with
$\gamma\in(0,\frac{2}{3}]$, then there exists $\lambda^{*}>0$ such that
problem (1.4) admits at least one solution for $\lambda\in(0,\lambda^{*})$ and
no solution for $\lambda>\lambda^{*}$. Moreover,

$\lambda^{*}\leq\frac{\int_{\Omega}a(x)dx}{\int_{\Omega}\frac{\mathbb{G}_{\Omega}[1](x)}{a(x)^{2}}dx}.$
(2.8)

Proof. Let $v_{0}\equiv 0$ in $\bar{\Omega}$ and

$v_{1}=\lambda\mathbb{G}_{\Omega}[\frac{1}{a^{2}}]>0,$

by (1.3) and Lemma 2.1,

$v_{1}=\lambda\mathbb{G}_{\Omega}[\frac{1}{a^{2}}]\leq\frac{\lambda}{\kappa^{2}}\mathbb{G}_{\Omega}[\rho^{-2\gamma}]\leq\frac{\lambda}{\kappa^{2}}c_{12}\varrho_{2-2\gamma},$
(2.9)

where $c_{12}>0$ depending on $\gamma$ and $\varrho_{2-2\gamma}$ is given by
(2.1).

For $0<\gamma\leq\frac{2}{3}$ and $\gamma\not=\frac{1}{2}$, we observe that
$\min\\{1,2-2\gamma\\}\geq\gamma$ and by (2.9),

$v_{1}(x)\leq\frac{\lambda}{\kappa^{2}}c_{12}\rho(x)^{\min\\{1,2-2\gamma\\}}\leq\frac{\lambda}{\kappa^{2}}c_{13}\rho(x)^{\gamma},\qquad
x\in\Omega.$

For $\gamma=\frac{1}{2}$, we see that $2-2\gamma=1$ and by (2.9),

$v_{1}(x)\leq\frac{\lambda}{\kappa^{2}}c_{12}\rho(x)\log\frac{1}{\rho(x)}\leq\frac{\lambda}{\kappa^{2}}c_{13}\rho(x)^{\gamma},\qquad
x\in\Omega.$

Then

$v_{1}(x)\leq\frac{\lambda}{\kappa^{2}}c_{13}\rho(x)^{\gamma},\qquad
x\in\Omega.$

Fixed $\mu\in(0,\kappa)$, there exists $\lambda_{1}>0$ such that

$\frac{\lambda}{\kappa^{2}}c_{13}\leq\mu<\kappa,\ \ \
\forall\lambda<\lambda_{1},$

then for any $\lambda<\lambda_{1}$,

$v_{1}(x)\leq\mu\rho(x)^{\gamma},\qquad\forall x\in\Omega.$

Let $v_{2}=\lambda\mathbb{G}_{\Omega}[\frac{1}{(a-v_{1})^{2}}]$, by the fact
that $a(x)\geq\kappa\rho(x)^{\gamma}>\mu\rho(x)^{\gamma}\geq v_{1}(x)>0$ for
$x\in\Omega$ and and Lemma 2.1, we have that

$v_{1}=\lambda\mathbb{G}_{\Omega}[\frac{1}{a^{2}}]\leq
v_{2}\leq\frac{\lambda}{(\kappa-\mu)^{2}}\mathbb{G}_{\Omega}[\rho^{-2\gamma}]\leq\frac{\lambda}{(\kappa-\mu)^{2}}c_{12}\varrho_{2-2\gamma}\leq\frac{\lambda}{(\kappa-\mu)^{2}}c_{14}\rho^{\gamma}\quad{\rm
in}\ \ \Omega$

for $\gamma\in(0,\frac{2}{3}]$. We observe that there exists
$\lambda_{2}\in(0,\lambda_{1}]$ such that

$\frac{\lambda}{(\kappa-\mu)^{2}}c_{14}\leq\mu<\kappa,\ \ \
\forall\lambda<\lambda_{2}.$

Then for any $\lambda<\lambda_{2}$, we obtain that

$v_{2}(x)\leq\mu\rho(x)^{\gamma},\qquad\forall x\in\Omega.$

Iterating above process, we have that

$v_{n}:=\lambda\mathbb{G}_{\Omega}[\frac{1}{(a-v_{n-1})^{2}}]\geq
v_{n-1},\quad n\in\mathbb{N}$

and

$v_{n}(x)\leq\mu\rho(x)^{\gamma},\qquad\forall x\in\Omega.$

Thus, the sequence $\\{v_{n}\\}_{n}$ converges, denoting by
$u_{\lambda}=\lim_{n\to\infty}v_{n}$, then $u_{\lambda}$ is a classical
solution of (1.4).

We claim that $u_{\lambda}$ is the minimal solution of (1.4), that is, for any
positive solution $u$ of (1.4), we always have $u_{\lambda}\leq u$. Indeed,
there holds $u\geq v_{0}$ and then

$u=\lambda\mathbb{G}_{\Omega}[\frac{1}{(a-u)^{2}}]\geq\lambda\mathbb{G}_{\Omega}[\frac{1}{a^{2}}]=v_{1}.$

We may show inductively that

$u\geq v_{n}$

for all $n\in\mathbb{N}$. The claim follows.

Similarly, if problem (1.4) has a super solution $u$ for $\lambda_{0}>0$, then
(1.4) admits a minimal solution $u_{\lambda}$ for all
$\lambda\in(0,\lambda_{0}]$. As a consequence, the mapping $\lambda\mapsto
u_{\lambda}$ is increasing. So we may define

$\lambda^{*}=\sup\\{\lambda>0:\,(\ref{eq 1.1})\ {\rm has\ a\ minimal\
solution\ for}\ \lambda\\},$

which is the largest $\lambda$ such that problem (1.4) has minimal positive
solution, and $\lambda^{*}>0$.

For $0<\lambda_{1}<\lambda_{2}<\lambda^{*}$, we know that $0\leq
u_{\lambda_{1}}\leq u_{\lambda_{2}}\leq a$ in $\Omega$, then

$\displaystyle-\Delta(u_{\lambda_{2}}-u_{\lambda_{1}})$ $\displaystyle=$
$\displaystyle\frac{\lambda_{2}}{(a-u_{\lambda_{2}})^{2}}-\frac{\lambda_{1}}{(a-u_{\lambda_{1}})^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{\lambda_{2}-\lambda_{1}}{(a-u_{\lambda_{1}})^{2}}\geq\frac{\lambda_{2}-\lambda_{1}}{a^{2}}>0,$

which implies that

$u_{\lambda_{2}}-u_{\lambda_{1}}\geq(\lambda_{2}-\lambda_{1})\mathbb{G}_{\Omega}[a^{-2}]>0.$
(2.10)

It infers that $u_{\lambda}<a$ in $\Omega$ for any $\lambda<\lambda^{*}$ and
by the interior regularity, we have that $u_{\lambda}\in
C_{loc}^{2,\gamma^{\prime}}(\Omega)$ for any $\gamma^{\prime}<\gamma$.

Now we prove that $\lambda^{*}<+\infty$. If not, then for any $\lambda>0$,
problem (1.4) has the minimal solution $u_{\lambda}$. Let
$A_{\delta}=\\{x\in\Omega:\,\rho(x)<\delta\\}$ and $n\in\mathbb{N}$,

$\eta_{n}=1\ \ {\rm in}\ \Omega\setminus{A_{1/n}},\quad\ \eta_{n}=0\ \ {\rm
in}\ A_{1/{2n}},\quad\ \eta_{n}\in C^{2}(\Omega),$ (2.11)

and $\xi_{n}=\mathbb{G}_{\Omega}[1]\eta_{n}$ in $\Omega$, we observe that
$\xi_{n}\in C_{c}^{2}(\Omega)$ and

$\displaystyle\int_{\Omega}u_{\lambda}(-\Delta\xi_{n})dx=\int_{\Omega}u_{\lambda}(\eta_{n}-\nabla\mathbb{G}_{\Omega}[1]\cdot\nabla\eta_{n}+(-\Delta\eta_{n})\mathbb{G}_{\Omega}[1])dx.$

By the fact that $|\nabla\eta_{n}|\leq c_{15}n$, $|(-\Delta)\eta_{n}|\leq
c_{16}n^{2}$ and $\mathbb{G}_{\Omega}[1]\leq c_{17}\rho$ in $\Omega$, then we
have that

$\int_{\Omega}u_{\lambda}(-\Delta\xi_{n})dx\leq\int_{\Omega}u_{\lambda}dx+c_{15}n\int_{A_{1/n}}u_{\lambda}dx+c_{18}n^{2}\int_{A_{1/n}}u_{\lambda}\rho
dx\leq c_{19},$ (2.12)

where $c_{15},c_{16},c_{17},c_{18},c_{19}>0$ are independent on $n$.

Since $u_{\lambda}$ is the minimal solution of (1.4), then

$\int_{\Omega}u_{\lambda}(-\Delta\xi_{n})dx=\int_{\Omega}\nabla
u_{\lambda}\cdot\nabla{\xi_{n}}dx=\int_{\Omega}(-\Delta
u_{\lambda})\xi_{n}dx=\int_{\Omega}\frac{\lambda\xi_{n}}{(a-u_{\lambda})^{2}}dx$

Passing to the limit of $n\to\infty$ and combining with (2.12), we see that

$\int_{\Omega}\frac{\lambda\mathbb{G}_{\Omega}[1]}{(a-u_{\lambda})^{2}}dx\leq
c_{20}.$

Thus,

$\displaystyle\int_{\Omega}a(x)dx\geq\int_{\Omega}u_{\lambda}(x)dx$
$\displaystyle=$ $\displaystyle\int_{\Omega}\mathbb{G}_{\Omega}[1](x)(-\Delta
u_{\lambda})(x)dx$ $\displaystyle=$
$\displaystyle\lambda\int_{\Omega}\frac{\mathbb{G}_{\Omega}[1](x)}{[a(x)-u_{\lambda}(x)]^{2}}dx\geq\lambda\int_{\Omega}\frac{\mathbb{G}_{\Omega}[1](x)}{a^{2}(x)}dx,$

which implies that

$\lambda\leq\frac{\int_{\Omega}a(x)dx}{\int_{\Omega}\frac{\mathbb{G}_{\Omega}[1](x)}{a^{2}(x)}dx}.$

Therefore,

$\lambda^{*}\leq\frac{\int_{\Omega}a(x)dx}{\int_{\Omega}\frac{\mathbb{G}_{\Omega}[1](x)}{a^{2}(x)}dx}.$

Similar proof as the claim, we know that problem (1.4) has the minimal
solution for $\lambda\in(0,\lambda^{*})$ and no solution for
$\lambda>\lambda^{*}$. This end the proof. $\Box$

Proof of Theorem 1.2. By contradiction. Under the assumptions of Theorem 1.2,
if there exists some $\lambda>0$ such that problem (1.4) has a solution
$u_{\lambda}$ satisfying $0<u_{\lambda}<a$ in $\Omega$, then

$\lambda\mathbb{G}_{\Omega}[\frac{1}{a^{2}}]\leq\lambda\mathbb{G}_{\Omega}[\frac{1}{(a-u_{\lambda})^{2}}]=u_{\lambda}<a\quad{\rm
in}\quad\Omega.$ (2.13)

On the other hand, by (1.5), we have that

$\lambda\mathbb{G}_{\Omega}[\frac{1}{a^{2}}]\geq\frac{\lambda}{c_{0}^{2}}\mathbb{G}_{\Omega}[\rho^{-2\gamma}].$

For $\gamma\in(\frac{2}{3},1)$, using Corollary 2.1 and (2.13), we deduce that

$\lim_{x\in\Omega,x\to\partial\Omega}a(x)\rho(x)^{-\gamma}=+\infty,$

which contradicts (1.5). This ends the proof. $\Box$

We next do the boundary decay estimate for $u_{\lambda}$. To this end, we
introduce the following lemma.

######  Lemma 2.2.

Assume that the function $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$
satisfies (1.3) with $\gamma\in(0,\frac{2}{3}]$ and $u$ is a super solution of
(1.4) with $\lambda>0$ such that

$u\leq\theta a\quad{\rm in}\quad\Omega$ (2.14)

for some $\theta\in(0,1)$, then (1.4) admits the minimal solution
$u_{\lambda}$ such that

$u_{\lambda}\leq c_{20}\varrho_{2-2\gamma}\quad{\rm in}\quad\Omega$ (2.15)

for some $c_{20}>0$, where $\varrho_{2-2\gamma}$ is defined by (2.1).

Proof. Since problem (1.4) admits a super solution $u$ satisfying (2.14) for
some $\lambda>0$, as in the proof of Proposition 2.1, we know that (1.4) has
the minimal solution $u_{\lambda}$ and then

$u_{\lambda}\leq u\leq\theta a,\qquad
u_{\lambda}=\lambda\mathbb{G}_{\Omega}[\frac{1}{(a-u_{\lambda})^{2}}]\leq\lambda\kappa^{-2}(1-\theta)^{-2}\mathbb{G}_{\Omega}[\rho^{-2\gamma}],$

by the fact of (1.3). Using Lemma 2.1 with $\tau=2-2\gamma$, we imply that
(2.15) holds. The proof is complete. $\Box$

######  Proposition 2.2.

Assume that the function $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$
satisfies (1.3) and(1.5) with $c_{0}\geq\kappa>0$ and
$\gamma\in(0,\frac{2}{3}]$. Then
$(i)$ for $\lambda\in(0,\lambda^{*})$, there exists $c_{21}\geq 1$ dependent
on $\gamma$ such that

$\frac{\lambda}{c_{21}}\varrho_{2-2\gamma}(x)\leq u_{\lambda}(x)\leq
c_{21}\lambda\rho(x)^{\gamma},\quad\forall x\in\Omega;$

$(ii)$ there exists $\lambda_{*}\leq\lambda^{*}$ such that for
$\lambda\in(0,\lambda_{*})$,

$\frac{\lambda}{c_{21}}\varrho_{2-2\gamma}(x)\leq u_{\lambda}(x)\leq
c_{21}\lambda\varrho_{2-2\gamma}(x),\quad\forall x\in\Omega$ (2.16)

where $\varrho_{2-2\gamma}$ is defined by (2.1).

Proof. Lower bound. By Proposition 2.1, we see that (1.4) admits the minimal
solution $u_{\lambda}$ for $\lambda\in(0,\lambda^{*})$, which is approximated
by an increasing sequence $\\{v_{n}\\}_{n}$ defined by

$v_{0}=0\quad{\rm and}\quad
v_{n}=\lambda\mathbb{G}_{\Omega}[\frac{1}{(a-v_{n-1})^{2}}].$

By the fact of (1.5) and Lemma 2.1 with $\tau=2-2\gamma$, we have that

$\displaystyle u_{\lambda}\geq
v_{1}=\lambda\mathbb{G}_{\Omega}[\frac{1}{a^{2}}]\geq\lambda
c_{0}^{2}\mathbb{G}_{\Omega}[\rho^{-2\gamma}]\geq
c_{22}\lambda\varrho_{2-2\gamma}\quad{\rm in}\quad\Omega$

for some $c_{22}\in(0,1)$ dependent on $\gamma$.

Upper bound. From the proof of Proposition 2.1, for $\lambda>0$ small, we see
that $u_{\lambda}(x)\leq\mu\rho(x)^{\gamma}$, $x\in\Omega$, for some
$\mu\in(0,\kappa)$, then there exists $\theta\in(0,1)$ such that

$u_{\lambda}(x)\leq\theta a(x),\quad x\in\Omega.$

By Lemma 2.2, we have that

$u_{\lambda}\leq c_{23}\varrho_{2-2\gamma}\quad{\rm in}\quad\Omega.$ (2.17)

Let us define

$\lambda_{*}=\sup\\{\lambda\in(0,\lambda^{*}):\
\limsup_{x\in\Omega,x\to\partial\Omega}u_{\lambda}(x)\rho(x)^{-\gamma}<\kappa\\},$

we observe that $\lambda_{*}\leq\lambda^{*}$ and (2.17) holds for all
$\lambda\in(0,\lambda_{*})$. This ends the proof. $\Box$

##  3\. Estimates for $\lambda^{*}$ and $\lambda_{*}$ when $\Omega=B_{1}(0)$

In this section, we do the estimates for $\lambda^{*}$ and $\lambda_{*}$ in
the case that $\Omega=B_{1}(0)$ and

$a(x)=\kappa(1-|x|^{2})^{\gamma}.$ (3.1)

In this case, the function $a$ represents the upper semi-sphere type shape in
$\mathbb{R}^{N}$. We have following monotonicity results.

######  Proposition 3.1.

Assume that $\Omega=B_{1}(0)$, the function $a$ satisfies (3.1) with
$\kappa>0$, $\gamma\in(0,\frac{2}{3}]$ and
$0<\lambda<\lambda^{*}(\kappa,\gamma)$. Then

$(i)$ the mappings: $\gamma\mapsto\lambda^{*}(\kappa,\gamma)$ and
$\gamma\mapsto\lambda_{*}(\kappa,\gamma)$ are decreasing for fixed $\kappa>0$;

$(ii)$ the mapping: $\kappa\mapsto\lambda^{*}(\kappa,\gamma)$ and
$\kappa\mapsto\lambda_{*}(\kappa,\gamma)$ are increasing for fixed
$\gamma\in(0,\frac{2}{3}]$.

Proof. Let $0<\gamma_{2}\leq\gamma_{1}\leq\frac{2}{3}$, for
$\lambda\in(0,\min\\{\lambda^{*}(\kappa,\gamma_{1}),\lambda^{*}(\kappa,\gamma_{2})\\})$,
denote by $u_{1},u_{2}$ the minimal solutions of (1.4) with $a=a_{1}$ and
$a=a_{2}$ respectively, where $a_{1}(x)=\kappa(1-|x|^{2})^{\gamma_{1}}$ and
$a_{2}(x)=\kappa(1-|x|^{2})^{\gamma_{2}}$. Since $\gamma_{2}\leq\gamma_{1}$,
then $a_{1}\leq a_{2}$ in $B_{1}(0)$ and for any
$\lambda\in(0,\lambda^{*}(\kappa,\gamma_{1}))$,

$-\Delta
u_{1}=\frac{\lambda}{(a_{1}-u_{1})^{2}}\geq\frac{\lambda}{(a_{2}-u_{1})^{2}}\quad{\rm
in}\ \ B_{1}(0),$

that is, $u_{1}$ is a super solution of (1.4) with $a=a_{2}$, for any
$\lambda\in(0,\lambda^{*}(\kappa,\gamma_{1}))$. By the same argument in
Proposition 2.1, problem (1.4) with $a=a_{2}$ admits the minimal solution for
any $\lambda\in(0,\lambda^{*}(\kappa,\gamma_{1}))$. Thus,

$\lambda^{*}(\kappa,\gamma_{2})\geq\lambda^{*}(\kappa,\gamma_{1}),$

by the definition of $\lambda^{*}(\kappa,\gamma_{2}))$. As a consequence, we
obtain that the mapping $\gamma\mapsto\lambda^{*}(\kappa,\gamma)$ is
decreasing for fixed $\kappa>0$.

It is similar to obtain the other assertions. $\Box$

######  Proposition 3.2.

Assume that $\Omega=B_{1}(0)$ and the function $a$ satisfies (3.1) with
$\kappa>0$, $\gamma\in(0,\frac{2}{3}]$. Then

$\lambda^{*}(\kappa,\gamma)\leq\frac{4N\kappa^{3}}{3}\mathcal{B}(\frac{1}{2},2-2\gamma),$
(3.2)

where $\mathcal{B}(\cdot,\cdot)$ is the Beta function.

Proof. From (2.8), we only need to do the estimates of $\int_{B_{1}(0)}a(x)dx$
and $\int_{B_{1}(0)}\frac{\mathbb{G}_{\Omega}[1](x)}{a^{2}(x)}dx$. Since $a$
satisfies (3.1), by direct computation, we have that

$\int_{B_{1}(0)}a(x)dx=\frac{2}{3}\kappa|S^{N-1}|,$ (3.3)

where $S^{N-1}$ is the unit sphere in $\mathbb{R}^{N}$.

On the other hand, we see that there exists $c_{27}>1$ such that

$\mathbb{G}_{B_{1}(0)}[1](x)=\frac{1}{2N}(1-|x|^{2}),\quad\forall x\in
B_{1}(0),$

then

$\displaystyle\int_{B_{1}(0)}\frac{\mathbb{G}_{\Omega}[1](x)}{a^{2}(x)}dx=\frac{1}{2N\kappa^{2}}\int_{B_{1}(0)}(1-|x|^{2})^{1-2\gamma}dx=\frac{|S^{N-1}|}{4N\kappa^{2}}\mathcal{B}(\frac{1}{2},2-2\gamma),$

which, combining with (3.3), (2.8), implies (3.2). $\Box$

From Proposition 3.1, the mapping $\gamma\mapsto\lambda_{*}(\kappa,\gamma)$ is
decreasing, so our interest is to evaluate the lower bound for $\lambda_{*}$
when $\gamma=\frac{2}{3}$.

######  Proposition 3.3.

Assume that $\Omega=B_{1}(0)$ and the function $a$ satisfies (3.1) with
$\kappa>0$ and $\gamma=\frac{2}{3}$. Then

$\lambda_{*}(\kappa,\frac{2}{3})\geq\left(\frac{2}{3}\right)^{5}\kappa^{3}.$
(3.4)

Proof. Let $w_{t}(r)=t\kappa(1-r^{2})^{2/3}$ with $t\in(0,1)$, then for $x\in
B_{1}(0)$,

$\displaystyle-\Delta w_{t}(|x|)$ $\displaystyle=$
$\displaystyle\frac{8}{9}t\kappa(1-|x|^{2})^{-4/3}+\frac{4}{3}t\kappa(N-\frac{2}{3})(1-|x|^{2})^{-1/3}$
$\displaystyle\geq$ $\displaystyle\frac{8}{9}t\kappa(1-|x|^{2})^{-4/3}$

and

$\frac{\lambda}{(a(x)-w(|x|))^{2}}\leq(1-t)^{-2}\kappa^{-2}\lambda(1-|x|^{2})^{-4/3},$

thus, if

$\lambda\leq\lambda_{t}:=\frac{8}{9}t(1-t)^{2}\kappa^{3},$ (3.5)

$w_{t}$ is a super solution of the equation (1.4). In fact, choose
$t=\frac{1}{3}$, we get the optimal
$\lambda_{t}=\frac{2^{5}}{3^{4}}\kappa^{3}$ and then by Lemma 2.2 to obtain
that problem (1.4) admits the minimal solution $u_{\lambda_{t}}$. Therefore,
$\lambda_{*}\geq\lambda_{t}$. The proof is complete. $\Box$

######  Lemma 3.1.

Assume that $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$ satisfies (1.3) and
(1.5) with $c_{0}\geq\kappa>0$, $\gamma\in(0,\frac{2}{3}]$, $\lambda_{*}$ is
given in Theorem 1.1 and $u_{\lambda}$ is the minimal solution of (1.4) with
$\lambda\in(0,\lambda_{*})$. Then $u_{\lambda}\in H^{1}_{0}(\Omega)$ and there
exists $c_{31}>0$ such that

$\int_{\Omega}\frac{u_{\lambda}}{(a-u_{\lambda})^{2}}dx\leq
c_{31}\lambda\qquad{\rm and}\qquad\int_{\Omega}|\nabla u_{\lambda}|^{2}dx\leq
c_{31}\lambda^{2}.$

Proof. For $\lambda\in(0,\lambda_{*})$ and
$\gamma\in(0,\frac{2}{3}]\setminus{\\{\frac{1}{2}\\}}$, by (2.16), we have
that

$u_{\lambda}(x)\leq c_{21}\lambda\rho(x)^{\min\\{1,2-2\gamma\\}},\quad\
x\in\Omega.$

Since $\kappa\rho(x)^{\gamma}\leq a(x)\leq c_{0}\rho(x)^{\gamma}$ for
$x\in\Omega$, then there exists $\theta_{1}\in(0,1)$ such that
$u_{\lambda}<\theta_{1}a$ in $A_{\delta}=\\{x\in\Omega:\,\rho(x)<\delta\\}$
for some $\delta>0$ small. Moreover, since $u_{\lambda}<a$ in $\Omega$, then
there exists $\theta_{2}\in(0,1)$ such that $u_{\lambda}\leq\theta_{2}a$ in
$\Omega\setminus{A_{\delta}}$. Let $\theta=\max\\{\theta_{1},\theta_{2}\\}$,
then we have that $u_{\lambda}\leq\theta a$ in $\Omega$ and then

$a-u_{\lambda}\geq(1-\theta)a\geq(1-\theta)\kappa\rho^{\gamma}\quad{\rm in}\ \
\Omega.$ (3.6)

Therefore,

$\int_{\Omega}\frac{u_{\lambda}}{(a-u_{\lambda})^{2}}dx\leq\int_{\Omega}\frac{c_{21}\lambda\rho(x)^{\min\\{1,2-2\gamma\\}}}{(1-\theta)^{2}\kappa^{2}\rho(x)^{2\gamma}}dx:=c_{24}\lambda<+\infty,$
(3.7)

by the fact that $\min\\{1,2-2\gamma\\}-2\gamma>-1$. Taking a sequence
$\\{\xi_{n}\\}_{n}\subset C_{c}^{2}(\Omega)$ which converges to $u_{\lambda}$
as $n\to\infty$, since $u_{\lambda}$ is the minimal solution of (1.4), then

$\int_{\Omega}\nabla
u_{\lambda}\cdot\nabla{\xi_{n}}dx=\int_{\Omega}\frac{\lambda\xi_{n}}{(a-u_{\lambda})^{2}}dx.$

Passing to the limit as $n\to\infty$, we obtain that

$\int_{\Omega}|\nabla u_{\lambda}|^{2}dx=\int_{\Omega}\frac{\lambda
u_{\lambda}}{(a-u_{\lambda})^{2}}dx\leq c_{24}\lambda^{2}.$

For $\lambda\in(0,\lambda_{*})$ and $\gamma=\frac{1}{2}$, by (2.16), we have
that

$u_{\lambda}(x)\leq c_{21}\lambda\rho(x)\log\frac{1}{\rho(x)},\quad\
x\in\Omega,$

similarly, there exists $\theta\in(0,1)$ such that $u_{\lambda}\leq\theta a$
in $\Omega$ and then (3.6) holds. Then

$\int_{\Omega}\frac{u_{\lambda}}{(a-u_{\lambda})^{2}}dx\leq\int_{\Omega}\frac{c_{21}\lambda}{(1-\theta)^{2}\kappa^{2}}\log\frac{1}{\rho(x)}dx\leq
c_{24}\lambda$ (3.8)

and then we have that $\int_{\Omega}|\nabla u_{\lambda}|^{2}dx\leq
c_{24}\lambda^{2}.$ The proof is complete. $\Box$

Proof of Theorem 1.1. The existence of the minimal solution for
$\lambda\in(0,\lambda^{*})$ and the nonexistence for $\lambda>\lambda^{*}$
follow by Proposition 2.1. The proof of Theorem 1.1 $(iii)$ sees Proposition
2.2 and Lemma 3.1. The estimates of $\lambda^{*}$ and $\lambda_{*}$ see
Proposition 3.1, Proposition 3.3 and Proposition 3.2. $\Box$

##  4\. Properties of minimal solution

###  4.1. Regularity

In this subsection, we study the solutions of (1.4) when
$\lambda=\lambda^{*}$.

######  Proposition 4.1.

Assume that the function $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$
satisfies (1.3) and (1.5) with $c_{0}\geq\kappa>0$, $\gamma\in(0,\frac{2}{3}]$
and $u_{\lambda^{*}}$ is given by (1.6). Then $u_{\lambda^{*}}$ is a weak
solution of (1.4) with $\lambda=\lambda^{*}$. Moreover, for any
$\beta\in(0,\gamma)$, there exists $c_{\beta}>0$ such that

$\lVert u_{\lambda^{*}}\rVert_{W^{1,\frac{N}{N-\beta}}(\Omega)}\leq c_{\beta}$
(4.1)

and

$\int_{\Omega}\frac{\rho^{1-\beta}}{(a-u_{\lambda^{*}})^{2}}dx\leq c_{\beta}.$
(4.2)

Proof. For any $\beta\in(0,\gamma)$ and $n\in\mathbb{N}$, denote
$\xi_{n}=\mathbb{G}_{\Omega}[\rho^{-1-\beta}]\eta_{n}$, where $\eta_{n}$ is
defined by (2.11), we observe that $\xi_{n}\in C_{c}^{2}(\Omega)$ and

$\displaystyle\int_{\Omega}u_{\lambda}(-\Delta\xi_{n})dx=\int_{\Omega}u_{\lambda}(\rho^{-1-\beta}\eta_{n}-\nabla\mathbb{G}_{\Omega}[\rho^{-1-\beta}]\cdot\nabla\eta_{n}+(-\Delta)\eta_{n}\mathbb{G}_{\Omega}[\rho^{-1-\beta}])dx.$

Using Lemma 2.1 with $\tau=1-\beta\in(1/3,1)$, we have that
$\mathbb{G}_{\Omega}[\rho^{-1-\beta}]\leq c_{25}\rho^{1-\beta}$ in $\Omega$.
Combining with the fact that $|\nabla\eta_{n}|\leq c_{16}n$,
$|(-\Delta)\eta_{n}|\leq c_{17}n^{2}$ and $0<u_{\lambda}(x)<a(x)\leq
c_{0}\rho(x)^{\gamma}$ for $x\in\Omega$, we obtain that

$\displaystyle\int_{\Omega}u_{\lambda}(-\Delta\xi_{n})dx$ $\displaystyle\leq$
$\displaystyle
c_{0}\int_{\Omega}\rho^{\gamma-1-\beta}dx+c_{26}n\int_{A_{1/n}}\rho^{\gamma-\beta}dx+c_{26}n^{2}\int_{A_{1/n}}\rho^{1-\beta+\gamma}dx$
$\displaystyle\leq$ $\displaystyle\bar{c}_{\beta},$

where $c_{26},\bar{c}_{\beta}>0$ independent on $n$ and $\bar{c}_{\beta}$
satisfies $\bar{c}_{\beta}\to+\infty$ as $\beta\to\gamma^{-}$. Thus,

$\int_{\Omega}\frac{\lambda\mathbb{G}_{\Omega}[\rho^{-1-\beta}]}{(a-u_{\lambda})^{2}}dx\leq\bar{c}_{\beta}$

and then

$\int_{\Omega}\frac{\lambda\rho^{1-\beta}}{(a-u_{\lambda})^{2}}dx\leq\frac{\bar{c}_{\beta}}{c_{25}}.$
(4.3)

By Theorem 1.1, we see that the mapping $\lambda\mapsto u_{\lambda}$ is
increasing and uniformly bounded by the function $a$, which is in
$L^{1}(\Omega)$, then

$u_{\lambda}\to u_{\lambda^{*}}\ \ \ {\rm in}\ \ L^{1}(\Omega)\quad\ {\rm as}\
\lambda\to\lambda^{*}$

and then for any $\xi\in C_{c}^{2}(\Omega)$, we have that

$\int_{\Omega}u_{\lambda}(-\Delta\xi)dx\to\int_{\Omega}u_{\lambda^{*}}(-\Delta\xi)dx\
\ \ {\rm as}\ \ \lambda\to\lambda^{*}.$ (4.4)

Moreover, we observe that the mapping
$\lambda\mapsto\frac{\lambda}{(a-u_{\lambda})^{2}}$ is increasing and by
(4.3),

$\frac{\lambda}{(a-u_{\lambda})^{2}}\to\frac{\lambda^{*}}{(a-u_{\lambda^{*}})^{2}}\
\ \ {\rm a.e.\ in}\ \ \Omega\quad{\rm as}\ \lambda\to\lambda^{*},$

Then (4.3) deduces that

$\frac{\lambda}{(a-u_{\lambda})^{2}}\to\frac{\lambda^{*}}{(a-u_{\lambda^{*}})^{2}}\
\ \ {\rm in}\ \ L^{1}(\Omega,\,\rho^{1-\beta}dx)\quad{\rm as}\
\lambda\to\lambda^{*}$

and

$\int_{\Omega}\frac{\lambda^{*}\rho^{1-\beta}}{(a-u_{\lambda}^{*})^{2}}dx\leq\frac{\bar{c}_{\beta}}{c_{25}},$
(4.5)

thus, for any $\xi\in C_{c}^{2}(\Omega)$, we have that

$\int_{\Omega}\frac{\lambda\xi}{(a-u_{\lambda})^{2}}dx\to\int_{\Omega}\frac{\lambda^{*}\xi}{(a-u_{\lambda^{*}})^{2}}dx\
\ \ {\rm as}\ \ \lambda\to\lambda^{*}.$ (4.6)

Since $u_{\lambda}$ is the minimal solution of (1.4),

$\int_{\Omega}u_{\lambda}(-\Delta\xi)dx=\int_{\Omega}\frac{\lambda\xi}{(a-u_{\lambda})^{2}}dx,\
\ \ \ \ \forall\,\xi\in C_{c}^{2}(\Omega),$

passing to the limit as $\lambda\to\lambda^{*}$, combining with (4.4) and
(4.6), then $u_{\lambda^{*}}$ is a weak solution of (1.4) with
$\lambda=\lambda^{*}$.

By [2, Theorem 2.6], for any $\beta^{\prime}\in(\beta,\gamma)$, there exists
$c_{27}>0$ such that

$\lVert|\nabla
u_{\lambda^{*}}|\rVert_{M^{\frac{N}{N-\beta^{\prime}}}(\Omega)}\leq
c_{27}\lVert(a-u_{\lambda^{*}})^{-2}\rVert_{L^{1}(\Omega,\,\rho^{1-\beta^{\prime}}dx)},$

where ${M^{\frac{N}{N-\beta^{\prime}}}(\Omega)}$ is the Marcinkiewicz space of
exponent $\frac{N}{N-\beta^{\prime}}$. Then, by (4.5), we have that

$\lVert|\nabla u_{\lambda^{*}}|\rVert_{L^{\frac{N}{N-\beta}}(\Omega)}\leq
c_{27}c_{25}^{-1}\bar{c}_{\beta}(\lambda^{*})^{-1}$

and then

$\lVert u_{\lambda}^{*}\rVert_{W^{1,\frac{N}{N-\beta}}(\Omega)}\leq c_{\beta}$

for some $c_{\beta}>0.$ Thus, (4.1) holds and (4.2) is obvious by (4.5). This
ends the proof. $\Box$

###  4.2. Stability

In this subsection, we introduce the stability of the minimal solution
$u_{\lambda}$ for problem (1.4). By the definition of $\lambda_{*}$, we
observe that for any $\lambda\in(0,\lambda_{*})$, there exists
$\theta\in(0,1)$ such that $u_{\lambda}\leq\theta a$. Therefore, by the fact
that $\gamma\in(0,\frac{2}{3}]$ and $a\geq\kappa\rho^{\gamma}$ in $\Omega$, we
have that

$\frac{1}{(a-u_{\lambda})^{3}}\leq\frac{1}{(1-\theta)^{3}a^{3}}\leq(1-\theta)^{-3}\kappa^{-3}\rho^{-3\gamma}\leq
c_{28}\rho^{-2}\quad{\rm in}\quad\Omega,$ (4.7)

where $c_{28}>0$ depends on $\theta,\kappa,\gamma$. This enables us to
consider the first eigenvalue of
$-\Delta-\frac{2\lambda}{(a-u_{\lambda})^{3}}$ in $H_{0}^{1}(\Omega)$, that
is,

$\mu_{1}(\lambda)=\inf_{\varphi\in
H_{0}^{1}(\Omega)}\frac{\int_{\Omega}(|\nabla\varphi|^{2}-\frac{2\lambda\varphi^{2}}{(a-u_{\lambda})^{3}})dx}{\int_{\Omega}\varphi^{2}\,dx}$

for $\lambda\in(0,\lambda_{*})$. It is well-known that $u_{\lambda}$ is stable
if $\mu_{1}(\lambda)>0$ and semi-stable if $\mu_{1}(\lambda)\geq 0$.

######  Lemma 4.1.

Assume that $\lambda\in(0,\lambda_{*})$, $a\in C^{\gamma}(\Omega)\cap
C(\bar{\Omega})$ satisfies (1.3) and (1.5) with $c_{0}\geq\kappa>0$,
$\gamma\in(0,\frac{2}{3}]$. Suppose that $u_{\lambda}$ is the minimal solution
of (1.4) and $v_{\lambda}$ is a super solution of (1.4).

If $\mu_{1}(\lambda)>0$, then

$u_{\lambda}\leq v_{\lambda}\quad{\rm in}\quad\Omega.$

If $\mu_{1}(\lambda)=0$, then

$u_{\lambda}=v_{\lambda}\quad{\rm in}\quad\Omega.$

Proof. We observe that $a>u_{\lambda}$ in $\Omega$ for
$\lambda\in(0,\lambda^{*})$ and $\mu_{1}(\lambda)$ is well-defined for
$\lambda\in(0,\lambda_{*})$, then it follows the procedure of the proof of
Lemma 4.1 in [4] just replacing $\frac{f}{(1-u)^{2}}$ by
$\frac{1}{(a-u)^{2}}$.

$\Box$

######  Proposition 4.2.

Assume that $\lambda\in(0,\lambda_{*})$, $a\in C^{\gamma}(\Omega)\cap
C(\bar{\Omega})$ satisfies (1.3) and (1.5) with $c_{0}\geq\kappa>0$,
$\gamma\in(0,\frac{2}{3}]$. Let $u_{\lambda}$ be the minimal solution of
(1.4), then $u_{\lambda}$ is stable.

Proof. We first claim that the mapping $\lambda\mapsto\mu_{1}(\lambda)$ is
locally Lipschitz continuous and strictly decreasing in $(0,\lambda_{*})$.
Observing that the mapping $\lambda\mapsto u_{\lambda}$ is strictly increasing
and so is $\lambda\mapsto\frac{2\lambda}{(a-u_{\lambda})^{3}}$, which implies
that the mapping $\lambda\mapsto\mu_{1}(\lambda)$ is strictly decreasing in
$(0,\lambda_{*})$. Let $0<\lambda_{1}<\lambda_{2}<\lambda_{*}$ and
$\phi_{\lambda_{2}}$ be the achieved function of $\mu_{1}(\lambda_{2})$ with
$\lVert\phi_{\lambda_{2}}\rVert_{L^{2}(\Omega)}=1$, then we have that

$\displaystyle 0<\mu_{1}(\lambda_{1})-\mu_{1}(\lambda_{2})$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}\left(|\nabla\phi_{\lambda_{2}}|^{2}-\frac{2\lambda_{1}\phi_{\lambda_{2}}^{2}}{(a-u_{\lambda_{1}})^{3}}\right)dx$
$\displaystyle-\int_{\Omega}\left(|\nabla\phi_{\lambda_{2}}|^{2}-\frac{2\lambda_{2}\phi_{\lambda_{2}}^{2}}{(a-u_{\lambda_{2}})^{3}}\right)dx$
$\displaystyle\leq$ $\displaystyle
2(\lambda_{2}-\lambda_{1})\int_{\Omega}\frac{\phi_{\lambda_{2}}^{2}}{(a-u_{\lambda_{2}})^{3}}dx,$

thus, the mapping $\lambda\mapsto\mu_{1}(\lambda)$ is locally Lipschitz
continuous.

We next prove that $u_{\lambda}$ is stable and $\mu_{1}(\lambda)>0$ for
$\lambda>0$ small. It follows by [10, Theorem 1] that there exists constant
$c_{29}>0$ such that

$\int_{\Omega}\varphi^{2}\rho^{-2}dx\leq
c_{29}\int_{\Omega}|\nabla\varphi|^{2}dx,\quad\forall\varphi\in
H^{1}_{0}(\Omega).$ (4.8)

For $\lambda<\lambda_{*}$, it follows from (4.7) and (4.8) that

$\int_{\Omega}\frac{2\lambda\varphi^{2}}{(a-u_{\lambda})^{3}}dx\leq 2\lambda
c_{28}\int_{\Omega}\varphi^{2}\rho^{-2}dx\leq 2\lambda
c_{28}c_{29}\int_{\Omega}|\nabla\varphi|^{2}dx,\quad\forall\varphi\in
H^{1}_{0}(\Omega).$

For $\lambda>0$ small such that $2\lambda c_{28}c_{29}<1$, we obtain that

$\int_{\Omega}\frac{2\lambda\varphi^{2}}{(a-u_{\lambda})^{3}}dx<\int_{\Omega}|\nabla\varphi|^{2}dx,\quad\forall\varphi\in
H_{0}^{1}(\Omega)\setminus\\{0\\},$

that is, $\mu_{1}(\lambda)>0$ and $u_{\lambda}$ is stable if $\lambda>0$
small.

Finally, we prove that $u_{\lambda}$ is stable for
$\lambda\in(0,\lambda_{*})$. We have obtained that $\mu_{1}(\lambda)>0$ for
$\lambda>0$ small and the mapping $\lambda\mapsto\mu_{1}(\lambda)$ is locally
Lipschitz continuous, strictly decreasing in $(0,\lambda_{*})$, so if there
exists $\lambda_{0}\in(0,\lambda_{*})$ such that $\mu_{1}(\lambda_{0})=0$,
then $\mu_{1}(\lambda)>0$ for $\lambda\in(0,\lambda_{0})$. Letting
$\lambda_{1}\in(\lambda_{0},\lambda_{*})$, the minimal solution
$u_{\lambda_{1}}$ is a super solution of

$\left\\{\begin{array}[]{lll}-\Delta u=\frac{\lambda_{0}}{(a-u)^{2}}&{\rm
in}\quad\ \Omega,\\\\[5.69054pt] \phantom{-\Delta}u=0&{\rm
on}\quad\partial\Omega\end{array}\right.$

and it infers from Lemma 4.1 that

$u_{\lambda_{1}}=u_{\lambda_{0}},$

which contradicts that the mapping $\lambda\mapsto u_{\lambda}$ is strictly
increasing. Therefore, $\mu_{1}(\lambda)>0$ and $u_{\lambda}$ is stable for
$\lambda\in(0,\lambda_{*})$. $\Box$

###  4.3. Particular case that $\gamma=\frac{2}{3}$

In this subsection, we study further on properties of the minimal solution
when $\gamma=\frac{2}{3}$.

######  Lemma 4.2.

Assume that $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$ satisfies (1.3) and
(1.5) with $c_{0}\geq\kappa>0$, $\gamma=\frac{2}{3}$. Then
$\lambda^{*}=\lambda_{*}$.

Proof. For $\lambda\in(0,\lambda^{*})$, by (2.10) and Lemma 2.1, it implies
that

$\displaystyle a-u_{\lambda}$ $\displaystyle\geq$
$\displaystyle(\lambda^{*}-\lambda)\mathbb{G}_{\Omega}[a^{-2}]$
$\displaystyle\geq$ $\displaystyle
c_{30}(\lambda^{*}-\lambda)\rho^{\gamma}\geq c_{31}(\lambda^{*}-\lambda)a,$

then there exists $\theta\in(0,1)$ such that $u_{\lambda}\leq\theta a$ in
$\Omega$. It follows by Lemma 2.2 and the definition of $\lambda_{*}$ that
$\lambda_{*}=\lambda^{*}$. $\Box$

######  Proposition 4.3.

Assume that $a\in C^{\gamma}(\Omega)\cap C(\bar{\Omega})$ satisfies (1.3) and
(1.5) with $c_{0}\geq\kappa>0$, $\gamma=\frac{2}{3}$ and $u_{\lambda^{*}}$ is
given by (1.6). Then $u_{\lambda^{*}}$ is a semi-stable weak solution of (1.4)
with $\lambda=\lambda^{*}$.

Proof. When $\gamma=\frac{2}{3}$, by Lemma 4.2 and Proposition 4.2, we have
that $u_{\lambda}$ is stable for $\lambda\in(0,\lambda^{*})$, that is,

$\int_{\Omega}\frac{2\lambda\varphi^{2}}{(a-u_{\lambda})^{3}}dx<\int_{\Omega}|\nabla\varphi|^{2}dx,\quad\
\forall\varphi\in H^{1}_{0}(\Omega)\setminus\\{0\\}.$

Let $\varphi=\mathbb{G}_{\Omega}[1]$, we have that

$\int_{\Omega}\frac{\rho^{2}}{(a-u_{\lambda})^{3}}dx<c_{39}\lambda^{-1}.$

Since the mapping $\lambda\mapsto\frac{\rho^{2}}{(a-u_{\lambda})^{3}}$ is
strictly increasing and bounded in $L^{1}(\Omega)$, then

$\frac{\rho^{2}}{(a-u_{\lambda})^{3}}\to\frac{\rho^{2}}{(a-u_{\lambda^{*}})^{3}}\quad{\rm
as}\quad\lambda\to\lambda^{*}\quad{\rm in}\quad L^{1}(\Omega)$

and for any $\varphi\in C_{c}^{2}(\Omega)$,

$\lim_{\lambda\to\lambda^{*}}\int_{\Omega}\frac{\lambda\varphi^{2}}{(a-u_{\lambda})^{3}}dx=\int_{\Omega}\frac{\lambda^{*}\varphi^{2}}{(a-u_{\lambda^{*}})^{3}}dx$

Therefore,

$\int_{\Omega}\frac{2\lambda^{*}\varphi^{2}}{(a-u_{\lambda^{*}})^{3}}dx\leq\int_{\Omega}|\nabla\varphi|^{2}dx,\quad\forall\varphi\in
C_{c}^{2}(\Omega),$

by the fact that $C_{c}^{2}(\Omega)$ is dense in $H^{1}_{0}(\Omega)$, then
$u_{\lambda^{*}}$ is semi-stable. $\Box$

We next improve the regularity of $u_{\lambda^{*}}$ and prove when $1\leq
N\leq 7$, the extremal solution $u_{\lambda^{*}}$ is a classical solution of
(1.4) with $\lambda=\lambda^{*}$. To this end, we need the following Lemma,
which is inspired by [4].

######  Lemma 4.3.

Assume that $\lambda\in(0,\lambda^{*})$, $\Omega=B_{1}(0)$ and
$a(x)=\kappa(1-|x|^{2})^{\frac{2}{3}}$ with $\kappa>0$. Let $u$ be a weak
solution of (1.4) satisfying for any compact set $K\subset B_{1}(0)$, there
exists $c_{32}>0$ such that

$\lVert\frac{1}{a-u}\rVert_{L^{\frac{3N}{2}}(K)}\leq c_{32}.$ (4.9)

Then there exists $c_{33}>0$ depending on $K$ such that

$\inf_{x\in K}(a(x)-u(x))>c_{33}.$ (4.10)

Proof. By (4.9), we have that

$\frac{1}{(a-u)^{2}}\in L^{\frac{3N}{4}}(K)$

and then $u\in W^{2,\frac{3N}{4}}(K)$ and by Sobolev’s Theorem, we deduce that
$u\in C^{\frac{2}{3}}(K^{\prime})$ with $K^{\prime}$ compact set in interior
point set of $K$. We next show that $u<a$ in $B_{1}(0)$. Indeed, if not, there
exists $x_{0}\in B_{1}(0)$ such that $u(x_{0})=a(x_{0})$. For compact set
$K\subset B_{1}(0)$ containing $x_{0}$, we have that

$\displaystyle|a(x)-u(x)|$ $\displaystyle\leq$
$\displaystyle|a(x)-a(x_{0})|+|u(x)-u(x_{0})|+|u(x_{0})-a(x_{0})|$
$\displaystyle=$ $\displaystyle|a(x)-a(x_{0})|+|u(x)-u(x_{0})|\leq
c_{34}|x-x_{0}|^{\frac{2}{3}},$

then

$\int_{K}\frac{1}{(a-u)^{\frac{3N}{2}}}dx\geq
c_{35}\int_{K}|x-x_{0}|^{-\frac{3N}{2}\cdot\frac{2}{3}}dx=\infty,$

which contradicts (4.9). Therefore, (4.10) holds. $\Box$

######  Proposition 4.4.

Assume that $1\leq N\leq 7$, $\Omega=B_{1}(0)$, the function $a$ satisfies
(3.1) with $\kappa>0$ and $\gamma=\frac{2}{3}$, $u_{\lambda^{*}}$ is given by
(1.6). Then $u_{\lambda^{*}}$ is a classical solution of (1.4) with
$\lambda=\lambda^{*}$.

Proof. Since the mapping $\lambda\mapsto u_{\lambda}$ is strictly increasing
and bounded by $a$, then from (1.5) and Lemma 4.3, we only have to improve the
regularity of $u_{\lambda^{*}}$ in any compact set of $B_{1}(0)$. For
$\lambda\in(0,\lambda^{*})$, we know that $u_{\lambda}$ is stable, then

$\int_{B_{1}(0)}\frac{2\lambda\varphi^{2}}{(a-u_{\lambda})^{3}}dx<\int_{\Omega}|\nabla\varphi|^{2}dx,\quad\forall\varphi\in
H^{1}_{0}(B_{1}(0))\setminus\\{0\\}.$ (4.11)

We claim that the minimal solutions $u_{\lambda}$ is radially symmetric for
$\lambda\in(0,\lambda^{*}]$. Indeed, the minimal solution $u_{\lambda}$ could
be approximated by the sequence of functions

$v_{n}=\lambda\mathbb{G}_{B_{1}(0)}[\frac{1}{(a-v_{n-1})^{2}}]\ \ \ {\rm
with}\ \ v_{0}=0.$

It follows by radially symmetry of $v_{n-1}$ and $a$ that $v_{n}$ is radially
symmetry and then $u_{\lambda}$ is radially symmetric. Then $u_{\lambda^{*}}$
is radially symmetric.

By (4.2), there exists a sequence $\\{r_{n}\\}_{n}\subset(0,1)$ such that

$\lim_{n\to+\infty}r_{n}=1\quad{\rm and}\quad
a(r_{n})-u_{\lambda^{*}}(r_{n})>0.$

Otherwise, there exists $r\in(0,1)$ such that

$a-u_{\lambda^{*}}=0\quad{\rm a.e.\ in}\ \ B_{1}(0)\setminus{B_{r}(0)},$

which contradicts (4.2).

Let us denote

$\varphi_{\theta}=\left\\{\begin{array}[]{lll}(a-u_{\lambda})^{\theta}-\epsilon_{\lambda}^{\theta}&{\rm
in}\quad B_{r_{n}}(0),\\\\[4.2679pt] \phantom{}0&{\rm in}\quad
B_{1}(0)\setminus B_{r_{n}}(0),\end{array}\right.$

where $\theta\in(-2-\sqrt{6},0)$ and
$\epsilon_{\lambda}=a(r_{n})-u_{\lambda}(r_{n})$. We observe that
$\varphi_{\theta}\in H_{0}^{1}(B_{1}(0))$. It follows by (4.11) with
$\varphi_{\theta}$, we have that

$\displaystyle\int_{B_{r_{n}}(0)}\frac{2\lambda[(a-u_{\lambda})^{\theta}-\epsilon_{\lambda}^{\theta}]^{2}}{(a-u_{\lambda})^{3}}dx$
$\displaystyle\leq$
$\displaystyle\int_{B_{r_{n}}(0)}|\nabla((a-u_{\lambda})^{\theta})|^{2}dx$
(4.12) $\displaystyle=$
$\displaystyle\theta^{2}\int_{B_{r_{n}}(0)}(a-u_{\lambda})^{2\theta-2}|\nabla(a-u_{\lambda})|^{2}dx.$

Since $u_{\lambda}$ is the minimal solution of (1.4), then

$-\Delta(a-u_{\lambda})=-\Delta a-\frac{\lambda}{(a-u_{\lambda})^{2}}\quad{\rm
in}\ \ B_{r_{n}}(0).$ (4.13)

Multiplying (4.13) by
$\frac{\theta^{2}}{1-2\theta}[(a-u_{\lambda})^{2\theta-1}-\epsilon_{\lambda}^{2\theta-1}]$
and applying integration by parts yields that

$\displaystyle\frac{\theta^{2}}{1-2\theta}\int_{B_{r_{n}}(0)}[-\Delta
a-\frac{\lambda}{(a-u_{\lambda})^{2}}][(a-u_{\lambda})^{2\theta-1}-\epsilon_{\lambda}^{2\theta-1}]dx$
$\displaystyle\qquad=\frac{\theta^{2}}{1-2\theta}\int_{B_{r_{n}}(0)}\nabla(a-u_{\lambda})\cdot\nabla\left((a-u_{\lambda})^{2\theta-1}\right)dx$
$\displaystyle\qquad\qquad=-\theta^{2}\int_{B_{r_{n}}(0)}(a-u_{\lambda})^{2\theta-2}|\nabla(a-u_{\lambda})|^{2}dx,$

then together with (4.12), we deduce that

$\displaystyle\int_{B_{r_{n}}(0)}\frac{2\lambda[(a-u_{\lambda})^{\theta}-\epsilon_{\lambda}^{\theta}]^{2}}{(a-u_{\lambda})^{3}}dx\leq\frac{\theta^{2}}{1-2\theta}\int_{B_{r_{n}}(0)}[\Delta
a+\frac{\lambda}{(a-u_{\lambda})^{2}}][(a-u_{\lambda})^{2\theta-1}-\epsilon_{\lambda}^{2\theta-1}]dx,$

thus,

$\displaystyle\lambda(2-\frac{\theta^{2}}{1-2\theta})\int_{B_{r_{n}}(0)}\frac{1}{(a-u_{\lambda})^{3-2\theta}}dx\leq\int_{B_{r_{n}}(0)}\frac{4\lambda\epsilon_{\lambda}^{\theta}}{(a-u_{\lambda})^{3-\theta}}dx-\int_{B_{r_{n}}(0)}\frac{2\lambda\epsilon_{\lambda}^{2\theta}}{(a-u_{\lambda})^{3}}dx$
$\displaystyle\ \ +\frac{\theta^{2}}{1-2\theta}\int_{B_{r_{n}}(0)}\frac{\Delta
a}{(a-u_{\lambda})^{1-2\theta}}dx-\frac{\theta^{2}\epsilon_{\lambda}^{2\theta-1}}{1-2\theta}\int_{B_{r_{n}}(0)}\Delta
adx-\frac{\theta^{2}\lambda}{1-2\theta}\int_{B_{r_{n}}(0)}\frac{\epsilon_{\lambda}^{2\theta-1}}{(a-u_{\lambda})^{2}}dx.$

Since the mapping $\lambda\mapsto u_{\lambda}$ is strictly increasing, we have
that

$\epsilon_{\lambda}=a(r_{n})-u_{\lambda}(r_{n})\geq
a(r_{n})-u_{\lambda^{*}}(r_{n}):=\varepsilon_{n}>0$

and it infers by $a(x)=\kappa(1-|x|^{2})^{\frac{2}{3}}$,

$|\Delta a|\leq C_{n}\quad{\rm in}\quad B_{r_{n}}(0)$

for some $C_{n}>0$, then letting $\theta\in(-2-\sqrt{6},0)$, we have that
$2-\frac{\theta^{2}}{1-2\theta}>0$ and by Hölder inequality, we obtain that

$\displaystyle\lambda(2-\frac{\theta^{2}}{1-2\theta})\int_{B_{r_{n}}(0)}\frac{1}{(a-u_{\lambda})^{3-2\theta}}dx$
$\displaystyle\leq\int_{B_{r_{n}}(0)}\frac{4\lambda\epsilon_{\lambda}^{\theta}}{(a-u_{\lambda})^{3-\theta}}dx+\frac{\theta^{2}}{1-2\theta}\int_{B_{r_{n}}(0)}\frac{C_{n}}{(a-u_{\lambda})^{1-2\theta}}dx+\frac{\theta^{2}\varepsilon_{n}^{2\theta-1}}{1-2\theta}C_{n}|B_{r_{n}}(0)|$
$\displaystyle\leq
4\lambda^{*}\varepsilon_{n}^{\theta}|B_{1}(0)|^{\frac{-\theta}{3-2\theta}}\left(\int_{B_{r_{n}}(0)}\frac{1}{(a-u_{\lambda})^{3-2\theta}}dx\right)^{\frac{3-\theta}{3-2\theta}}$
$\displaystyle\quad+C_{n}|B_{1}(0)|^{\frac{2}{3-2\theta}}\left(\int_{B_{r_{n}}(0)}\frac{1}{(a-u_{\lambda})^{3-2\theta}}dx\right)^{\frac{1-2\theta}{3-2\theta}}+\frac{\theta^{2}\varepsilon_{n}^{2\theta-1}}{1-2\theta}C_{n}|B_{1}(0)|$

thus, there exists $c_{44}>0$ independent on $\lambda$ such that

$\int_{B_{r_{n}}(0)}\frac{1}{(a-u_{\lambda})^{3-2\theta}}dx\leq c_{44}.$
(4.14)

When $1\leq N\leq 7$, $\frac{3N}{2}\leq 3-2\theta$ for some
$\theta\in(-2-\sqrt{6},0)$, then by Lemma 4.3, we have that $u_{\lambda}$ has
uniformly in $C^{2,\beta}_{loc}(B_{1}(0))$ with $\beta<\gamma$, then
$u_{\lambda^{*}}$ is a classical solution of (1.4) with $\lambda=\lambda^{*}$
and $a-u_{\lambda^{*}}>0$ in $B_{1}(0)$. $\Box$

Proof of Theorem 1.3. Proposition 4.1 shows that $u_{\lambda^{*}}$ is a weak
solution of (1.4) with $\lambda=\lambda^{*}$. The stability of $u_{\lambda}$
follows by Proposition 4.2 for $\lambda\in(0,\lambda_{*})$. When
$\gamma=\frac{2}{3}$, the stability and regularity of $u_{\lambda^{*}}$ see
Proposition 4.3 and Proposition 4.4. $\Box$

Acknowledgements: H. Chen is supported by NSFC, No: 11401270. F. Zhou is
supported by NSFC, No: 11271133.

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