# A Nonlinear elliptic PDE involving two Hardy-Sobolev critical exponents in
domains with curve singularity

El Hadji Abdoulaye THIAM  H. E. A. T. : Université Iba Der Thiam de Thies, UFR
des Sciences et Techniques, département de mathématiques, Thies.
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$ and
$\Gamma$ be a closed curve contained in $\Omega$. We study existence of
positive solutions $u\in H^{1}_{0}\left(\Omega\right)$ to the equation

$-\Delta
u+hu=\lambda\rho^{-s_{1}}_{\Gamma}u^{2^{*}_{s_{1}}-1}+\rho^{-s_{2}}_{\Gamma}u^{2^{*}_{s_{2}}-1}\qquad\textrm{
in }\Omega$ (0.1)

where $h$ is a continuous function and $\rho_{\Gamma}$ is the distance
function to $\Gamma$. We prove the existence of a mountain pass solution for
this Euler-Lagrange equation depending on the local geometry of the curve and
the potential $h$.
In this paper, we also study existence, symmetry and decay estimates of the
positive entire solutions of (0.1) with $\Omega=\mathbb{R}^{N}$ and $\Gamma$
the real line.

Key Words: Two Hardy-Sobolev exponents; Curvature; Positive mountain Pass
solution; Curve singularity.

##  1\. Introduction

For $N\geq 3$, the famous Caffarelli-Kohn-Nirenberg inequality asserts that:
there exists a positive constant $C=C_{N,a,b}$ only depending on $N,a,b$, such
that

$C\left(\int_{\mathbb{R}^{N}}|x|^{-bq}|u|^{q}dx\right)^{2/q}\leq\int_{\mathbb{R}^{N}}|x|^{-2a}|\nabla
u|^{2}dx\qquad\forall u\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{N}),$ (1.1)

where $N\geq 3$, $-\infty<a<\frac{N-2}{2},0\leq b-a\leq 1$ and
$q=\frac{2N}{N-2+2(b-a)},$ see for instance [9]. Note that the case $b=a+1$
and $p=2$, (1.1) corresponds to the following Hardy inequality:

$\displaystyle\left(\frac{N-2}{2}\right)^{2}\int_{\mathbb{R}^{N}}|x|^{-2}|u|^{2}dx\leq\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx\qquad\forall u\in\mathcal{D}^{1,2}({\mathbb{R}^{N}}),$ (1.2)

where $\mathcal{D}^{1,2}({\mathbb{R}^{N}})$ denotes the completion of
$\mathcal{C}^{\infty}_{c}(\mathbb{R}^{N})$ with respect to the norm

$u\longmapsto\sqrt{\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx}.$

The constant $\left(\displaystyle\frac{N-2}{2}\right)^{2}$ is sharp and never
achieved in $\mathcal{D}^{1,2}({\mathbb{R}^{N}})$. The case $a=b=0$ and
$p=\frac{2N}{N-2}$corresponds to the famous Sobolev inequality:

$S_{N,0}\left(\int_{\mathbb{R}^{N}}|u|^{2^{*}}dx\right)^{2/2^{*}}\leq\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx\qquad\forall u\in\mathcal{D}^{1,2}({\mathbb{R}^{N}}),$ (1.3)

where the best constant

$S_{N,0}=\displaystyle\frac{N(N-2)}{4}\omega_{N}^{2/N}$

is achieved in $\mathcal{D}^{1,2}({\mathbb{R}^{N}})$. Here
$\omega_{N}=|S^{N-1}|$ is the volume of the N-sphere and $\displaystyle
2^{*}:=2^{*}(0)=\frac{2N}{N-2}$ is the critical Sobolev exponent. By Hölder’s
inequality, we get the interpolation between the Hardy and the Sobolev
inequalities, called Hardy-Sobolev inequality given by

$S_{N,s}\left(\int_{\mathbb{R}^{N}}|x|^{-s}|u|^{2^{*}(s)}dx\right)^{2/2^{*}(s)}\leq\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx\qquad\forall u\in\mathcal{D}^{1,2}({\mathbb{R}^{N}}),$ (1.4)

where for $s\in[0,2]$, we have $\displaystyle 2^{*}(s)=\frac{2(N-s)}{N-2}$ is
the critical Hardy-Sobolev exponent. We refer to [15] for more details about
Hardy-Sobolev inequality. The value of the best constant is

$S_{N,s}:=(N-2)(N-s)\left[\frac{w_{N-1}}{2-s}\frac{\Gamma^{2}(N-\frac{s}{2-s})}{\Gamma(\frac{2(N-s)}{2-s})}\right]^{\frac{2-s}{N-s}},$

where $\Gamma$ is the Gamma Euler function. It was computed by Lieb [27] when
$s\in(0,2)$. The ground state solution is given, up to dilation, by

$w(x)=C_{N,s}(1+|x|^{2-s})^{\frac{2-N}{2-s}},$

for some positive known constant $C_{N,s}$.

The Caffarelli-Kohn-Nirenberg’s inequality on domains and related problems
have been studied these last years. For instance, we let $\Omega$ be a domain
of $\mathbb{R}^{N}$ and consider the equation

$\begin{cases}-\text{div}(|x|^{-2a}\nabla u)=|x|^{-bq}u^{q-1},\quad
u>0\qquad&\textrm{ in $\Omega$}\\\ \ u=0&\textrm{ on
$\partial\Omega$}.\end{cases}$ (1.5)

To study (1.5), one could let $w(x)=|x|^{-a}u(x)$. Direct computations show
that

$\int_{\Omega}|x|^{-2a}|\nabla u|^{2}dx=\int_{\Omega}|\nabla
w|^{2}dx-a(n-2-a)\int_{\Omega}|x|^{-2}w^{2}dx.$

Then solutions of (1.5) can be obtained by minimizing the following quotient

$S^{N}_{a,b}(\Omega):=\inf_{u\in\mathcal{D}^{1,2}_{a}(\Omega)\setminus\\{0\\}}\frac{\displaystyle\int_{\Omega}|\nabla
w|^{2}dx-a(n-2-a)\int_{\Omega}|x|^{-2}w^{2}dx}{\left(\displaystyle\int_{\Omega}|x|^{-bq}|u|^{q}dx\right)^{2/q}},$
(1.6)

where $\mathcal{D}^{1,2}_{a}(\Omega)$ be the completion of
$\mathcal{C}^{\infty}_{c}(\Omega)$ with respect to the norm

$u\longmapsto\sqrt{\int_{\Omega}|x|^{-2a}|\nabla u|^{2}dx}.$

The question related to the attainability of the best constant
$S^{N}_{a,b}(\Omega)$ in (1.6) is studied by many authors. For more
developments related to that, we refer the readers to [4, 5, 8, 10, 11, 13,
15, 16, 23, 27, 29, 30] and references therein.

When $0\in\partial\Omega$, the existence of minimizers for
$S^{N}_{a,b}(\Omega)$ was first studied by Ghoussoub-Kang [15] and Ghoussoub-
Robert[16]. Later Chern and Lin [10] proved the existence of minimizer
provided the mean curvature of the boundary at the origin is negative and
($a<b<a+1$ and $N\geq 3$) or ($b=a>0$ and $N\geq 4$). The case $a=0$ and
$0<b<1$ was first studied by [16] before the generalization in [10]. More
generally questions related to Partial Differential Equations involving
multiples Hardy-Sobolev critical exponents have been investigated these last
decades. In particular, we let $\Omega$ be a domain of $\mathbb{R}^{N}$ such
that $0\in\partial\Omega$ and consider the equation

$\begin{cases}\displaystyle-\Delta
u=\lambda\frac{u^{2^{*}_{s_{1}}-1}(x)}{|x|^{s_{1}}}+\frac{u^{2^{*}_{s_{2}}-1}}{|x|^{s_{2}}}\qquad&\textrm{
in $\Omega$}\\\ \ u(x)>0&\textrm{ in $\Omega$},\end{cases}$ (1.7)

where $0\leq s_{2}<s_{1}<2$, $\lambda\in\mathbb{R}$ and for $i=1,2$,
$2^{*}_{s_{1}}:=\frac{2(N-s_{i})}{N-2}$ it the critical Hard-Sobolev exponent.
When $s_{2}=0$ and $\lambda<0$, then equation (1.7) has no nontrivial
solution. For $\lambda>0$, $0<s_{1}<2$ and $s_{2}=0$, then using variational
methods, Hsia Lin and Wadade [25] proved existence of solutions provided
$N\geq 4$ and the mean curvature at the origin is negative. For the case
$N=3$, $\lambda\in\mathbb{R}$ and $0<s_{2}<s_{1}<2$, the equation (1.7) has a
least-energy solution provided the mean curvature at the origin is negative,
see [28].

Concerning the existence and non-existence of solution related to equation
(1.7) in the half-space $\Omega=\mathbb{R}^{N}_{+}$, we refer to Bartsch-Peng
and Zhang [4] for the case $0<s_{2}<s_{1}=2$ and
$\lambda<\left(\frac{N-2}{2}\right)^{2}$; to Musina [31] when $N\geq 4$,
$s_{2}=0$, $s_{1}=2$ and $0<\lambda<\left(\frac{N-2}{2}\right)^{2}$ and to
Hsia, Lin and Wadade [25] when $s_{2}=0$, $0<s_{1}<2$ and $\lambda>0$.

In this paper, we are concerned with the effect of the local geometry of the
singularity $\Gamma$ in the existence of solutions of the following non-linear
partial differential equation involving two Hardy-Sobolev critical exponents.
More precisely, letting $h$ be a continuous function and $\lambda$ be a real
parameter, we consider

$\begin{cases}\displaystyle-\Delta
u(x)+hu(x)=\lambda\frac{u^{2^{*}_{s_{1}}-1}(x)}{\rho_{\Gamma}^{s_{1}}(x)}+\frac{u^{2^{*}_{s_{2}}-1}(x)}{\rho_{\Gamma}^{s_{2}}(x)}\qquad&\textrm{
in $\Omega$}\\\ \\\ u(x)>0\qquad\textrm{ and }\qquad u(x)=0&\textrm{ on
$\partial\Omega$},\end{cases}$ (1.8)

where $\rho_{\Gamma}(x):=\inf_{y\in\Gamma}|y-x|$ is the distance function to
the curve $\Gamma$, $0<s_{2}<s_{1}<2$, $2^{*}_{s_{1}}:=\frac{2(N-s_{1})}{N-2}$
and $2^{*}_{s_{2}}:=\frac{2(N-s_{2})}{N-2}$ are two critical Hardy-Sobolev
exponents. To study the equation (1.8), we consider the following non-linear
functional $\Psi:H^{1}_{0}(\Omega)\to\mathbb{R}$ defined by:

$\Psi(u):=\frac{1}{2}\int_{\Omega}|\nabla
u|^{2}dx+\frac{1}{2}\int_{\Omega}h(x)u^{2}dx-\frac{\lambda}{2^{*}_{s_{1}}}\int_{\Omega}\frac{|u|^{2^{*}_{s_{1}}}}{\rho_{\Gamma}^{s_{1}}(x)}dx-\frac{1}{2^{*}_{s_{2}}}\int_{\Omega}\frac{|u|^{2^{*}_{s_{2}}}}{\rho_{\Gamma}^{s_{2}}(x)}dx.$
(1.9)

It is easy to verify that there exists a positive constant $r>0$ and $u_{0}\in
H^{1}_{0}(\Omega)$ such that $\|u_{0}\|_{H^{1}_{0}(\Omega)}>r$ and

$\inf_{\|u\|_{H^{1}_{0}(\Omega)}=r}\Psi(u)>\Psi(0)\geq\Phi(u_{0}),$

see for instance Lemma 4.5 below. Then the point $(0,\Psi(0))$ is separated
from the point $(u_{0},\Psi(u_{0}))$ by a ring of mountains. Set

$c^{*}:=\inf_{P\in\mathcal{P}}\max_{v\in P}\Psi(v),$ (1.10)

where $\mathcal{P}$ is the class of continuous paths in $H^{1}_{0}(\Omega)$
connecting $0$ to $u_{0}$. Since $2^{*}_{s_{2}}>2^{*}_{s_{1}}$, the function
$t\longmapsto\Psi(tv)$ has the unique maximum for $t\geq 0$. Furthermore, we
have

$c^{*}:=\inf_{u\in H^{1}_{0}(\Omega),u\geq 0,u\neq 0}\max_{t\geq 0}\Psi(tu).$

Due to the fact that the embedding of $H^{1}_{0}(\Omega)$ into the weighted
Lebesgue spaces $L^{2^{*}_{si}}(\rho_{\Gamma}^{-si}dx)$ is not compact, the
functional $\Psi$ does not satisfy the Palais-Smale condition. Therefore, in
general $c^{*}$ might not be a critical value for $\Psi$.

To recover compactness, we study the following non-linear problem: let
$x=(y,z)\in\mathbb{R}\times\mathbb{R}^{N-1}$ and consider

$\begin{cases}\displaystyle-\Delta
u=\lambda\frac{u^{2^{*}_{s_{1}}-1}(x)}{|z|^{s_{1}}}+\frac{u^{2^{*}_{s_{2}}-1}}{|z|^{s_{2}}}\qquad&\textrm{
in $\mathbb{R}^{N}$}\\\ \ u(x)>0&\textrm{ in $\mathbb{R}^{N}$}.\end{cases}$
(1.11)

To obtain solutions of (1.11), we consider the functional
$\Phi:\mathcal{D}^{1,2}(\mathbb{R}^{N})$ defined by

$\Phi(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\lambda}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|z|^{-s_{1}}|u|^{2^{*}_{s_{1}}}dx-\frac{1}{2^{*}_{s_{2}}}\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|u|^{2^{*}_{s_{2}}}dx.$

Next, we define

$\beta^{*}:=\inf_{u\in D^{1,2}(\mathbb{R}^{N}),u\geq 0,u\neq 0}\max_{t\geq
0}\Phi(tu).$

Then we get compactness provided

$c^{*}<\beta^{*},$

see Proposition 4.3 below. So it is important to study existence, symmetry and
decay estimates of non-trivial solution
$w\in\mathcal{D}^{1,2}(\mathbb{R}^{N})$ of (1.11). Then we have the following
results.

######  Theorem 1.1.

Let $N\geq 3$, $0\leq s_{2}<s_{1}<2$, $\lambda\in\mathbb{R}$. Then equation

$\begin{cases}\displaystyle-\Delta
u=\lambda\frac{u^{2^{*}_{s_{1}}-1}(x)}{|z|^{s_{1}}}+\frac{u^{2^{*}_{s_{2}}-1}}{|z|^{s_{2}}}\qquad&\textrm{
in $\mathbb{R}^{N}$}\\\ \ u(x)>0&\textrm{ in $\mathbb{R}^{N}$}\end{cases}$
(1.12)

has a positive ground state solution $w\in\mathcal{D}^{1,2}(\mathbb{R}^{N})$.
Moreover $w$ depend only on $|y|$ and $|z|$. In other words, there exists a
function $\theta:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ such
that

$w(x)=\theta(|y|,|z|).$

Next we have the following decay estimates of the solution $w$ and its higher
order derivatives.

######  Theorem 1.2.

Let $w$ be a solution of the Euler-Lagrange equation (1.12). Then

  * (i)

there exists two positive constants $c_{1}<c_{2}$ such that:

$\frac{c_{1}}{1+|x|^{N-2}}\leq u(x)\leq\frac{c_{2}}{1+|x|^{N-2}},\qquad\forall
x\in\mathbb{R}^{N}.$

  * (ii)

For $|x|=|(t,z)|\leq 1$

$|\nabla w(x)|+|x||D^{2}w(x)|\leq C_{2}|z|^{1-s_{1}}$

  * (iii)

For $|x|=|(t,z)|\geq 1$

$|\nabla w(x)|+|x||D^{2}w(x)|\leq C_{2}\max(1,|z|^{-s_{1}})|x|^{1-N}.$

These two theorems will play a crucial role in the following which is our main
result. Then we have

######  Theorem 1.3.

Let $N\geq 4$, $0\leq s_{2}<s_{1}<2$ and $\Omega$ be a bounded domain of
$\mathbb{R}^{N}$. Consider $\Gamma$ a smooth closed curve contained in
$\Omega$. Let $h$ be a continuous function such that the linear operator
$-\Delta+h$ is coercive. Then there exists a positive constant
$A^{N}_{s_{1},s_{2}}$, only depending on $N$, $s_{1}$ and $s_{2}$ with the
property that if there exists $y_{0}\in\Gamma$ such that

$A^{N}_{s_{1},s_{2}}|\kappa(y_{0})|^{2}+h(y_{0})<0\qquad\textrm{ for $N\geq
4$},$ (1.13)

then $c^{*}<\beta^{*}$, where $\kappa:\Gamma\to\mathbb{R}^{N}$ is the
curvature vector of $\Gamma$. Moreover there exists $u\in
H^{1}_{0}(\Omega)\setminus\\{0\\}$ non-negative solution of

$-\Delta
u(x)+hu(x)=\lambda\frac{u^{2^{*}_{s_{1}}-1}(x)}{\rho_{\Gamma}^{s_{1}}(x)}+\frac{u^{2^{*}_{s_{2}}-1}(x)}{\rho_{\Gamma}^{s_{2}}(x)}\qquad\textrm{
in $\Omega$}.$

The effect of curvatures in the study of Hardy-Sobolev inequalities have been
intensively studied in the recent years. For each of these works, the sign of
the curvatures at the point of singularity plays important roles for the
existence a solution. The first paper, to our knowledge, being the one of
Ghoussoub and Kang [15] who considered the Hardy-Sobolev inequality with
singularity at the boundary. For more results in this direction, see the works
of Ghoussoub and Robert in [17, 18, 19, 20], Demyanov and Nazarov [12], Chern
and Lin [10], Lin and Li [28], the author, Fall and Minlend in [14] and the
references there in. The Hardy-Sobolev inequality with interior singularity on
Riemannian manifolds have been studied by Jaber [26] and the author [34]. Here
also the impact of the scalar curvature at the point singularity plays an
important role for the existence of minimizers in higher dimensions $N\geq 4$.
The paper [26] contains also existence result under positive mass condition
for $N=3$. We point out theat the 3-dimensional version of this paper is
presented in [36]. The existence of solution does not depends on the local
geometry of the singularity but on the regular part of the Green function of
the operator $-\Delta+h$.

The proof of Theorem 1.3 relies on test function methods. Namely to build
appropriate test functions allowing to compare $c^{*}$ and $\beta^{*}$. While
it always holds that $c^{*}\leq\beta^{*}$, our main task is to find a function
for which $c^{*}<\beta^{*}$, see Section 5. This then allows to recover
compactness and thus every minimizing sequence for $c^{*}$ converges to a
minimizer, up to a subsequence. Building these approximates solutions requires
to have sharp decay estimates of a minimizer $w$ for $\beta^{*}$, see Section
2. Section 3 is devoted to the local parametrization and computation of the
local metric.

##  2\. Proof of Theorem 1.1 and Theorem 1.2

######  Theorem 2.1.

Let $N\geq 3$, $x:=(y,z)\in\mathbb{R}\times\mathbb{R}^{N-1}$,
$0<s_{2}<s_{1}<2$ and $\lambda\in\mathbb{R}$. Then there exists
$w\in\mathcal{D}^{1,2}(\mathbb{R}^{N})$ positive, satisfying

$-\Delta
w=\lambda\frac{w^{2^{*}_{s_{1}}-1}}{|z|^{s_{1}}}+\frac{w^{2^{*}_{s_{2}}-1}}{|z|^{s_{2}}}\qquad\textrm{
in }\mathbb{R}^{N}.$

###### Proof.

By Ekland variational principle there exits a minimizing sequence
$(w_{n})_{n}$ for $\beta^{*}$ such that

$\beta^{*}=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
w_{n}|^{2}dx-\frac{\lambda}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|z|^{-s_{1}}|w_{n}|^{2^{*}_{s_{1}}}dx-\frac{1}{2^{*}_{s_{2}}}\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|w_{n}|^{2^{*}_{s_{2}}}dx+o(1)$
(2.1)

and for $\eta\in\mathcal{C}^{\infty}(\mathbb{R}^{N})$,

$\int_{\mathbb{R}^{N}}\nabla w_{n}\cdot\nabla\eta
dx-\lambda\int_{\mathbb{R}^{N}}|z|^{-s_{1}}|w_{n}|^{2^{*}_{s_{1}}-2}w_{n}\eta
dx-\int_{\mathbb{R}^{N}}|z|^{-s_{1}}|w_{n}|^{2^{*}_{s_{1}}-2}w_{n}\eta
dx=o(1).$ (2.2)

By (2.1) and (2.2), we have

$\beta^{*}=\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\mathbb{R}^{N}}|\nabla
w_{n}|^{2}dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|w_{n}|^{2^{*}_{s_{2}}}dx+o(1)$

By continuity, there exists $r_{n}>0$, such that

$\frac{\beta^{*}}{2}:=\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{B_{r_{n}}}|\nabla
w_{n}|^{2}dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{B_{r_{n}}}|z|^{-s_{2}}|w_{n}|^{2^{*}_{s_{2}}}dx+o(1).$
(2.3)

Define

$v_{n}(x)=r_{n}^{\frac{N-2}{2}}w(r_{n}x).$

Then by the change of variable formula $\tilde{x}=r_{n}x$, it easy follows
that

$\int_{\mathbb{R}^{N}}|\nabla w_{n}|^{2}dx=\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx;\quad\int_{\mathbb{R}^{N}}|z|^{-s_{i}}|w_{n}|^{2^{*}_{s_{i}}}dx=\int_{\mathbb{R}^{N}}|z|^{-s_{i}}|v_{n}|^{2^{*}_{s_{i}}}dx,$

for all $i=1,2$. Moreover we have

$\int_{B_{r_{n}}}|\nabla w_{n}|^{2}dx=\int_{B_{1}}|\nabla
v_{n}|^{2}dx\qquad\textrm{and}\qquad\int_{B_{r_{n}}}|z|^{-s_{2}}|w_{n}|^{2^{*}_{s_{2}}}=\int_{B_{1}}|z|^{-s_{2}}|v_{n}|^{2^{*}_{s_{2}}}.$
(2.4)

Therefore $(v_{n})_{n}$ is also a minimizing sequence. In particular
${v_{n}}\rightharpoonup v$ for some $v$ in
$\mathcal{D}^{1,2}(\mathbb{R}^{N})$. We wish to show that $v\neq 0$. If not
then $v_{n}\to 0$ in $L^{2}_{loc}(\mathbb{R}^{N})$. Let $\varphi\in
C^{\infty}_{c}({B_{1}})$ such that $\varphi\equiv 1$ on $B_{\frac{1}{2}}$.
Using $\varphi^{2^{*}_{s_{1}}}v_{n}$ as test function in (2.2) and using
integration by parts, we obtain

$\displaystyle\beta^{*}-\frac{1}{2}$
$\displaystyle\int_{\mathbb{R}^{N}}|\nabla(\varphi
v_{n})|^{2}dx+\frac{1}{2^{*}_{s_{2}}}\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|\varphi
v_{n}|^{2^{*}_{s_{2}}}dx$
$\displaystyle\leq-\frac{1}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}\nabla
v_{n}\cdot\nabla(v_{n}\varphi^{2^{*}_{s_{1}}})dx+\frac{1}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|v_{n}|^{2^{*}_{s_{2}}-2}v^{2}_{n}\varphi^{2^{*}_{s_{1}}}dx+o(1)$
$\displaystyle=-\frac{1}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|\nabla(\varphi
v_{n})|dx+\frac{1}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|v_{n}|^{2^{*}_{s_{2}}-2}v^{2}_{n}\varphi^{2^{*}_{s_{1}}}dx+o(1)$
$\displaystyle\leq-\frac{1}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|\nabla(\varphi
v_{n})|dx+\frac{1}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|\varphi
v_{n}|^{2^{*}_{s_{2}}}dx+o(1).$

Therefore

$\beta^{*}\leq\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\mathbb{R}^{N}}|\nabla(\varphi
v_{n})|dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|\varphi
v_{n}|^{2^{*}_{s_{2}}}dx+o(1).$ (2.5)

Moreover by (2.3) and (2.4), we have

$\frac{\beta^{*}}{2}:=\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{B_{1}}|\nabla
w_{n}|^{2}dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{B_{1}}|z|^{-s_{2}}|w_{n}|^{2^{*}_{s_{2}}}dx+o(1).$
(2.6)

Hence combining (2.5) and (2.6), we obtain

$\displaystyle\beta^{*}$
$\displaystyle\leq\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\mathbb{R}^{N}}|\nabla(\varphi
v_{n})|^{2}dx+\left(\frac{1}{2^{*}_{s_{2}}}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|\varphi
v_{n}|^{2^{*}_{s_{2}}}dx+o(1)$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{B_{1}}|\nabla(\varphi
v_{n})|^{2}dx+\left(\frac{1}{2^{*}_{s_{2}}}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{B_{1}}|z|^{-s_{2}}|\varphi
v_{n}|^{2^{*}_{s_{2}}}dx+o(1)=\frac{\beta^{*}}{2}+o(1).$

Then taking the limit as $n\to\infty$, we obtain

$0<\beta^{*}\leq\frac{\beta^{*}}{2}.$

which is false. Therefore $v\neq 0$ is a minimizer. Standard arguments show
that $v^{+}=\max(v,0)$ is also a minimizer and the proof is completed. ∎

Next we will establish symmetry and decay estimates properties of positive
solutions $u\in\mathcal{D}^{1,2}(\mathbb{R}^{N})$ of the following Euler-
Lagrange equations

$\displaystyle-\Delta
u=\lambda\frac{u^{2^{*}_{s_{1}}-1}}{|z|^{s_{1}}}+\frac{u^{2^{*}_{s_{2}}-1}}{|z|^{s_{2}}}\quad\textrm{
in $\mathbb{R}^{N}$}$ (2.7)

where for $N\geq 3$, we have $x:=(y,z)\in\mathbb{R}\times\mathbb{R}^{N-1}$,
$0<s_{2}<s_{1}<2$ and $2^{*}_{s_{i}}:=\frac{2(N-s_{i})}{N-2}\quad(i=1,2)$ are
two Hardy-Sobolev critical exponents. Next, rewrite equation (2.7) as follows

$-\Delta u=\frac{f(x)}{|z|^{s_{1}}}u+\frac{g(x)}{|z|^{s_{1}}},$

where $f,g\in L^{P}_{loc}(\mathbb{R}^{N})$ for some $p>\frac{N}{2-s_{1}}$.
Then the following result follows from [[21], Lemma 3.2 and Lemma 3.3].

######  Proposition 2.2.

Let $u$ is a solution of the Euler-Lagrange equation (2.7). We assume that

$\begin{cases}\displaystyle s_{1}<1+\frac{1}{N}\qquad&\textrm{ if $N\geq 4$
}\\\ \\\ \displaystyle s_{1}<\frac{3}{2}&\textrm{ if $N=3$}.\end{cases}$

Then $u\in\mathcal{C}^{\infty}$ in the $z$ variable while, in the $y$
variable, it is $\mathcal{C}^{1,\alpha}$ for all $\alpha<1-s_{1}$ if $s_{1}<1$
and $\mathcal{C}^{0,\alpha}(\mathbb{R}^{N})$ for all $\alpha<2-s_{1}$ if
$1\leq s_{1}<2$.

This then allows to prove the following symmetry and decay estimates result.

######  Proposition 2.3.

Let $u$ be a solution of the Euler-Lagrange equation (2.7). Then

  * (i)

the function $u$ depends only on $|y|$ and $|z|$

  * (ii)

there exists two constants $0<c_{1}<c_{2}$ such that:

$\frac{c_{1}}{1+|x|^{N-2}}\leq u(x)\leq\frac{c_{2}}{1+|x|^{N-2}},\qquad\forall
x\in\mathbb{R}^{N}.$ (2.8)

###### Proof.

The proof of the symmetry is based on the moving plane method, see for
instance [2, 3, 7, 6, 33] and references therein. We let
$x=(y,t)\in\mathbb{R}\times\mathbb{R}^{N-1}$. For $\mu>0$, we define

$\Omega_{\mu}=\\{x=(y,z)\in\mathbb{R}\times\mathbb{R}^{N}:y>\mu\\}$

and for all $x\in\Omega_{\mu}$, we set $x_{\mu}=(2\mu-y,z)\in\Omega_{\mu}.$
Next, we set

$w_{\mu}(x):=u_{\mu}(x)-u(x)=u(x_{\mu})-u(x)\qquad\textrm{ in }\Omega_{\mu}.$

Then $w_{\mu}\in H^{1}_{0}(\Omega_{\mu})$.
Step 1: We first prove that:

$w_{\mu}\geq 0\qquad\textrm{in }\Omega_{\mu}$ (2.9)

for $\mu$ large enough. Thanks to (2.7), we have

$-\Delta
w_{\mu}(x)=\lambda\left(\frac{u^{2^{*}_{s_{1}}-1}(x)}{|z|^{s_{1}}}-\frac{u_{\mu}^{2^{*}_{s_{1}}-1}(x)}{|z|^{s_{1}}}\right)+\left(\frac{u^{2^{*}_{s_{2}}-1}(x)}{|z|^{s_{2}}}-\frac{u_{\mu}^{2^{*}_{s_{2}}-1}(x)}{|z|^{s_{2}}}\right)\qquad\textrm{
in }\Omega_{\mu}.$ (2.10)

We multiply (2.10) by $w_{\mu}^{-}:=\min\\{w_{\mu},0\\}$ and we integrate by
parts to get

$\displaystyle\int_{\Omega_{\mu}}|\nabla w_{\mu}^{-}|^{2}dx$
$\displaystyle=\lambda\int_{\Omega_{\mu}}w_{\mu}^{-}(x)\left(\frac{u^{2^{*}_{s_{1}}-1}(x)}{|z|^{s_{1}}}-\frac{u_{\mu}^{2^{*}_{s_{1}}-1}(x)}{|z|^{s_{1}}}\right)dx+\int_{\Omega_{\mu}}w_{\mu}^{-}(x)\left(\frac{u^{2^{*}_{s_{2}}-1}(x)}{|z|^{s_{2}}}-\frac{u_{\mu}^{2^{*}_{s_{2}}-1}(x)}{|z|^{s_{2}}}\right)dx$
$\displaystyle\leq|\lambda|\int_{\Omega_{\mu}}\frac{w_{\mu}^{-}(x)}{|z|^{s_{1}}}\left(u^{2^{*}_{s_{1}}-1}(x)-u_{\mu}^{2^{*}_{s_{1}}-1}(x)\right)dx+\int_{\Omega_{\mu}}\frac{w_{\mu}^{-}(x)}{|z|^{s_{2}}}\left(u^{2^{*}_{s_{2}}-1}(x)-u_{\mu}^{2^{*}_{s_{2}}-1}(x)\right)dx.$

We have $u_{\mu}(x)\leq u(x)$ in $\Omega_{\mu}$. Then using the convexity of
the function $t\longmapsto t^{2^{*}_{s_{i}}}$ ($i=1,2$) on $(0,\infty)$, we
get

$u_{\mu}^{2^{*}_{s_{i}}-1}(x)-u^{2^{*}_{s_{i}}-1}(x)\leq(2^{*}_{s_{i}}-1)u^{2^{*}_{s_{i}}-2}(x)\left(u(x)-u_{\mu}(x)\right)=(1-2^{*}_{s_{i}})u^{2^{*}_{s_{i}}-2}(x)w_{\mu}^{-}(x)\qquad\textrm{in
$\Omega_{\mu}$}.$

Therefore

$\int_{\Omega_{\mu}}|\nabla
w_{\mu}^{-}|^{2}dx\leq|\lambda|(2^{*}_{s_{1}}-1)\int_{\Omega_{\mu}}\frac{|w_{\mu}^{-}(x)|^{2}}{|z|^{s_{1}}}|u(x)|^{2^{*}_{s_{1}}-2}dx+(2^{*}_{s_{2}}-1)\int_{\Omega_{\mu}}\frac{|w_{\mu}^{-}(x)|^{2}}{|z|^{s_{2}}}|u(x)|^{2^{*}_{s_{2}}-2}dx.$

Next, by Holder’s inequality, we have

$\int_{\Omega_{\mu}}\frac{|w_{\mu}^{-}(x)|^{2}}{|z|^{s_{i}}}|u(x)|^{2^{*}_{s_{i}}-2}dx\leq\left(\int_{\Omega_{\mu}}\frac{|w_{\mu}^{-}(x)|^{2^{*}_{s_{i}}}}{|z|^{s_{i}}}dx\right)^{{2/2^{*}_{s_{i}}}}\left(\int_{M_{\mu}\cap\Omega_{\mu}}\frac{|u(x)|^{2^{*}_{s_{i}}}}{|z|^{s_{i}}}dx\right)^{\frac{2^{*}_{s_{i}}-2}{2^{*}_{s_{i}}}},$

where $i=1,2$ and

$M_{\mu}:=\\{x\in\Omega_{\mu}:\quad u(x)>u_{\mu}(x)\\}.$

Since

$\lim_{\mu\to\infty}\int_{M_{\mu}\cap\Omega_{\mu}}\frac{|u(x)|^{2^{*}_{s_{1}}}}{|z|^{s_{1}}}dx=\lim_{\mu\to\infty}\int_{M_{\mu}\cap\Omega_{\mu}}\frac{|u(x)|^{2^{*}_{s_{2}}}}{|z|^{s_{2}}}dx=0,$

we deduce that

$\int_{\Omega_{\mu}}|\nabla
w_{\mu}^{-}|^{2}dx<S_{N,0}\left(\int_{\Omega_{\mu}}|w_{\mu}^{-}|^{\frac{2N}{N-2}}dx\right)^{\frac{N-2}{2N}},$

where $S_{N,0}$ is the Sobolev best constant. As a consequence, for $\mu$
large enough, we have $w_{\mu}^{-}=0$ and hence

$u_{\mu}(x)\geq u(x).$

Next, we let

$\mu^{*}:=\inf\\{\mu>0:\quad u(x)\leq u_{\mu^{\prime}}(x)\quad\textrm{ for all
$x\in\Omega_{\mu^{\prime}}$ and all $\mu^{\prime}>\mu$}\\}<\infty.$

Step 2: Then we will prove that $\mu^{*}=0$.
By contradiction, we assume that $\lambda^{*}>0$. Then

$-\Delta w_{\mu^{*}}(x)=c_{\mu^{*}}(x)w_{\mu^{*}}(x)\qquad\textrm{ in
}\Omega_{\mu^{*}},$

where

$c_{\mu^{*}}:=\begin{cases}\displaystyle\left(\frac{u_{\mu^{*}}^{2^{*}_{s_{1}}-1}(x)-u^{2^{*}_{s_{1}}-1}(x)}{|z|^{s_{1}}}-\frac{u_{\mu^{*}}^{2^{*}_{s_{2}}-1}(x)-u^{2^{*}_{s_{2}}-1}(x)}{|z|^{s_{2}}}\right)w_{\mu^{*}}^{-1}\qquad&\textrm{
if $w_{\lambda^{*}}\neq 0$}\\\ \\\ 0&\textrm{ if $w_{\mu^{*}}=0$}.\end{cases}$

Clearly $c_{\mu^{*}}\in L^{\infty}(\Omega_{\mu^{*}})$. Moreover
$w_{\mu^{*}}\geq 0\quad\textrm{ in $\partial\Omega_{\mu^{*}}$}.$ Then applying
the maximum principle, we have

$w_{\mu^{*}}>0\quad\textrm{ in $\Omega_{\mu^{*}}$}.$

Let $D$ be any smooth compact set in $\Omega_{\mu^{*}}$ such that
$|\Omega_{\mu}\setminus D|$ is sufficiently small for any $\mu$ near
$\mu^{*}$. Since

$w_{\mu^{*}}(x)\geq\delta>0\qquad\textrm{ in $D$},$

we have, by continuity,

$w_{\mu}(x)\geq 0\qquad\textrm{ in D},$

for all $\mu$ near $\mu^{*}$. In particular

$w_{\mu}(x)\geq 0\qquad\textrm{ on $\partial(\Omega_{\mu}\setminus D)$}.$

Using again the maximum principle, we have

$w_{\mu}(x)\geq 0\qquad\textrm{ in }\Omega_{\mu}\setminus D$

and thus $w_{\mu}(x)\geq 0$ in $\Omega_{\mu}$, contrary to the definition of
$\mu^{*}$. We therefore conclude that $\mu^{*}=0.$ Consequently

$u(-y,z)\geq u(y,z)\qquad\forall y>0.$

We apply the same arguments as before on the function $v(y,z)=u(-y,z)$ in
$\mathbb{R}^{N}$ leads to

$u(-y,z)\leq u(y,z).$

Therefore

$u(-y,z)=u(y,z)\qquad\textrm{ in $\mathbb{R}^{N}$}.$

Hence the solution $u$ of (2.7) is symmetric with respect to $y$.
Step 3: Repeating the same arguments as before for the functions

$x\longmapsto w(\mathcal{R}_{N-1}y,z)\quad\textrm{and}\quad x\longmapsto
w(y,\mathcal{R}_{N-1}z)$

where $\mathcal{R}_{N-1}\in O(N-1)$ is a $(N-1)$-dimensional rotation, we
conclude that $w$ only depends on $|y|$ and $|z|$ and $w$ is strictly
decreasing in $|y|$. This then ends the proof of (i).

For the decay estimate, we write the Euler-Lagrange equation (2.7), as follow

$-\Delta w(x)=A(x)w\qquad\textrm{in }\mathbb{R}^{N},$

where

$A(x)=\lambda\frac{w^{2^{*}_{s_{1}}-2}(x)}{|z|^{s_{1}}}+\frac{w^{2^{*}_{s_{2}}-2}(x)}{|z|^{s_{2}}}.$

For $x\neq 0$, we let $v(x)=|x|^{2-N}w(x|x|^{-2}),$ the Kelvin transformation
of $w$. It also satisfies (2.7). Therefore, using the fact that a solution of
(2.7) is bounded, we can find two constants $0<c_{1}<c_{2}$ such that

$\frac{c_{1}}{1+|x|^{N-2}}\leq w(x)\leq\frac{c_{1}}{1+|x|^{N-2}},\qquad\forall
x\in\mathbb{R}^{N}.$

This then ends the proof. ∎

We close this section by proving the following decay properties of $w$
involving its higher derivatives.

######  Proposition 2.4.

Let $w$ be a ground state for $S_{N,\sigma}$ then there exist positive
constant $C$, only depending on $N$ and $s_{1}$ and $s_{2}$, such that

For $|x|=|(t,z)|\leq 1$

$|\nabla w(x)|+|x||D^{2}w(x)|\leq C_{2}|z|^{1-s_{1}}$

For $|x|=|(t,z)|\geq 1$

$|\nabla w(x)|+|x||D^{2}w(x)|\leq C_{2}\max(1,|z|^{-s_{1}})|x|^{1-N}.$

###### Proof.

Let $\theta:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a function
such that

$w(x)=w(y,z)=\theta(|y|,|z|).$

Using polar coordinates, the function $\theta=\theta(t,\rho)$ verifies

$\rho^{2-N}(\rho^{N-2}\theta_{2})_{2}+\theta_{11}=\lambda\rho^{-s_{1}}\theta^{2^{*}_{s_{1}}-1}+\rho^{-s_{2}}\theta^{2^{*}_{s_{2}}-1}\qquad\textrm{
for $t,\rho\in\mathbb{R}_{+}$},$ (2.11)

where $\theta_{1}$ and $\theta_{2}$ are respectively the derivatives of
$\theta$ with respect to the first and the second variables. Then integrating
this identity in the $\rho$ variable, we therefore get, for every $\rho>0$,

$\displaystyle\theta_{2}(t,\rho)=-\frac{1}{\rho^{N-2}}\int_{0}^{\rho}r^{N-2}\theta_{11}(t,r)dr$
$\displaystyle+\frac{\lambda}{\rho^{N-2}}\int_{0}^{\rho}r^{N-2}r^{-s_{1}}\theta^{2^{*}_{s_{1}}-1}(t,r)dr$
$\displaystyle\frac{1}{\rho^{N-2}}\int_{0}^{\rho}r^{N-2}r^{-s_{2}}\theta^{2^{*}_{s_{2}}-1}(t,r)dr.$

Next differentiating with respect to the first variable, we get

$\displaystyle\theta_{12}(t,\rho)=\frac{-1}{\rho^{N-2}}\int_{0}^{\rho}r^{N-2}\theta_{111}(t,r)dr$
$\displaystyle+\frac{\lambda}{\rho^{N-2}}\int_{0}^{\rho}r^{N-2}r^{-s_{1}}\theta_{1}(t,r)\theta^{2^{*}_{s_{1}}-2}(t,r)dr$
$\displaystyle+\frac{1}{\rho^{N-2}}\int_{0}^{\rho}r^{N-2}r^{-s_{2}}\theta_{1}(t,r)\theta^{2^{*}_{s_{2}}-2}(t,r)dr.$

By Proposition 2.2 and the fact that $2^{*}_{s_{2}}>2^{*}_{s_{1}}\geq 2$, we
obtain

$|\theta_{2}(t,\rho)|+|\theta_{12}(t,\rho)|\leq
C\left(\rho+\rho^{1-s_{1}}+\rho^{1-s_{2}}\right)\leq
C\rho^{1-s_{1}}\qquad\textrm{ for $|(t,\rho)|\leq 1$}.$ (2.12)

Now using this in (2.11), we get

$|\theta_{22}|\leq C\rho^{-s_{1}},\quad\textrm{ for $|(t,\rho)|\leq 1$}.$
(2.13)

By (2.12) and (2.13), we obtain

$|\theta_{2}(t,\rho)|+|\theta_{12}(t,\rho)|+\rho|\theta_{22}|\leq
C\rho^{1-s_{1}}.$

Therefore, it easy follows that

$|\nabla w(x)|+|x||D^{2}w(x)|\leq C_{2}|z|^{1-s_{1}},\qquad\textrm{ for all
$|x|=|(t,z)|\leq 1$}$

and for $|x|=|(t,z)|\geq 1$ that

$|\nabla w(x)|+|x||D^{2}w(x)|\leq C_{2}\max(1,|z|^{-s_{1}})|x|^{1-N}.$

This then completes the proof. ∎

##  3\. Local Parametrization and metric

Let $\Gamma\subset\mathbb{R}^{N}$ be a smooth closed curve. Let
$(E_{1};\dots;E_{N})$ be an orthonormal basis of $\mathbb{R}^{N}$. For
$y_{0}\in\Gamma$ and $r>0$ small, we consider the curve
$\gamma:\left(-r,r\right)\to\Gamma$, parameterized by arclength such that
$\gamma(0)=y_{0}$. Up to a translation and a rotation, we may assume that
$\gamma^{\prime}(0)=E_{1}$. We choose a smooth orthonormal frame field
$\left(E_{2}(y);...;E_{N}(y)\right)$ on the normal bundle of $\Gamma$ such
that $\left(\gamma^{\prime}(y);E_{2}(y);...;E_{N}(y)\right)$ is an oriented
basis of $\mathbb{R}^{N}$ for every $y\in(-r,r)$, with $E_{i}(0)=E_{i}$.
We fix the following notation, that will be used a lot in the paper,

$Q_{r}:=(-r,r)\times B_{\mathbb{R}^{N-1}}(0,r),$

where $B_{\mathbb{R}^{k}}(0,r)$ denotes the ball in $\mathbb{R}^{k}$ with
radius $r$ centered at the origin. Provided $r>0$ small, the map
$F_{y_{0}}:Q_{r}\to\Omega$, given by

$(y,z)\mapsto F_{y_{0}}(y,z):=\gamma(y)+\sum_{i=2}^{N}z_{i}E_{i}(y),$

is smooth and parameterizes a neighborhood of $y_{0}=F_{y_{0}}(0,0)$. We
consider $\rho_{\Gamma}:\Gamma\to\mathbb{R}$ the distance function to the
curve given by

$\rho_{\Gamma}(y)=\min_{\overline{y}\in\mathbb{R}^{N}}|y-\overline{y}|.$

In the above coordinates, we have

$\rho_{\Gamma}\left(F_{y_{0}}(x)\right)=|z|\qquad\textrm{ for every
$x=(y,z)\in Q_{r}.$}$ (3.1)

Clearly, for every $t\in(-r,r)$ and $i=2,\dots N$, there are real numbers
$\kappa_{i}(y)$ and $\tau^{j}_{i}(y)$ such that

$E_{i}^{\prime}(y)=\kappa_{i}(y)\gamma^{\prime}(y)+\sum_{{j=2}}^{N}\tau_{i}^{j}(y)E_{j}(y).$
(3.2)

The quantity $\kappa_{i}(y)$ is the curvature in the $E_{i}(y)$-direction
while $\tau_{i}^{j}(y)$ is the torsion from the osculating plane spanned by
$\\{\gamma^{\prime}(y);E_{j}(y)\\}$ in the direction $E_{i}$. We note that
provided $r>0$ small, $\kappa_{i}$ and $\tau_{i}^{j}$ are smooth functions on
$(-r,r)$. Moreover, it is easy to see that

$\tau^{j}_{i}(y)=-\tau^{i}_{j}(y)\qquad\textrm{ for $i,j=2,\dots,N$. }$ (3.3)

The curvature vector is $\kappa:\Gamma\to\mathbb{R}^{N}$ is defined as
$\kappa(\gamma(y)):=\sum_{i=2}^{N}\kappa_{i}(y)E_{i}(y)$ and its norm is given
by $|\kappa\gamma(y)|:=\sqrt{\sum_{i=2}^{N}\kappa^{2}_{i}(y)}$. Next, we
derive the expansion of the metric induced by the parameterization $F_{y_{0}}$
defined above. For $x=(y,z)\in Q_{r}$, we define

$g_{11}(x)={\partial_{y}F_{y_{0}}}(x)\cdot{\partial_{y}F_{y_{0}}}(x),\quad
g_{1i}(x)={\partial_{y}F_{y_{0}}}(x)\cdot{\partial_{z_{i}}F_{y_{0}}}(x),\quad
g_{ij}(x)={\partial_{z_{j}}F_{y_{0}}}(x)\cdot{\partial_{z_{i}}F_{y_{0}}}(x).$

We have the following result.

######  Lemma 3.1.

There exits $r>0$, only depending on $\Gamma$ and $N$, such that for ever
$x=(t,z)\in Q_{r}$

$\begin{cases}\displaystyle
g_{11}(x)=1+2\sum_{i=2}^{N}z_{i}\kappa_{i}(0)+2y\sum_{i=2}^{N}z_{i}\kappa^{\prime}_{i}(0)+\sum_{ij=2}^{N}z_{i}z_{j}\kappa_{i}(0)\kappa_{j}(0)+\sum_{ij=2}^{N}z_{i}z_{j}\beta_{ij}(0)+O\left(|x|^{3}\right)\\\
\ \displaystyle
g_{1i}(x)=\sum_{j=2}^{N}z_{j}\tau^{i}_{j}(0)+y\sum_{j=2}^{N}z_{j}\left(\tau^{i}_{j}\right)^{\prime}(0)+O\left(|x|^{3}\right)\\\
\ \displaystyle g_{ij}(x)=\delta_{ij},\end{cases}$ (3.4)

where $\beta_{ij}(y):=\sum_{l=2}^{N}\tau_{i}^{l}(y)\tau_{j}^{l}(y).$

As a consequence we have the following result.

######  Lemma 3.2.

There exists $r>0$ only depending on $\Gamma$ and $N$, such that for every
$x\in Q_{r}$, we have

$\sqrt{|g|}(x)=1+\sum_{i=2}^{N}z_{i}\kappa_{i}(0)+y\sum_{i=2}^{N}z_{i}\kappa_{i}^{\prime}(0)+\frac{1}{2}\sum_{ij=2}^{N}z_{i}z_{j}\kappa_{i}(0)\kappa_{j}(0)+O\left(|x|^{3}\right),$
(3.5)

where $|g|$ stands for the determinant of $g$. Moreover $g^{-1}(x)$, the
matrix inverse of $g(x)$, has components given by

$\begin{cases}\displaystyle
g^{11}(x)=1-2\sum_{i=2}^{N}z_{i}\kappa_{i}(0)-2y\sum_{i=2}^{N}z_{i}\kappa_{i}^{\prime}(0)+3\sum_{ij=2}^{N}z_{i}z_{j}\kappa_{i}(0)\kappa_{j}(0)+O\left(|x|^{3}\right)\\\
\displaystyle
g^{i1}(x)=-\sum_{j=2}^{N}z_{j}\tau^{i}_{j}(0)-y\sum_{j=2}^{N}z_{j}\left(\tau^{i}_{j}\right)^{\prime}(0)+2\sum_{j=2}^{N}z_{l}z_{j}\kappa_{l}(0)\tau^{i}_{j}(0)+O\left(|x|^{3}\right)\\\
\displaystyle
g^{ij}(x)=\delta_{ij}+\sum_{lm=2}^{N}z_{l}z_{m}\tau^{j}_{l}(0)\tau^{i}_{m}(0)+O\left(|x|^{3}\right).\end{cases}$
(3.6)

We will also need the following estimates result.

######  Lemma 3.3.

Let $v\in{\mathcal{D}}^{1,2}(\mathbb{R}^{N})$, $N\geq 3,$ satisfy
$v(y,z)=\overline{\theta}(|y|,|z|)$, for some some function
$\overline{\theta}:\mathbb{R}_{+}\times\mathbb{R}_{+}\to\mathbb{R}$. Then for
$0<r<R$, we have

$\displaystyle\int_{{Q}_{R}\setminus{Q}_{r}}|\nabla v|^{2}_{g}\sqrt{|g|}dx$
$\displaystyle=\int_{{Q}_{R}\setminus{Q}_{r}}|\nabla
v|^{2}dx+\frac{|\kappa(x_{0})|^{2}}{N-1}\int_{{Q}_{R}\setminus{Q}_{r}}|z|^{2}\left|{\partial_{y}v}\right|^{2}dx$
$\displaystyle+\frac{|\kappa(x_{0})|^{2}}{2(N-1)}\int_{{Q}_{R}\setminus{Q}_{r}}|z|^{2}|\nabla
v|^{2}dx+O\left(\int_{{Q}_{R}\setminus{Q}_{r}}|x|^{3}|\nabla v|^{2}dx\right).$

For the proofs of these Lemma 3.1, Lemma 3.2 and Lemma 3.3, we refer to the
paper of the author and Fall [22]. See also [35] for a generalization.

##  4\. Existence Result in domains

The aim of this section is to prove the following result.

######  Proposition 4.1.

Let $N\geq 4$, $0\leq s_{2}<s_{1}<2$ and $\Omega$ be a bounded domain of
$\mathbb{R}^{N}$. Consider $\Gamma$ a smooth closed curve contained in
$\Omega$. Let $h$ be a continuous function such that the linear operator
$-\Delta+h$ is coercive. We assume that

$c^{*}:=\sup_{t\geq 0}\Psi(v)<\beta^{*}.$ (4.1)

Then there exists a positive function $u\in H^{1}_{0}(\Omega)$ solution of the
Euler-Lagrange equation

$-\Delta
u+hu=\lambda\rho^{-s_{1}}_{\Gamma}u^{2^{*}_{s_{1}}-1}+\rho^{-s_{2}}_{\Gamma}u^{2^{*}_{s_{2}}-1}\qquad\textrm{
in }\Omega.$ (4.2)

The proof of Proposition 4.1 is divided into various preliminaries results. We
start by the following.

######  Lemma 4.2.

For $N\geq 3$, we let $\Omega$ be an open subset of $\mathbb{R}^{N}$ and let
$\Gamma\subset\Omega$ be a smooth closed curve contained in $\Omega$. Then for
every $r>0$, there exists $c_{r}>0$, only depending on
$\Omega,\Gamma,N,\sigma$ and $r$, such that for every $u\in H^{1}_{0}(\Omega)$

$\left(\frac{1}{2}-\frac{1}{2^{*}_{\sigma}}\right)\int_{\Omega}|\nabla
u|^{2}dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{\Omega}\rho_{\Gamma}^{-s_{2}}dx+c_{r}\int_{\Omega}|u|^{2}dy\geq
c^{*}.$

where, for $0\leq s_{2}<s_{1}<2$, $2^{*}_{s_{1}}=\frac{2(N-s_{1})}{N-2}$ and
$2^{*}_{s_{2}}=\frac{2(N-s_{2})}{N-2}$.

###### Proof.

We let $r>0$ small. We can cover a tubular neighborhood of $\Gamma$ by a
finite number of sets $\left(T^{y_{i}}_{r}\right)_{1\leq i\leq m}$ given by

$T^{y_{i}}_{r}:=F_{y_{i}}\left(Q_{r}\right),\qquad\textrm{ with
$y_{i}\in\Gamma$}.$

We refer to Section 3 for the parameterization $F_{y_{i}}:Q_{r}\to\Omega$. Let
$\left(\varphi_{i}\right)_{1\leq i\leq m}$ be a partition of unity
subordinated to this covering such that

$\sum_{i}^{m}\varphi_{i}=1\qquad\textrm{and}\qquad|\nabla\varphi_{i}^{\frac{1}{2^{*}_{\sigma}}}|\leq
C\qquad\textrm{ in }U:=\displaystyle\cup_{i=1}^{m}T^{y_{i}}_{r},$ (4.3)

for some positive constant $C$. We define

$\psi_{i}(y):=\varphi_{i}^{\frac{1}{2^{*}_{\sigma}}}(y)u(y)\qquad\textrm{ and
}\qquad\widetilde{\psi_{i}}(x)=\psi_{i}(F_{y_{i}}(x)).$ (4.4)

Then, we have

$\displaystyle\displaystyle\int_{\Omega}\rho^{-\sigma}_{\Gamma}|u|^{2^{*}_{\sigma}}dy\geq\int_{U}\rho^{-\sigma}_{\Gamma}\left|u\right|^{2^{*}_{\sigma}}dy=\sum_{i}^{m}\int_{T^{y_{i}}_{r}}\rho^{-\sigma}_{\Gamma}|\psi_{i}|^{2^{*}_{\sigma}}dy.$
(4.5)

By change of variables and Lemma 3.2, we have

$\displaystyle\int_{T^{y_{i}}_{r}}\rho^{-\sigma}_{\Gamma}|\psi_{i}|^{2^{*}_{\sigma}}dy=\int_{Q_{r}}|z|^{-\sigma}|\widetilde{\psi}_{i}|^{2^{*}_{\sigma}}\sqrt{|g|}(x)dx\geq\left(1-cr\right)\int_{Q_{r}}|z|^{-\sigma}|\widetilde{\psi}_{i}|^{2^{*}_{\sigma}}dx,$
(4.6)

for some positive constant $c$. By (4.5) and (4.6) and the summing over
$i=1,\cdots,m$, we obtain

$\int_{\Omega}\rho^{-\sigma}_{\Gamma}|u|^{2^{*}_{\sigma}}dy\geq\left(1-cr\right)\sum_{i=1}^{m}\int_{Q_{r}}|z|^{-\sigma}|\widetilde{\psi}_{i}|^{2^{*}_{\sigma}}dx=(1-cr)\int_{U}|z|^{-\sigma}|\tilde{u}(x)|^{2^{*}_{\sigma}}dx,$
(4.7)

with $\tilde{u}:=u(F_{y_{i}}(x))$. Next, we have

$\int_{\Omega}|\nabla u|^{2}dx\geq\int_{U}|\nabla
u|^{2}dy=\sum_{i}^{m}\int_{T^{y_{i}}_{r}}|\nabla\psi_{i}|^{2}dy.$ (4.8)

By change of variables, Lemma 3.2, (4.3) and (4.4), we have

$\displaystyle\int_{T^{y_{i}}_{r}}|\nabla\psi_{i}|^{2}dy$
$\displaystyle=\int_{Q_{r}}|\nabla\widetilde{\psi}_{i}|^{2}\sqrt{|g|}(x)dx\geq\left(1-cr\right)\int_{Q_{r}}|\nabla\widetilde{\psi}_{i}|^{2}dx$
$\displaystyle\geq\left(1-c^{\prime}r\right)\int_{T^{y_{i}}_{r}}|\nabla(\varphi_{i}^{\frac{1}{2^{*}_{\sigma}}}u)|^{2}dy=\int_{T^{y_{i}}_{r}}\varphi_{i}^{\frac{2}{2^{*}_{\sigma}}}|\nabla\tilde{u}|^{2}dy-
c_{r}\int_{\Omega}|u|^{2}dy,$

for some positive constants $c$ and $c_{r}$. Therefore

$\int_{T^{y_{i}}_{r}}|\nabla\psi_{i}|^{2}dy\geq\int_{T^{y_{i}}_{r}}|\nabla\tilde{u}|^{2}dy-
c_{r}\int_{\Omega}|u|^{2}dy.$ (4.9)

Hence combining (4.8) and (4.9), we obtain

$\int_{\Omega}|\nabla u|^{2}dx\geq\int_{U}|\nabla\tilde{u}|^{2}dy-
c_{r}\int_{\Omega}|u|^{2}dy.$ (4.10)

Thanks to (4.7),(4.10) and the definition of $\beta^{*}$, we get

$\displaystyle\left(\frac{1}{2}-\frac{1}{2^{*}_{\sigma}}\right)\int_{\Omega}|\nabla
u|^{2}dx$
$\displaystyle+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{\Omega}\rho_{\Gamma}^{-s_{2}}dx+c_{r}\int_{\Omega}|u|^{2}dy$
$\displaystyle\geq\left(\frac{1}{2}-\frac{1}{2^{*}_{\sigma}}\right)\int_{U}|\nabla\tilde{u}|^{2}dy+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)(1-cr)\int_{U}|\tilde{u}(x)|^{2^{*}_{\sigma}}dx\geq\beta^{*}.$

This then ends the proof. ∎

######  Lemma 4.3.

Let $\alpha<\beta^{*}$ and $(u_{n})_{n}\subset H^{1}_{0}(\Omega)$ be a Palais-
Smale sequence for $\Psi$ at level $\alpha$. Then, up to a subsequence, there
exists $u\in H^{1}_{0}(\Omega)$ such that

$\begin{cases}u_{n}\to u\qquad\textrm{ strongly in $H^{1}_{0}(\Omega)$}\\\ \\\
\Psi(u)=\alpha\\\ \\\ \Psi^{\prime}(u)=0.\end{cases}$

###### Proof.

Let $\alpha<\beta^{*}$ and $(u_{n})_{n}\subset H^{1}_{0}(\Omega)$ be a Palais-
Smale sequence for $\Psi$ at level $\alpha$. That is

$\alpha=\frac{1}{2}\int_{\Omega}|\nabla
u_{n}|^{2}dx+\frac{1}{2}\int_{\Omega}hu_{n}^{2}dx-\frac{\lambda}{2^{*}_{s_{1}}}\int_{\Omega}\rho^{-s_{1}}_{\Gamma}|u_{n}|^{2^{*}_{s_{1}}}dx-\frac{1}{2^{*}_{s_{2}}}\int_{\Omega}\rho^{-s_{2}}_{\Gamma}|u_{n}|^{2^{*}_{s_{2}}}dx+o(1)$
(4.11)

and

$\int_{\Omega}\nabla u_{n}\nabla\varphi dx+\int_{\Omega}hu_{n}\varphi
dx-\lambda\int_{\Omega}\rho^{-s_{1}}_{\Gamma}|u_{n}|^{2^{*}_{s_{1}}-2}u_{n}\varphi
dx-\frac{1}{2^{*}_{s_{2}}}\int_{\Omega}\rho^{-s_{2}}_{\Gamma}|u_{n}|^{2^{*}_{s_{2}}-2}u_{n}\varphi
dx+o(1),$ (4.12)

for all $\varphi\in H^{1}_{0}(\Omega)$ as $n\to\infty$. Combining (4.11) and
(4.12), we obtain

$\alpha=\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\Omega}|\nabla
u_{n}|^{2}dx+\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\Omega}hu_{n}^{2}dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{\Omega}\rho^{-s_{2}}_{\Gamma}|u_{n}|^{2^{*}_{s_{2}}}dx+o(1).$
(4.13)

Now we use the fact that $\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}$ and
$\frac{1}{2}-\frac{1}{2^{*}_{s_{2}}}$ are positive and the coercivity of the
linear operator $-\Delta+h$, we obtain

$\frac{\alpha}{\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)}+o(1)\geq\int_{\Omega}|\nabla
u_{n}|^{2}dx+\int_{\Omega}hu_{n}^{2}dx\geq\|u_{n}\|_{H^{1}(\Omega)}.$

Consequently, up to a subsequence, there exists $u\in H^{1}_{0}(\Omega)$ such
that $u_{n}$ converges weakly to $u$ in $H^{1}_{0}(\Omega)$ and strongly to
$L^{p}(\Omega)$ for all $2\leq p<2^{*}_{0}$. We assume by contradiction that
$u=0$. Therefore, by (4.13), we obatin

$\alpha=\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\Omega}|\nabla
u_{n}|^{2}dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{\Omega}\rho^{-s_{2}}_{\Gamma}|u_{n}|^{2^{*}_{s_{2}}}dx+o(1).$
(4.14)

Moreover by Lemma 4.2, we get

$\left(\frac{1}{2}-\frac{1}{2^{*}_{s_{1}}}\right)\int_{\Omega}|\nabla
u_{n}|^{2}dx+\left(\frac{1}{2^{*}_{s_{1}}}-\frac{1}{2^{*}_{s_{2}}}\right)\int_{\Omega}\rho^{-s_{2}}_{\Gamma}|u_{n}|^{2^{*}_{s_{2}}}dx+o(1)\geq\beta^{*}.$
(4.15)

Hence by (4.14) and (4.15), we obtain

$\alpha\geq\beta^{*},$

which contradicts the fact that $\alpha<\beta^{*}$. Then $u\neq 0$ and

$u_{n}\to u\qquad\textrm{in $H^{1}_{0}(\Omega)$}.$

This then ends the proof. ∎

Next, we will need the following so-called mountain pass lemma due to
Ambrosetti and Robinowitz, see [1]. Then we have

######  Lemma 4.4.

(Mountain Pass Lemma)
Let $(X,\|\cdot\|_{X})$ be a Banach space and $\Psi:X\to\mathbb{R}$ a
functional of class $\mathcal{C}^{1}$. Wa assume that

  1. 1.

$\Psi(0)=0$;

  2. 2.

there exist $\lambda,r>0$ such that $\Psi(u)\geq\lambda$ for all $u\in X$, we
have $\|u\|_{X}=r$;

  3. 3.

there exists $u_{0}\in X$ such that

$\lim_{t\to+\infty}\sup\Psi(tu_{0})<0.$

Consider $t_{0}>0$ sufficiently large such that $\|t_{0}u_{0}\|_{X}>r$ and
$\Psi(t_{0}u_{0})<0$. Define $\beta\leq\lambda$ as

$\beta:=\inf_{\gamma\in\mathcal{P}}\sup_{t\in[0,1]}\Psi(\gamma(t)),$

where

$\mathcal{P}=\\{\gamma\in\mathcal{C}^{0}([0,1];X)\textrm{ such that
}\gamma(0)=0\textrm{ and }\gamma(1)=t_{0}u_{0}\\}.$

Then, there exists a sequence $(u_{n})_{n}\subset X$ such that
$\Psi(u_{n})\to\beta$ and $\Psi^{\prime}(u_{n})\to 0$ strongly in
$X^{\prime}$. Moreover, we have that

$\beta\leq\sup_{t\geq 0}\Psi(tu_{0}).$

######  Lemma 4.5.

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, $\Gamma$ be a closed
curve included in $\Omega$ and $h$ be a continuous function such that the
linear operator is $-\Delta+h$ is coercive. Let $u_{0}\in
H^{1}_{0}(\Omega)\setminus\\{0\\}$. Then there exists $c_{0}$ a positive
constant depending on $u_{0}$ and $(u_{n})_{n}\subset H^{1}_{0}(\Omega)$ a a
Palais-Smale sequence for $\Psi$ at level $c_{0}$. Moreover

$c_{0}\leq\sup_{t\geq 0}\Psi(tu_{0}).$

###### Proof.

We let $t\in\mathbb{R}$. Recall that for all $u\in H^{1}_{0}(\Omega)$, we have

$\Psi(tu):=\frac{t^{2}}{2}\int_{\Omega}(|\nabla
u|^{2}+hu^{2})dx-\lambda\frac{|t|^{2^{*}_{s_{1}}}}{2^{*}_{s_{1}}}\int_{\Omega}\rho_{\Gamma}^{-s_{1}}|u|^{2^{*}_{s_{1}}}dx-\frac{|t|^{2^{*}_{s_{2}}}}{2^{*}_{s_{2}}}\int_{\Omega}\rho_{\Gamma}^{-s_{2}}|u|^{2^{*}_{s_{2}}}dx$

Then $\Psi\in\mathcal{C}^{1}(H_{0}^{1}(\Omega),\mathbb{R})$. Since
$0<s_{2}<s_{1}<2$ and the fact that the function $t\longmapsto
2^{*}_{s}:=\frac{2(N-s)}{N-2}$ is decreasing, we have

$\lim_{t\to\infty}\Psi(tu)=-\infty.$ (4.16)

Moreover, using the fact that $2^{*}_{s_{1}},2^{*}_{s_{2}}>2$, then there
exists sufficiently positive numbers $\lambda,r$ such that

$\inf_{\|u\|=r}\Psi(u)\geq\lambda.$

Therefore by the Mountain pass Lemma (4.4), we get the desired result. ∎

###### Proof.

of Proposition 4.1.
Let $u_{0}\in H^{1}_{0}(\Omega)$ be a non-negative, non-vanishing function
such that

$\sup_{t\geq 0}\Psi(tu_{0})<\beta^{*}.$

Then by Lemma 4.5, there exists $c_{0}>0$ depending on $u_{0}$ and a Palais-
Smale sequence $(u_{n})_{n}\subset H^{1}_{0}(\Omega)$ for $\Psi$ at level
$c_{0}$ such that

$c_{0}\leq\sup_{t\geq 0}\Psi(tu_{0})<\beta^{*}.$

By Lemma 4.3, there exists $u\in{\mathcal{H}}^{1}_{0}(\Omega)\setminus\\{0\\}$
such that, up to a subsequence,

$u_{n}\longmapsto u\quad\textrm{strongly in $H^{1}_{0}(\Omega)$ as
$n\to\infty$ and }\Psi^{\prime}(u)=0.$

The last equality corresponds exactly to the Euler-Lagrange equation (4.2).
This then ends the proof. ∎

##  5\. Existence of solution in domains: Proof of Theorem 1.3

Next, we let $w\in\mathcal{D}^{1,2}(\mathbb{R}^{N})$ be a positive ground
state solution of

$-\Delta
w=\lambda|z|^{-s_{1}}w^{2^{*}_{s_{1}}-1}+|z|^{-s_{2}}w^{2^{*}_{s_{2}}-1}\qquad\textrm{
in }\mathbb{R}^{N}$ (5.1)

and

$\beta^{*}=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx-\frac{\lambda}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|z|^{-s_{1}}|w|^{2^{*}_{s_{1}}}dx-\frac{1}{2^{*}_{s_{2}}}\int_{\mathbb{R}^{N}}|z|^{-s_{2}}|w|^{2^{*}_{s_{2}}}dx.$

In what follows, we define

$A_{N,s_{1},s_{2}}:=\frac{\displaystyle\int_{\mathbb{R}^{N}}|z|^{2}|\partial_{y}w|^{2}dx+\int_{\mathbb{R}^{N}}|z|^{2}|\nabla
w|^{2}dx-\frac{\lambda}{2^{*}_{s_{1}}}\int_{\mathbb{R}^{N}}|z|^{2-s_{1}}|w|^{2^{*}_{s_{1}}}dx-\frac{1}{2^{*}_{s_{2}}}\int_{\mathbb{R}^{N}}|z|^{2-s_{2}}|w|^{2^{*}_{s_{2}}}dx}{\displaystyle
2(N-1)\int_{\mathbb{R}^{N}}w^{2}dx},$

for $N\geq 5$ and $A_{4,s_{1},s_{2}}:=3/2.$ Then we have the following result.

######  Proposition 5.1.

For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$. We
assume that

$A^{N}_{s_{1},s_{2}}|\kappa(y_{0})|^{2}+h(y_{0})<0,$ (5.2)

for some positive constant. Then there exists $u\in
H^{1}_{0}(\Omega)\setminus\\{0\\}$ such that

$c^{*}:=\max_{t\geq 0}\Psi(tu)<\beta^{*}.$

Let $\Omega$ a bounded domain of $\mathbb{R}^{N}$ and $\Gamma\subset\Omega$ be
a smooth closed curve. We let
$\eta\in\mathcal{C}^{\infty}_{c}\left(F_{y_{0}}\left({Q}_{2r}\right)\right)$
be such that

$0\leq\eta\leq 1\qquad\textrm{ and }\qquad\eta\equiv 1\quad\textrm{in
}{Q}_{r}.$

For $\varepsilon>0$, we consider the test function
$u_{\varepsilon}:\Omega\to\mathbb{R}$ given by

$u_{\varepsilon}(y):=\varepsilon^{\frac{2-N}{2}}\eta(F^{-1}_{y_{0}}(y))w\left(\varepsilon^{-1}{F^{-1}_{y_{0}}(y)}\right).$
(5.3)

In particular, for every $x=(t,z)\in\mathbb{R}\times\mathbb{R}^{N-1}$, we have

$u_{\varepsilon}\left(F_{y_{0}}(x)\right):=\varepsilon^{\frac{2-N}{2}}\eta\left(x\right)\theta\left({|t|}/{\varepsilon},{|z|}/{\varepsilon}\right).$
(5.4)

It is clear that $u_{\varepsilon}\in H^{1}_{0}(\Omega).$ Moreover, for $t\geq
0$, we have

$\Psi(tu_{\varepsilon})=\frac{t^{2}}{2}\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}+h(x)u_{\varepsilon}^{2}dx-\lambda\frac{t^{2^{*}_{s_{1}}}}{2^{*}_{s_{1}}}\int_{\Omega}\rho_{\Gamma}^{-s_{1}}|u_{\varepsilon}|^{2^{*}_{s_{1}}}dx-\frac{t^{2^{*}_{s_{2}}}}{2^{*}_{s_{2}}}\int_{\Omega}\rho_{\Gamma}^{-s_{2}}|u_{\varepsilon}|^{2^{*}_{s_{2}}}dx.$
(5.5)

To simplify the notations, we will write $F$ in the place of $F_{y_{0}}$.
Recalling (5.3), we write

$u_{\varepsilon}(y)=\varepsilon^{\frac{2-N}{2}}\eta(F^{-1}(y))W_{\varepsilon}(y),$

where $W_{\varepsilon}(y)=w\left(\frac{F^{-1}(y)}{\varepsilon}\right)$.

######  Lemma 5.2.

As $\varepsilon\to 0$, we have

$\displaystyle\int_{\Omega}$ $\displaystyle|\nabla
u_{\varepsilon}|^{2}dy+\int_{\Omega}h(x)u_{\varepsilon}^{2}(x)dx=\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{N-1}\int_{\mathbb{R}^{N}}|z|^{2}\left|{\partial_{t}w}\right|^{2}dx$
$\displaystyle+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{\mathbb{R}^{N}}|z|^{2}|\nabla
w|^{2}dx+\varepsilon^{2}h(y_{0})\int_{\mathbb{R}^{N}}w^{2}(x)dx+O\left(\varepsilon^{N-2}\right)\qquad\textrm{
for $N\geq 5$}.$

For $N=4$, there exists $C>0$, we have

$\displaystyle\int_{\Omega}$ $\displaystyle|\nabla
u_{\varepsilon}|^{2}dy+\int_{\Omega}h(x)u_{\varepsilon}^{2}(x)dx\leq\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx+C\varepsilon^{2}\left(\frac{3}{2}|\kappa(y_{0})|^{2}+h(y_{0})\right)|\ln(\varepsilon)|+O(\varepsilon^{2}).$

###### Proof.

We have

$|\nabla u_{\varepsilon}|^{2}=\varepsilon^{2-N}\left(\eta^{2}|\nabla
W_{\varepsilon}|^{2}+\eta^{2}|\nabla W_{\varepsilon}|^{2}+\frac{1}{2}\nabla
W_{\varepsilon}^{2}\cdot\nabla\eta^{2}\right).$

Then integrating by parts, we get

$\displaystyle\displaystyle\int_{\Omega}|\nabla u_{\varepsilon}|^{2}dy$
$\displaystyle\displaystyle=\varepsilon^{2-N}\int_{F\left({Q}_{2r}\right)}\eta^{2}|\nabla
W_{\varepsilon}|^{2}dy+\varepsilon^{2-N}\int_{F\left({Q}_{2r}\right)\setminus
F\left({Q}_{r}\right)}W_{\varepsilon}^{2}\left(|\nabla\eta|^{2}-\frac{1}{2}\Delta\eta^{2}\right)dy$
$\displaystyle\displaystyle=\varepsilon^{2-N}\int_{F\left({Q}_{2r}\right)}\eta^{2}|\nabla
W_{\varepsilon}|^{2}dy-\varepsilon^{2-N}\int_{F\left({Q}_{2r}\right)\setminus
F\left({Q}_{r}\right)}W_{\varepsilon}^{2}\eta\Delta\eta dy$
$\displaystyle\displaystyle=\varepsilon^{2-N}\int_{F\left({Q}_{2r}\right)}\eta^{2}|\nabla
W_{\varepsilon}|^{2}dy+O\left(\varepsilon^{2-N}\int_{F\left({Q}_{2r}\right)\setminus
F\left({Q}_{r}\right)}W_{\varepsilon}^{2}dy\right).$ (5.6)

By the change of variable $y=\frac{F(x)}{\varepsilon}$ and (5.4), we can apply
Lemma 3.3, to get

$\displaystyle\displaystyle\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}dy\displaystyle=\int_{{Q}_{r/\varepsilon}}|\nabla
w|^{2}_{g_{\varepsilon}}\sqrt{|g_{\varepsilon}|}dx+O\left(\varepsilon^{2}\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}w^{2}dx+\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|\nabla
w|^{2}dx\right)$ $\displaystyle=\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{N-1}\int_{{Q}_{r/\varepsilon}}|z|^{2}\left|{\partial_{t}w}\right|^{2}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{{Q}_{r/\varepsilon}}|z|^{2}|\nabla
w|^{2}dx$
$\displaystyle+O\left(\varepsilon^{3}\int_{{Q}_{r/\varepsilon}}|x|^{3}|\nabla
w|^{2}dx+\varepsilon^{2}\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|w|^{2}dx+\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}|\nabla
w|^{2}dx+\varepsilon^{2}\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|z|^{2}|\nabla
w|^{2}dx\right).$

By Proposition 2.4, we have, for $N\geq 4$, that

$\displaystyle\varepsilon^{3}\int_{{Q}_{r/\varepsilon}}|x|^{3}|\nabla
w|^{2}dx$
$\displaystyle+\varepsilon^{2}\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|w|^{2}dx+\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}|\nabla
w|^{2}dx$
$\displaystyle+\varepsilon^{2}\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|z|^{2}|\nabla
w|^{2}dx=O(\varepsilon^{N-2})$

and

$\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}w^{2}dx+\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}|z|^{2}\left|{\partial_{t}w}\right|^{2}dx+\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}|z|^{2}|\nabla
w|^{2}dx=O(\varepsilon^{N-4})\qquad\forall N\geq 5.$

Therefore if $N\geq 5$, we have

$\displaystyle\displaystyle\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}dy\displaystyle=\int_{\mathbb{R}^{N}}|\nabla w|^{2}dx$
$\displaystyle+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{N-1}\int_{\mathbb{R}^{N}}|z|^{2}\left|{\partial_{t}w}\right|^{2}dx$
$\displaystyle+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{\mathbb{R}^{N}}|z|^{2}|\nabla
w|^{2}dx+O\left(\varepsilon^{N-2}\right).$ (5.7)

For $N=4,$ we have

$\displaystyle\displaystyle\int_{\Omega}|\nabla
u_{\varepsilon}|^{2}dy\displaystyle$
$\displaystyle\leq\int_{\mathbb{R}^{N}}|\nabla
w|^{2}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2}\int_{{Q}_{r/\varepsilon}}|z|^{2}\left|\nabla
w\right|^{2}dx+O\left(\varepsilon^{2}\right).$ (5.8)

Next, by the change of variable formula $y=\frac{F(x)}{\varepsilon}$,(5.4) and
the continuity of the function $h$, we have

$\int_{\Omega}h(x)u_{\varepsilon}^{2}(x)dx=\varepsilon^{2}h(y_{0})\int_{Q_{r/\varepsilon}}w^{2}(x)dx+\varepsilon^{2}\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}w^{2}(x)dx.$

Using again Proposition 2.3, we get

$\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}w^{2}(x)dx=O\left(\varepsilon^{N-2}\right).$

Moreover for $N\geq 5$, we have

$\int_{\mathbb{R}^{N}\setminus
Q_{r/\varepsilon}}w^{2}(x)dx=O\left(\varepsilon^{N-2}\right)$

Therefore

$\int_{\Omega}u_{\varepsilon}^{2}(x)dx=\varepsilon^{2}h(y_{0})\int_{\mathbb{R}^{N}}w^{2}(x)dx+o\left(\varepsilon^{2}\right).$
(5.9)

If $N=4$, we have

$\int_{\Omega}u_{\varepsilon}^{2}(x)dx=\varepsilon^{2}h(y_{0})\int_{Q_{r/\varepsilon}}w^{2}(x)dx+O\left(\varepsilon^{2}\right).$
(5.10)

Next, we assume that $N=4$ and we let $\eta_{\varepsilon}(x)=\eta(\varepsilon
x)$. We multiply (5.1) by $|z|^{2}\eta_{\varepsilon}w$ and integrate by parts
to get

$\displaystyle\lambda\int_{Q_{2r/\varepsilon}}\eta_{\varepsilon}|z|^{2-s_{1}}w^{2^{*}_{s_{1}}}dx+\int_{Q_{2r/\varepsilon}}\eta_{\varepsilon}|z|^{2-s_{2}}w^{2^{*}_{s_{2}}}dx\displaystyle=\int_{Q_{2r/\varepsilon}}\nabla
w\cdot\nabla\left(\eta_{\varepsilon}|z|^{2}w\right)dx$
$\displaystyle\displaystyle=\int_{Q_{2r/\varepsilon}}\eta_{\varepsilon}|z|^{2}|\nabla
w|^{2}dx+\frac{1}{2}\int_{Q_{2r/\varepsilon}}\nabla
w^{2}\cdot\nabla\left(|z|^{2}\eta_{\varepsilon}\right)dx\int_{Q_{2r/\varepsilon}}\eta_{\varepsilon}|z|^{2}|\nabla
w|^{2}dx-\frac{1}{2}\int_{Q_{2r/\varepsilon}}w^{2}\Delta\left(|z|^{2}\eta_{\varepsilon}\right)dx$
$\displaystyle\displaystyle=\int_{Q_{2r/\varepsilon}}\eta_{\varepsilon}|z|^{2}|\nabla
w|^{2}dx-3\int_{Q_{2r/\varepsilon}}w^{2}\eta_{\varepsilon}dx=\displaystyle\quad-\frac{1}{2}\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}w^{2}(|z|^{2}\Delta\eta_{\varepsilon}+4\nabla\eta_{\varepsilon}\cdot
z)dx.$

We then deduce that

$\displaystyle\lambda\int_{Q_{2r/\varepsilon}}|z|^{2-s_{1}}w^{2^{*}_{s_{1}}}dx$
$\displaystyle+\int_{Q_{2r/\varepsilon}}|z|^{2-s_{2}}w^{2^{*}_{s_{2}}}dx=\int_{Q_{r/\varepsilon}}|z|^{2}|\nabla
w|^{2}dx-(N-1)\int_{Q_{r/\varepsilon}}w^{2}dx$
$\displaystyle+O\left(\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z|^{2-\sigma}w^{2^{*}_{\sigma}}dx+\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z|^{2}|\nabla w|^{2}dx+\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}w^{2}dx\right)$
$\displaystyle+O\left(\varepsilon\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z||\nabla
w|dx+\varepsilon^{2}\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z|^{2}w^{2}dx\right).$

By Proposition 2.3, we have

$\lambda\int_{Q_{2r/\varepsilon}}|z|^{2-s_{1}}w^{2^{*}_{s_{1}}}dx+\int_{Q_{2r/\varepsilon}}|z|^{2-s_{2}}w^{2^{*}_{s_{2}}}dx=O(1)$

and

$\displaystyle+\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z|^{2-\sigma}w^{2^{*}_{\sigma}}dx$
$\displaystyle+\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z|^{2}|\nabla w|^{2}dx+\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}w^{2}dx$
$\displaystyle+\varepsilon\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z||\nabla
w|dx+\varepsilon^{2}\int_{Q_{2r/\varepsilon}\setminus
Q_{r/\varepsilon}}|z|^{2}w^{2}dx=O(\varepsilon^{2}).$

Therefore

$\displaystyle\int_{Q_{r/\varepsilon}}|z|^{2}|\nabla
w|^{2}dx=3\int_{Q_{r/\varepsilon}}w^{2}dx+O(1).$ (5.11)

To finish, we use Proposition 2.3 to get

$\displaystyle\int_{Q_{r/\varepsilon}}w^{2}dx\leq
C\int_{Q_{r/\varepsilon}}\frac{dx}{1+|x|^{4}}=C|S^{3}|\int_{0}^{r\varepsilon}\frac{t^{3}dt}{1+t^{4}}\leq
C(1+|\ln(\varepsilon)|),$ (5.12)

where $C$ is a positive constant that may change from an inequality to
another. Thus the result follows immediately from (5), (5.8), (5.9), (5.10)
(5.11) and (5.12). This then ends the proof. ∎

######  Lemma 5.3.

Let $s\in(0,2)$. Then we have

$\int_{\Omega}\rho^{-s}_{\Gamma}|u_{\varepsilon}|^{2^{*}_{s}}dx=\int_{\mathbb{R}^{N}}|z|^{-s}w^{2^{*}_{s}}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{\mathbb{R}^{N}}|z|^{2-s}w^{2^{*}_{s}}dx+O\left(\varepsilon^{N-s}\right).$

###### Proof.

Let $s\in[0,2)$. Then by the change of variable $y=\frac{F(x)}{\varepsilon}$,
(3.1) and (3.5), we get

$\displaystyle\displaystyle\int_{\Omega}\rho^{-s}_{\Gamma}|u_{\varepsilon}|^{2^{*}_{s}}dy=\int_{{Q}_{r/\varepsilon}}|z|^{-s}w^{2^{*}_{s}}\sqrt{|g_{\varepsilon}|}dx+O\left(\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|z|^{-s}(\eta(\varepsilon
x)w)^{2^{*}_{s}}dx\right)$
$\displaystyle\displaystyle=\int_{{Q}_{r/\varepsilon}}|z|^{-s}w^{2^{*}_{s}}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{{Q}_{r/\varepsilon}}|z|^{2-s}w^{2^{*}_{s}}dx$
$\displaystyle+O\left(\varepsilon^{3}\int_{{Q}_{r/\varepsilon}}|x|^{3}|z|^{-s}w^{2^{*}_{s}}dx+\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|z|^{-s}w^{2^{*}_{s}}dx\right)$
$\displaystyle\displaystyle=\int_{\mathbb{R}^{N}}|z|^{-s}w^{2^{*}_{s}}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{{Q}_{r/\varepsilon}}|z|^{2-s}w^{2^{*}_{s}}dx$
$\displaystyle\quad+O\left(\varepsilon^{3}\int_{{Q}_{r/\varepsilon}}|x|^{3}|z|^{-s}w^{2^{*}_{s}}dx+\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}|z|^{-s}w^{2^{*}_{s}}dx+\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|z|^{-s}w^{2^{*}_{s}}dx\right).$

By Proposition 2.3, we have

$\varepsilon^{3}\int_{{Q}_{r/\varepsilon}}|x|^{3}|z|^{-s}w^{2^{*}_{s}}dx+\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}|z|^{-s}w^{2^{*}_{s}}dx+\int_{{Q}_{2r/\varepsilon}\setminus{Q}_{r/\varepsilon}}|z|^{-s}w^{2^{*}_{s}}dx=O\left(\varepsilon^{N-s}\right)$

and

$\int_{\mathbb{R}^{N}\setminus{Q}_{r/\varepsilon}}|z|^{2-s}w^{2^{*}_{s}}dx=O\left(\varepsilon^{N-2-s}\right)\qquad\forall
N\geq 4.$ (5.13)

Therefore

$\int_{\Omega}\rho^{-s}_{\Gamma}|u_{\varepsilon}|^{2^{*}_{s}}dx=\int_{\mathbb{R}^{N}}|z|^{-s}w^{2^{*}_{s}}dx+\varepsilon^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{\mathbb{R}^{N}}|z|^{2-s}w^{2^{*}_{s}}dx+O\left(\varepsilon^{N-s}\right),$
(5.14)

as $\varepsilon\to 0$. This then ends the proof. ∎

Now we are in position to prove Proposition 5.1.

###### Proof.

of Proposition 5.1
Recall that, for all $t\geq 0$ and all $u\in H^{1}_{0}(\Omega)$, we have

$\Psi(tu):=\frac{t^{2}}{2}\int_{\Omega}|\nabla
u|^{2}dx+\frac{1}{2}\int_{\Omega}h(x)u^{2}dx-t^{2^{*}_{s_{1}}}\frac{\lambda}{2^{*}_{s_{1}}}\int_{\Omega}\frac{|u|^{2^{*}_{s_{1}}}}{\rho_{\Gamma}^{s_{1}}(x)}dx-t^{2^{*}_{s_{2}}}\frac{1}{2^{*}_{s_{2}}}\int_{\Omega}\frac{|u|^{2^{*}_{s_{2}}}}{\rho_{\Gamma}^{s_{2}}(x)}dx.$

Then by Lemma 5.2 and Lemma 5.3, we have

$\displaystyle J\left(tu_{\varepsilon}\right)=\Psi(tw)$
$\displaystyle+\varepsilon^{2}t^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\left(\int_{{Q}_{r/\varepsilon}}|z|^{2}\left|{\partial_{t}w}\right|^{2}dx+\int_{{Q}_{r/\varepsilon}}|z|^{2}|\nabla
w|^{2}dx\right)$
$\displaystyle+\varepsilon^{2}t^{2}h(y_{0})\int_{{Q}_{r/\varepsilon}}w^{2}dx+\varepsilon^{2}\lambda\frac{t^{2^{*}_{s_{1}}}}{2^{*}_{s_{1}}}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{{Q}_{r/\varepsilon}}|z|^{2-s_{1}}w^{2^{*}_{s_{1}}}dx$
$\displaystyle+\varepsilon^{2}\frac{t^{2^{*}_{s_{2}}}}{2^{*}_{s_{2}}}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{{Q}_{r/\varepsilon}}|z|^{2-s_{2}}w^{2^{*}_{s_{2}}}dx+O\left(\varepsilon^{N-2}\right)\qquad\textrm{
for $N\geq 5$}.$

For $N=4$, there exists $C>0$, we have

$\displaystyle
J(tu_{\varepsilon})\leq\Psi(tw)+C\varepsilon^{2}t^{2}\left(\frac{3}{2}|\kappa(y_{0})|^{2}+h(y_{0})\right)|\ln(\varepsilon)|+O(\varepsilon^{2}).$

Since $2^{*}_{s_{2}}>2^{*}_{s_{1}}$, $J(tu_{\varepsilon})$ has a unique
maximum, we have

$\max_{t\geq 0}\Psi(tw)=\Psi(w)=\beta^{*}.$

Therefore, the maximum of $J(tu_{\varepsilon})$ occurs at
$t_{\varepsilon}:=1+o_{\varepsilon}(1)$. Next setting

$\displaystyle\mathcal{G}(tw):$
$\displaystyle=\displaystyle\varepsilon^{2}t^{2}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\left(\int_{\mathbb{R}^{N}}|z|^{2}\left|{\partial_{t}w}\right|^{2}dx+\int_{\mathbb{R}^{N}}|z|^{2}|\nabla
w|^{2}dx\right)$
$\displaystyle+\varepsilon^{2}t^{2}h(y_{0})\int_{\mathbb{R}^{N}}w^{2}dx+\varepsilon^{2}\displaystyle\lambda\frac{t^{2^{*}_{s_{1}}}}{2^{*}_{s_{1}}}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{\mathbb{R}^{N}}|z|^{2-s_{1}}w^{2^{*}_{s_{1}}}dx$
$\displaystyle+\varepsilon^{2}\frac{t^{2^{*}_{s_{2}}}}{2^{*}_{s_{2}}}\frac{|\kappa(y_{0})|^{2}}{2(N-1)}\int_{\mathbb{R}^{N}}|z|^{2-s_{2}}w^{2^{*}_{s_{2}}}dx+o(\varepsilon^{2})\quad\textrm{
for $N\geq 5$},$

and

$\mathcal{G}(tw)=C\varepsilon^{2}|\ln(\varepsilon)|t^{2}\left(\frac{3}{2}|\kappa(y_{0})|^{2}+h(y_{0})\right)+O(\varepsilon^{2})\qquad\textrm{
for $N=4$}.$

Thanks to assumption (5.2), we have

$\mathcal{G}(w)<0.$

Therefore

$\max_{t\geq
0}J(tu_{\varepsilon}):=J(t_{\varepsilon}u_{\varepsilon})\leq\Psi(t_{\varepsilon}w)+\varepsilon^{2}\mathcal{G}(t_{\varepsilon}w)\leq\Psi(t_{\varepsilon}w)<\Psi(w)=\beta^{*}$

We thus get the desired result. ∎

###### Proof.

of Theorem 1.3 The proof of Theorem 1.3 is a direct consequence of Proposition
4.1 and Proposition 5.1. ∎

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