A compactness result for a system with weight and boundary singularity

TL;DR

This paper investigates the blow-up behavior and compactness properties of solutions to an elliptic system with boundary singularities and weighted conditions, providing new insights into the system's solution structure.

Contribution

It presents a novel compactness result for elliptic systems with boundary singularities and weighted conditions, extending previous understanding of solution behavior.

Findings

Established blow-up behavior for solutions with boundary singularities.

Proved a compactness theorem for systems with Hölderian weights and boundary singularities.

Demonstrated conditions under which solutions remain bounded or blow up.

Abstract

We give blow-up behavior for solutions to an elliptic system with Dirichlet condition, and, weight and boundary singularity. Also, we have a compactness result for this elliptic system with regular H{\"o}lderian weight and boundary singularity and Lipschitz condition.

Equations

$$(P)\left\{\begin{split}-\Delta u&=|x|^{2\beta}Ve^{v}\,\,&\text{in}\,\,&\Omega\subset{\mathbb{R}}^{2},\\ -\Delta v&=We^{u}\,\,&\text{in}\,\,&\Omega\subset{\mathbb{R}}^{2},\\ u&=0\,\,&\text{in}\,\,&\partial\Omega,\\ v&=0\,\,&\text{in}\,\,&\partial\Omega.\end{split}\right.$$

$$0\le V\le b_1<+\infty ,e^u\in L^1(\Omega )\mathrm{and}u\in W_0^{1,1}(\Omega ),$$

$$0\le W\le b_2<+\infty ,|x{|}^{2\beta }{e}^v\in L^1(\Omega )\mathrm{and}v\in W_0^{1,1}(\Omega ),$$

$$0\in \partial \Omega ,\beta \ge 0.$$

$$\underset{K}{\sup }{u}_i\le c=c(a,b,K,\Omega ).$$

$${\int }_{\Omega }{e}^{u_i}dy\le C,$$

$$\underset{K}{\sup }{u}_i\le c=c(b,C,K,\Omega ).$$

$$E={\int }_{\Omega }{e}^{u_i}dy.$$

$${\int }_{\Omega }{e}^{u_i}dy\le C,$$

$$\underset{\Omega }{\sup }{u}_i\to +\infty .$$

$$C\left(\frac{a}{b}\right)\underset{K}{\sup }{u}_i+\underset{\Omega }{\inf }{u}_i\le c=c(a,b,K,\Omega ).$$

$${\int }_{\Omega }{e}^{u_i}dx\le C_1,{\int }_{\Omega }|x{|}^{2\beta }{e}^{v_i}dx\le C_2,$$

$$||u_i|{|}_{L^{\infty }}\le C_3=C_3(\beta ,C_1,C_2,\Omega )?$$

$$||v_i|{|}_{L^{\infty }}\le C_4=C_4(\beta ,C_1,C_2,\Omega )?$$

$$0\le V_i\le b_1,0\le W_i\le b_2,$$

$$||\nabla V_i|{|}_{L^{\infty }}\le A_1,||\nabla W_i|{|}_{L^{\infty }}\le A_2,\beta \ge 0.$$

$$0\le V_i\le b_1,\mathrm{and}{\int }_{\Omega }{e}^{u_i}dx\le C_1,\forall i,$$

$$0\le W_i\le b_2,\mathrm{and}{\int }_{\Omega }|x{|}^{2\beta }{e}^{v_i}dx\le C_2,\forall i,$$

$${\int }_{\partial \Omega }{\partial }_{\nu }{u}_i\phi \to {\int }_{\partial \Omega }{\partial }_{\nu }u\phi +\sum _{j=1}^N{\alpha }_j\phi (x_j),{\alpha }_j\ge 4\pi ,$$

$$u_i\to u\mathrm{in}{C}_{loc}^1(\bar{\Omega }-\{x_1,\dots ,x_N\}).$$

$${\int }_{\partial \Omega }{\partial }_{\nu }{v}_i\phi \to {\int }_{\partial \Omega }{\partial }_{\nu }v\phi +\sum _{j=1}^N{\beta }_j\phi (x_j),{\beta }_j\ge 4\pi ,$$

$$v_i\to v\mathrm{in}{C}_{loc}^1(\bar{\Omega }-\{x_1,\dots ,x_N\}).$$

$$x_1=0\in \partial \Omega ,\beta \ge 0,$$

$$0\le V_i\le b_1,||\nabla V_i|{|}_{L^{\infty }}\le A_1,\mathrm{and}{\int }_{\Omega }{e}^{u_i}\le C_1,$$

$$0\le W_i\le b_2,||\nabla W_i|{|}_{L^{\infty }}\le A_2,\mathrm{and}{\int }_{\Omega }|x{|}^{2\beta }{e}^{v_i}\le C_2,$$

$$||u_i|{|}_{L^{\infty }}\le C_3(b_1,b_2,\beta ,A_1,A_2,C_1,C_2,\Omega ),$$

$$||v_i|{|}_{L^{\infty }}\le C_4(b_1,b_2,\beta ,A_1,A_2,C_1,C_2,\Omega ),$$

$$u_i,v_i\in W_0^{1,1}(\Omega ).$$

$$u_i\in W^{2,k}(\Omega )\cap C^{1,ϵ}(\bar{\Omega }).$$

$$v_i,u_i\in W_0^{1,1}(\Omega ).$$

Full text

Mathematics Subject Classification: 35J60 35B45 35B50

Keywords: blow-up, boundary, system, Dirichlet condition, a priori estimate, analytic domain, regular weight, boundary singularity, Lipschitz condition.

1. Introduction and Main Results

We set $\Delta=\partial_{11}+\partial_{22}$ on open analytic domain $\Omega$ of ${\mathbb{R}}^{2}$.

We consider the following equation:

[TABLE]

Here, we assume that:

[TABLE]

[TABLE]

and,

[TABLE]

When $u=v$ and $\beta=0$, the above system is reduced to an equation which was studied by many authors, with or without the boundary condition, also for Riemann surfaces, see [1-17], one can find some existence and compactness results, also for a system.

Among other results, we can see in [6] the following important Theorems ($\beta=0$):,

Theorem A.(Brezis-Merle [6]).Consider the case of one equation; if $(u_{i})_{i}=(v_{i})_{i}$ and $(V_{i})_{i}=(W_{i})_{i}$ are two sequences of functions relatively to the problem $(P)$ with, $0<a\leq V_{i}\leq b<+\infty$, then, for all compact set $K$ of $\Omega$,

[TABLE]

Theorem B (Brezis-Merle [6]).Consider the case of one equation and assume that $(u_{i})_{i}$ and $(V_{i})_{i}$ are two sequences of functions relatively to the previous problem $(P)$ with, $0\leq V_{i}\leq b<+\infty$, and,

[TABLE]

then, for all compact set $K$ of $\Omega$,

[TABLE]

Next, we call energy the following quantity:

[TABLE]

The boundedness of the energy is a necessary condition to work on the problem $(P)$ as showed in $[6]$, by the following counterexample ($\beta=0$):

Theorem C (Brezis-Merle [6]).Consider the case of one equation, then there are two sequences $(u_{i})_{i}$ and $(V_{i})_{i}$ of the problem $(P)$ with, $0\leq V_{i}\leq b<+\infty$, and,

[TABLE]

and

[TABLE]

When $\beta=0$, the above system have many properties in the constant and the Lipschitzian cases. Indeed we have (when $\beta=0$):

In [12], Dupaigne-Farina-Sirakov proved (by an existence result of Montenegro, see [16]) that the solutions of the above system when $V$ and $W$ are constants can be extremal and this condition imply the boundedness of the energy and directly the compactness. Note that in [11], if we assume (in particular) that $\nabla\log V$ and $\nabla\log W$ and $V>a>0$ or $W>a^{\prime}>0$ and $V,W$ are nonegative and uniformly bounded then the energy is bounded and we have a compactness result.

Note that in the case of one equation (and $\beta=0$), we can prove by using the Pohozaev identity that if $+\infty>b\geq V\geq a>0$, $\nabla V$ is uniformely Lipschitzian that the energy is bounded when $\Omega$ is starshaped. In [15] Ma-Wei, using the moving-plane method showed that this fact is true for all domain $\Omega$ with the same assumptions on $V$. In [11] De Figueiredo-do O-Ruf extend this fact to a system by using the moving-plane method for a system.

Theorem C, shows that we have not a global compactness to the previous problem with one equation, perhaps we need more information on $V$ to conclude to the boundedness of the solutions. When $\nabla\log V$ is Lipschitz function and $\beta=0$, Chen-Li and Ma-Wei see [7] and [15], showed that we have a compactness on all the open set. The proof is via the moving plane-Method of Serrin and Gidas-Ni-Nirenberg. Note that in [11], we have the same result for this system when $\nabla\log V$ and $\nabla\log W$ are uniformly bounded. We will see below that for a system we also have a compactness result when $V$ and $W$ are Lipschitzian and $\beta\geq 0$.

Now consider the case of one equation. In this case our equation have nice properties.

If we assume $V$ with more regularity, we can have another type of estimates, a $\sup+\inf$ type inequalities. It was proved by Shafrir see [17], that, if $(u_{i})_{i},(V_{i})_{i}$ are two sequences of functions solutions of the previous equation without assumption on the boundary and, $0<a\leq V_{i}\leq b<+\infty$, then we have the following interior estimate:

[TABLE]

Now, if we suppose $(V_{i})_{i}$ uniformly Lipschitzian with $A$ the Lipschitz constant, then, $C(a/b)=1$ and $c=c(a,b,A,K,\Omega)$, see [5].

Here we are interested by the case of a system of this type of equation. First, we give the behavior of the blow-up points on the boundary, with weight and boundary singularity, and in the second time we have a proof of compactness of the solutions to Gelfand-Liouville type system with weight and boundary singularity and Lipschitz condition.

Here, we write an extention of Brezis-Merle Problem (see [6]) to a system:

Problem. Suppose that $V_{i}\to V$ and $W_{i}\to W$ in $C^{0}(\bar{\Omega})$, with, $0\leq V_{i}$ and $0\leq W_{i}$. Also, we consider two sequences of solutions $(u_{i}),(v_{i})$ of $(P)$ relatively to $(V_{i}),(W_{i})$ such that,

[TABLE]

is it possible to have:

[TABLE]

and,

[TABLE]

In this paper we give a caracterization of the behavior of the blow-up points on the boundary and also a proof of the compactness theorem when $V_{i}$ and $W_{i}$ are uniformly Lipschitzian and $\beta\geq 0$. For the behavior of the blow-up points on the boundary, the following condition are enough,

[TABLE]

The conditions $V_{i}\to V$ and $W_{i}\to W$ in $C^{0}(\bar{\Omega})$ are not necessary.

But for the proof of the compactness for the system, we assume that:

[TABLE]

Our main result are:

** Theorem 1.1****.**

Assume that $\max_{\Omega}u_{i}\to+\infty$ and $\max_{\Omega}v_{i}\to+\infty$ Where $(u_{i})$ and $(v_{i})$ are solutions of the probleme $(P)$ with ($\beta\geq 0$), and:

[TABLE]

and,

[TABLE]

then; after passing to a subsequence, there is a finction $u$, there is a number $N\in{\mathbb{N}}$ and $N$ points $x_{1},x_{2},\ldots,x_{N}\in\partial\Omega$, such that,

[TABLE]

$\text{for\, any}\,\,\varphi\in C^{0}(\partial\Omega)$, and,

[TABLE]

[TABLE]

$\text{for\, any}\,\,\varphi\in C^{0}(\partial\Omega)$, and,

[TABLE]

In the following theorem, we have a proof for the global a priori estimate which concern the problem $(P)$.

** Theorem 1.2****.**

Assume that $(u_{i}),(v_{i})$ are solutions of $(P)$ relatively to $(V_{i}),(W_{i})$ with the following conditions:

[TABLE]

and,

[TABLE]

[TABLE]

We have,

[TABLE]

and,

[TABLE]

2. Proof of the theorems

Proof of theorem 1.1:**

We have:

[TABLE]

Since $e^{u_{i}}\in L^{1}(\Omega)$ by the corollary 1 of Brezis-Merle’s paper (see [6]) we have $e^{v_{i}}\in L^{k}(\Omega)$ for all $k>2$ and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:

[TABLE]

And,

We have:

[TABLE]

Since $|x|^{2\beta}e^{v_{i}}\in L^{1}(\Omega)$ by the corollary 1 of Brezis-Merle’s paper (see [6]) we have $e^{u_{i}}\in L^{k}(\Omega)$ for all $k>2$ and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:

[TABLE]

Since $|x|^{2\beta}V_{i}e^{v_{i}}$ and $W_{i}e^{u_{i}}$ are bounded in $L^{1}(\Omega)$, we can extract from those two sequences two subsequences which converge to two nonegative measures $\mu_{1}$ and $\mu_{2}$. (This procedure is similar to the procedure of Brezis-Merle, we apply corollary 4 of Brezis-Merle paper, see [6]).

If $\mu_{1}(x_{0})<4\pi$, by a Brezis-Merle estimate for the first equation, we have $e^{u_{i}}\in L^{1+\epsilon}$ around $x_{0}$, by the elliptic estimates, for the second equation, we have $v_{i}\in W^{2,1+\epsilon}\subset L^{\infty}$ around $x_{0}$, and , returning to the first equation, we have $u_{i}\in L^{\infty}$ around $x_{0}$.

If $\mu_{2}(x_{0})<4\pi$, then $u_{i}$ and $v_{i}$ are also locally bounded around $x_{0}$.

Thus, we take a look to the case when, $\mu_{1}(x_{0})\geq 4\pi$ and $\mu_{2}(x_{0})\geq 4\pi$. By our hypothesis, those points $x_{0}$ are finite.

We will see that inside $\Omega$ no such points exist. By contradiction, assume that, we have $\mu_{1}(x_{0})\geq 4\pi$. Let us consider a ball $B_{R}(x_{0})$ which contain only $x_{0}$ as nonregular point. Thus, on $\partial B_{R}(x_{0})$, the two sequence $u_{i}$ and $v_{i}$ are uniformly bounded. Let us consider:

[TABLE]

By the maximum principle we have:

[TABLE]

and $z_{i}\to z$ almost everywhere on this ball, and thus,

[TABLE]

and,

[TABLE]

but, $z$ is a solution in $W_{0}^{1,q}(B_{R}(x_{0}))$, $1\leq q<2$, of the following equation:

[TABLE]

with, $\mu_{1}\geq 4\pi$ and thus, $\mu_{1}\geq 4\pi\delta_{x_{0}}$ and then, by the maximum principle in $W_{0}^{1,q}(B_{R}(x_{0}))$:

[TABLE]

thus,

[TABLE]

which is a contradiction. Thus, there is no nonregular points inside $\Omega$

Thus, we consider the case where we have nonregular points on the boundary, we use two estimates:

[TABLE]

and,

[TABLE]

We have the same computations, as in the case of one equation.

We consider a points $x_{0}\in\partial\Omega$ such that:

[TABLE]

We consider a test function on the boundary $\eta$ we extend $\eta$ by a harmonic function on $\Omega$, we write the equation:

[TABLE]

with,

[TABLE]

[TABLE]

with,

[TABLE]

By the Brezis-Merle estimate, we have uniformly, $e^{u_{i}}\in L^{1+\epsilon}$ around $x_{0}$, by the elliptic estimates, for the second equation, we have $v_{i}\in W^{2,1+\epsilon}\subset L^{\infty}$ around $x_{0}$, and , returning to the first equation, we have $u_{i}\in L^{\infty}$ around $x_{0}$.

We have the same thing if we assume:

[TABLE]

Thus, if $\mu_{1}(x_{0})<4\pi$ or $\mu_{2}(x_{0})<4\pi$, we have for $R>0$ small enough:

[TABLE]

By our hypothesis the set of the points such that:

[TABLE]

is finite, and, outside this set $u_{i}$ and $v_{i}$ are locally uniformly bounded. By the elliptic estimates, we have the $C^{1}$ convergence to $u$ and $v$ on each compact set of $\bar{\Omega}-\{x_{1},\ldots x_{N}\}$.

Indeed,

By the Stokes formula we have,

[TABLE]

We use the weak convergence in the space of Radon measures to have the existence of a nonnegative Radon measure $\mu_{1}$ such that,

[TABLE]

We take an $x_{0}\in\partial\Omega$ such that, $\mu_{1}({x_{0}})<4\pi$. For $\epsilon>0$ small enough set $I_{\epsilon}=B(x_{0},\epsilon)\cap\partial\Omega$ on the unt disk or one can assume it as an interval. We choose a function $\eta_{\epsilon}$ such that,

[TABLE]

We take a $\tilde{\eta}_{\epsilon}$ such that,

[TABLE]

Remark:* We use the following steps in the construction of $\tilde{\eta}_{\epsilon}$:*

We take a cutoff function $\eta_{0}$ in $B(0,2)$ or $B(x_{0},2)$:

1- We set $\eta_{\epsilon}(x)=\eta_{0}(|x-x_{0}|/\epsilon)$ in the case of the unit disk it is sufficient.

2- Or, in the general case: we use a chart $(f,\tilde{\Omega})$ with $f(0)=x_{0}$ and we take $\mu_{\epsilon}(x)=\eta_{0}(f(|x|/\epsilon))$ to have connected sets $I_{\epsilon}$ and we take $\eta_{\epsilon}(y)=\mu_{\epsilon}(f^{-1}(y))$. Because $f,f^{-1}$ are Lipschitz, $|f(x)-x_{0}|\leq k_{2}|x|\leq 1$ for $|x|\leq 1/k_{2}$ and $|f(x)-x_{0}|\geq k_{1}|x|\geq 2$ for $|x|\geq 2/k_{1}>1/k_{2}$, the support of $\eta$ is in $I_{(2/k_{1})\epsilon}$.

[TABLE]

3- Also, we can take: $\mu_{\epsilon}(x)=\eta_{0}(|x|/\epsilon)$ and $\eta_{\epsilon}(y)=\mu_{\epsilon}(f^{-1}(y))$, we extend it by [math] outside $f(B_{1}(0))$. We have $f(B_{1}(0))=D_{1}(x_{0})$, $f(B_{\epsilon}(0))=D_{\epsilon}(x_{0})$ and $f(B_{\epsilon}^{+})=D_{\epsilon}^{+}(x_{0})$ with $f$ and $f^{-1}$ smooth diffeomorphism.

[TABLE]

And, $H_{1}(J^{\prime}_{\epsilon})\leq C_{1}H_{1}(I_{2\epsilon})=C_{1}4\epsilon$, since $f$ is Lipschitz. Here $H_{1}$ is the Hausdorff measure.

We solve the Dirichlet Problem:

[TABLE]

and finaly we set $\tilde{\eta}_{\epsilon}=-\bar{\eta}_{\epsilon}+\eta_{\epsilon}$. Also, by the maximum principle and the elliptic estimates we have :

[TABLE]

with $C_{1}$ depends on $\Omega$.

We use the following estimate, see [8],

[TABLE]

We deduce from the last estimate that, $(v_{i})$ converge weakly in $W_{0}^{1,q}(\Omega)$, almost everywhere to a function $v\geq 0$ and $\int_{\Omega}|x|^{2\beta}e^{v}<+\infty$ (by Fatou lemma). Also, $V_{i}$ weakly converge to a nonnegative function $V$ in $L^{\infty}$.

We deduce from the last estimate that, $(u_{i})$ converge weakly in $W_{0}^{1,q}(\Omega)$, almost everywhere to a function $u\geq 0$ and $\int_{\Omega}e^{u}<+\infty$ (by Fatou lemma). Also, $W_{i}$ weakly converge to a nonnegative function $W$ in $L^{\infty}$.

The function $u,v$ are in $W_{0}^{1,q}(\Omega)$ solutions of :

[TABLE]

And,

[TABLE]

According to the corollary 1 of Brezis-Merle’s result, see [6], we have $e^{ku}\in L^{1}(\Omega),k>1$. By the elliptic estimates, we have $v\in C^{1}(\bar{\Omega})$.

According to the corollary 1 of Brezis-Merle’s result, see [6], we have $e^{kv}\in L^{1}(\Omega),k>1$. By the elliptic estimates, we have $u\in C^{1}(\bar{\Omega})$.

For two vectors $f$ and $g$ we denote by $f\cdot g$ the inner product of $f$ and $g$.

We can write:

[TABLE]

[TABLE]

We use the interior esimate of Brezis-Merle, see [6],

Step 1:* Estimate of the integral of the first term of the right hand side of $(\ref{(1)})$.*

We use the Green formula between $\tilde{\eta}_{\epsilon}$ and $u$, we obtain,

[TABLE]

We have,

[TABLE]

We use the Green formula between $u_{i}$ and $\tilde{\eta}_{\epsilon}$ to have:

[TABLE]

From $(\ref{(2)})$ and $(\ref{(3)})$ we have for all $\epsilon>0$ there is $i_{0}=i_{0}(\epsilon)$ such that, for $i\geq i_{0}$,

[TABLE]

Step 2:* Estimate of integral of the second term of the right hand side of $(\ref{(1)})$.*

Let $\Sigma_{\epsilon}=\{x\in\Omega,d(x,\partial\Omega)=\epsilon^{3}\}$ and $\Omega_{\epsilon^{3}}=\{x\in\Omega,d(x,\partial\Omega)\geq\epsilon^{3}\}$, $\epsilon>0$. Then, for $\epsilon$ small enough, $\Sigma_{\epsilon}$ is hypersurface.

The measure of $\Omega-\Omega_{\epsilon^{3}}$ is $k_{2}\epsilon^{3}\leq meas(\Omega-\Omega_{\epsilon^{3}})=\mu_{L}(\Omega-\Omega_{\epsilon^{3}})\leq k_{1}\epsilon^{3}$.

Remark*: for the unit ball ${\bar{B}(0,1)}$, our new manifold is ${\bar{B}(0,1-\epsilon^{3})}$.*

( Proof of this fact; let’s consider $d(x,\partial\Omega)=d(x,z_{0}),z_{0}\in\partial\Omega$, this imply that $(d(x,z_{0}))^{2}\leq(d(x,z))^{2}$ for all $z\in\partial\Omega$ which it is equivalent to $(z-z_{0})\cdot(2x-z-z_{0})\leq 0$ for all $z\in\partial\Omega$, let’s consider a chart around $z_{0}$ and $\gamma(t)$ a curve in $\partial\Omega$, we have;

$(\gamma(t)-\gamma(t_{0})\cdot(2x-\gamma(t)-\gamma(t_{0}))\leq 0$* if we divide by $(t-t_{0})$ (with the sign and tend $t$ to $t_{0}$), we have $\gamma^{\prime}(t_{0})\cdot(x-\gamma(t_{0}))=0$, this imply that $x=z_{0}-s\nu_{0}$ where $\nu_{0}$ is the outward normal of $\partial\Omega$ at $z_{0}$))*

With this fact, we can say that $S=\{x,d(x,\partial\Omega)\leq\epsilon\}=\{x=z_{0}-s\nu_{z_{0}},z_{0}\in\partial\Omega,\,\,-\epsilon\leq s\leq\epsilon\}$. It is sufficient to work on $\partial\Omega$. Let’s consider a charts $(z,D=B(z,4\epsilon_{z}),\gamma_{z})$ with $z\in\partial\Omega$ such that $\cup_{z}B(z,\epsilon_{z})$ is cover of $\partial\Omega$ . One can extract a finite cover $(B(z_{k},\epsilon_{k})),k=1,...,m$, by the area formula the measure of $S\cap B(z_{k},\epsilon_{k})$ is less than a $k\epsilon$ (a $\epsilon$-rectangle). For the reverse inequality, it is sufficient to consider one chart around one point of the boundary.

We write,

[TABLE]

Step 2.1:* Estimate of $\int_{\Omega-\Omega_{\epsilon^{3}}}|\nabla(u_{i}-u)\cdot\nabla\tilde{\eta}_{\epsilon}|dx$.*

First, we know from the elliptic estimates that $||\nabla\tilde{\eta}_{\epsilon}||_{L^{\infty}}\leq C_{1}/\epsilon^{2}$, $C_{1}$ depends on $\Omega$

We know that $(|\nabla u_{i}|)_{i}$ is bounded in $L^{q},1<q<2$, we can extract from this sequence a subsequence which converge weakly to $h\in L^{q}$. But, we know that we have locally the uniform convergence to $|\nabla u|$ (by Brezis-Merle’s theorem), then, $h=|\nabla u|$ a.e. Let $q^{\prime}$ be the conjugate of $q$.

We have, $\forall f\in L^{q^{\prime}}(\Omega)$

[TABLE]

If we take $f=1_{\Omega-\Omega_{\epsilon^{3}}}$, we have:

[TABLE]

Then, for $i\geq i_{1}(\epsilon)$,

[TABLE]

Thus, we obtain,

[TABLE]

The constant $C_{1}$ does not depend on $\epsilon$ but on $\Omega$.

Step 2.2:* Estimate of $\int_{\Omega_{\epsilon^{3}}}|\nabla(u_{i}-u)\cdot\nabla\tilde{\eta}_{\epsilon}|dx$.*

We know that, $\Omega_{\epsilon}\subset\subset\Omega$, and ( because of Brezis-Merle’s interior estimates) $u_{i}\to u$ in $C^{1}(\Omega_{\epsilon^{3}})$. We have,

[TABLE]

We write,

[TABLE]

For $\epsilon>0$, we have for $i\in{\mathbb{N}}$, $i\geq\max\{i_{1},i_{2},i_{3}\}$,

[TABLE]

From $(\ref{(4)})$ and $(\ref{(7)})$, we have, for $\epsilon>0$, there is $i_{3}=i_{3}(\epsilon)\in{\mathbb{N}},i_{3}=\max\{i_{0},i_{1},i_{2}\}$ such that,

[TABLE]

We choose $\epsilon>0$ small enough to have a good estimate of $(\ref{(1)})$.

Indeed, we have:

[TABLE]

with $||g_{i,\epsilon}||_{L^{1}(\Omega)}\leq 4\pi-\dfrac{\epsilon_{0}}{2}.$

We can use Theorem 1 of [6] to conclude that there are $q\geq\tilde{q}>1$ such that:

[TABLE]

where, $V_{\epsilon}(x_{0})$ is a neighberhood of $x_{0}$ in $\bar{\Omega}$. Here we have used that in a neighborhood of $x_{0}$ by the elliptic estimates, $1-C\epsilon\leq\tilde{\eta}_{\epsilon}\leq 1$.

Thus, for each $x_{0}\in\partial\Omega-\{\bar{x}_{1},\ldots,\bar{x}_{m}\}$ there is $\epsilon_{x_{0}}>0,q_{x_{0}}>1$ such that:

[TABLE]

Now, we consider a cutoff function $\eta\in C^{\infty}({\mathbb{R}}^{2})$ such that

[TABLE]

We write

[TABLE]

Because, by Poincaré and Gagliardo-Nirenberg-Sobolev inequalities:

[TABLE]

with, $q^{*}=2q/(2-q)>2>1$.

By the elliptic estimates, $(v_{i})_{i}$ is uniformly bounded in $L^{\infty}(V_{\epsilon}(x_{0}))$. Finaly, we have, for some $\epsilon>0$ small enough,

[TABLE]

Now, we consider a cutoff function $\eta\in C^{\infty}({\mathbb{R}}^{2})$ such that

[TABLE]

We write

[TABLE]

By the elliptic estimates, $(u_{i})_{i}$ is uniformly bounded in $L^{\infty}(V_{\epsilon}(x_{0}))$ and also in $C^{0,\theta}$ norm.

If we repeat this procedure another time, we have a boundedness of $(u_{i})_{i}$ and $(v_{i})_{i}$ in the $C^{1,\theta}$ norm, because they are bounded in $W^{2,q}\subset W^{1,q^{*}}$ norms with $2q/(2-q)=q^{*}>2$.

We have the same computations and conclusion if we consider a regular point $x_{0}$ for the measure $\mu_{2}$.

We have proved that, there is a finite number of points $\bar{x}_{1},\ldots,\bar{x}_{m}$ such that the squence $(u_{i})_{i}$ and $(v_{i})_{i}$ are locally uniformly bounded (in $C^{1,\theta},\theta>0$) in $\bar{\Omega}-\{\bar{x}_{1},\ldots,\bar{x}_{m}\}$.

Proof of theorem 1.2:**

Without loss of generality, we can assume that $0=x_{1}$ is a blow-up point. Since the boundary is an analytic curve $\gamma_{1}(t)$, there is a neighborhood of [math] such that the curve $\gamma_{1}$ can be extend to a holomorphic map such that $\gamma_{1}^{\prime}(0)\not=0$ (series) and by the inverse mapping one can assume that this map is univalent around [math]. In the case when the boundary is a simple Jordan curve the domain is simply connected. In the case that the domains has a finite number of holes it is conformally equivalent to a disk with a finite number of disks removed. Here we consider a general domain. Without loss of generality one can assume that $\gamma_{1}(B_{1}^{+})\subset\Omega$ and also $\gamma_{1}(B_{1}^{-})\subset(\bar{\Omega})^{c}$ and $\gamma_{1}(-1,1)\subset\partial\Omega$ and $\gamma_{1}$ is univalent. This means that $(B_{1},\gamma_{1})$ is a local chart around [math] for $\Omega$ and $\gamma_{1}$ univalent. (This fact holds if we assume that we have an analytic domain, (below a graph of an analytic function), we have necessary the condition $\partial\bar{\Omega}=\partial\Omega$ and the graph is analytic, in this case $\gamma_{1}(t)=(t,\varphi(t))$ with $\varphi$ real analytic and an example of this fact is the unit disk around the point $(0,1)$ for example).

By this conformal transformation, we can assume that $\Omega=B_{1}^{+}$, the half ball, and $\partial^{+}B_{1}^{+}$ is the exterior part, a part which not contain [math] and on which $u_{i}$ converge in the $C^{1}$ norm to $u$. Let us consider $B_{\epsilon}^{+}$, the half ball with radius $\epsilon>0$. Also, one can consider a $C^{1}$ domain (a rectangle between two half disks) and by charts its image is a $C^{1}$ domain) We know that:

[TABLE]

Thus we can use integrations by parts (Stokes formula). The Pohozaev identity applied around the blow-up [math]:

[TABLE]

Thus,

[TABLE]

After integration by parts, we obtain:

[TABLE]

[TABLE]

[TABLE]

Also, for $u$ and $v$, we have:

[TABLE]

[TABLE]

[TABLE]

If, we take the difference, we obtain:

[TABLE]

But,

[TABLE]

and,

[TABLE]

a contradiction.