\documentclass[reqno]{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 121, pp. 1--22.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/121\hfil Concentration phenomena] {Concentration phenomena for fourth-order elliptic equations with critical exponent} \author[M. Hammami\hfil EJDE-2004/121\hfilneg] {Mokhless Hammami} \address{Mokhless Hammami \hfill\break D{\'e}partement de Math{\'e}matiques, Facult{\'e} des Sciences de Sfax, Route Soukra, 3018, Sfax, Tunisia} \email{Mokhless.Hammami@fss.rnu.tn} \date{} \thanks{Submitted August 25, 2004. Published October 14, 2004.} \subjclass[2000]{35J65, 35J40, 58E05} \keywords{Fourth order elliptic equations; critical Sobolev exponent; \hfill\break\indent blowup solution} \begin{abstract} We consider the nonlinear equation $$\Delta ^2u= u^{\frac{n+4}{n-4}}-\varepsilon u$$ with $u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$. Where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\geq 9$, and $\varepsilon$ is a small positive parameter. We study the existence of solutions which concentrate around one or two points of $\Omega$. We show that this problem has no solutions that concentrate around a point of $\Omega$ as $\varepsilon$ approaches $0$. In contrast to this, we construct a domain for which there exists a family of solutions which blow-up and concentrate in two different points of $\Omega$ as $\varepsilon$ approaches $0$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{pro}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \section{Introduction and statement of results} This paper concerns the concentration phenomena for the following nonlinear equation under Navier boundary conditions: $$\label{Pe} \begin{gathered} \Delta ^2 u = u^p-\varepsilon u,\quad u>0 \quad \mbox{in } \Omega \\ \Delta u= u =0 \quad \mbox{on } \partial \Omega , \end{gathered}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\geq 9$, $\varepsilon$ is a small positive parameter and $p+1=2n/(n-4)$ is the critical Sobolev exponent of the embedding $H^2(\Omega )\cap H^1_0(\Omega ) \hookrightarrow L^{p+1}(\Omega)$. In the last decades, there have been many works in the study of concentration phenomena for second order elliptic equations with critical exponent; see for example \cite{AP,BLR,BEGR,BP,CY,DFM1,DFM2,H,KR,MiP,MP,MMP,R1,R2,R3,R4} and the references therein. In sharp contrast to this, very little is known for fourth order elliptic equations. For $\varepsilon =0$, the situation is complex, Van Der Vorst showed in \cite{V1} that if $\Omega$ is starshaped \eqref{Pe} has no solution whereas Ebobisse and Ould Ahmedou proved in \cite{EA} that \eqref{Pe} has a solution provided that some homology group of $\Omega$ is nontrivial. This topological condition is sufficient, but not necessary, as examples of contractible domains $\Omega$ on which a solution exists show \cite{GGS}. For $-\lambda_1 (\Omega)<\varepsilon <0$, Van der Vost has shown in \cite{V2} that \eqref{Pe} has a solution, generalizing to \eqref{Pe} the famous Brezis-Nirenberg's result \cite{BN} concerning the corresponding second order elliptic equation, where $\lambda_1(\Omega)$ denotes the first eigenvalue of $\Delta^2$ under the Navier boundary condition. Recently, also for $\varepsilon < 0$, El Mehdi and Selmi \cite{ES} have constructed a solution of \eqref{Pe} which concentrates around a critical point of Robin's function, However, as far as the author know, the case of $\varepsilon>0$ has not been considered before and this is precisely the first aim of the present paper. More precisely, our goal is to study the existence of solutions of \eqref{Pe} which concentrate in one or two points of $\Omega$. The similar problems in the case of Laplacian have been considered by Musso and Pistoia \cite{MMP}. Compared with the second order case, further technical problems arise which are overcome by careful and delicate expansions of the Euler functional associated to \eqref{Pe} and its gradient near a neighborhood of highly concentrated functions. Such expansions, which are of self interest, are highly nontrivial and use the techniques developed by Bahri \cite{B} and Rey \cite{R1} in the framework of the {\it Theory of critical points at infinity}. To state our results, we need to introduce some notations. We denote by $G$ the Green's function of $\Delta ^2$, that is, for all $x\in\Omega$, \begin{gather*} \Delta ^2 G(x,.) = c'_n\delta _x \quad \mbox{in } \Omega \\ \Delta G(x,.)= G(x,.) =0 \quad \mbox{on } \partial \Omega , \end{gather*} where $\delta _x$ denotes the Dirac mass at $x$ and $c'_n=(n-4)(n-2)|S^{n-1}|$. We also denote by $H$ the regular part of $G$, that is, $$H(x,y)=|x-y|^{4-n} -G(x,y),\quad \mbox{for } (x,y)\in\Omega\times\Omega.$$ For $\lambda >0$ and $x \in \mathbb{R}^n$, let $$\label{e:11} \delta _ {x,\lambda }(y) = \frac {c_n\lambda ^{\frac{n-4}{2}}}{(1+\lambda^2|y-x|^2)^{\frac{n-4}{2}}},\quad c_n=[(n-4)(n-2)n(n+2)]^{(n-4)/8}\,.$$ It is well known \cite{Lin} that $\delta _{x,\lambda}$ are the only solutions of $\Delta ^2 u = u^{\frac{n+4}{n-4}},\quad u>0 \mbox{ in } \mathbb{R}^n$ with $u\in L^{p+1}(\mathbb{R}^n)$ and $\Delta u \in L^2(\mathbb{R}^n)$. They are also the only minimizers of the Sobolev inequality on the whole space; that is, $$\label{e:12} S =\inf\{\|\Delta u\|^{2}_{L^2(\mathbb{R}^n)}\|u\|^{-2}_{L^{\frac{2n}{n-4}}(\mathbb{R}^n)} : \Delta u\in L^2 ,u\in L^{\frac{2n}{n-4}} ,u\neq 0 \}.$$ We denote by $P\delta _{x,\lambda}$ the projection of the $\delta_{x,\lambda}$'s onto $H^2(\Omega )\cap H^1_0(\Omega)$, defined by $$\Delta ^2 P\delta _{x,\lambda}=\Delta ^2\delta _{x,\lambda} \mbox{ in } \Omega \quad \mbox{and}\quad \Delta P\delta _{x,\lambda}=P\delta _{x,\lambda}=0\mbox{ on } \partial \Omega,$$ and we set $$\varphi_{x,\lambda} = \delta _{x,\lambda}- P\delta _{x,\lambda}.$$ The space $\mathcal{H}(\Omega) := H^2(\Omega )\cap H^1_0(\Omega)$ is equipped with the norm $\|\cdot\|$ and its corresponding inner product $(.,.)$ defined by \begin{gather} \|u\|=\Big(\int_\Omega |\Delta u|^2\Big)^{1/2},\quad u\in \mathcal{H}(\Omega ) \label{e:13},\\ (u,v)=\int_\Omega \Delta u\Delta v,\quad u,v \in \mathcal{H}(\Omega ). \label{e:14} \end{gather} For $x \in \Omega$, $\lambda >0$, let $$E_{x,\lambda}=\{ v\in \mathcal{H}(\Omega) : (v,P\delta _{x,\lambda})=(v,\frac{\partial P\delta _{x,\lambda}}{\partial\lambda})=(v,\frac{\partial P\delta _{x,\lambda}}{\partial x_j})=0,\,j=1,\dots ,n\},$$ where the $x_j$ is the $j$-th component of $x$. Now we state our first result. \begin{thm}\label{t:11} There does not exist any solution of \eqref{Pe} of the form $$\label{e:18} u_\varepsilon = \alpha_\varepsilon P\delta _{x_\varepsilon , \lambda _\varepsilon}+v_\varepsilon,$$ where $$\label{e:19} v_\varepsilon \in E_{x_\varepsilon, \lambda_\varepsilon},\quad x_\varepsilon \in \Omega \quad \mbox{and as } \varepsilon\to 0, \,\,\alpha_\varepsilon \to 1,\,\, \|v_\varepsilon\|\to 0,\,\, \lambda_\varepsilon d(x_\varepsilon,\partial\Omega) \to +\infty.$$ \end{thm} On the contrary, if $\Omega$ is a domain with small hole", we prove the existence of a family of solutions which blow-up and concentrate in two points. Namely, we have the following result. \begin{thm}\label{t:12} Let $D$ be a bounded smooth domain in $\mathbb{R}^n$ which contains the origin $0$. There exists $r_0>0$ such that, if $00$ such that problem \eqref{Pe} has a solution $u_\varepsilon$ for any $0<\varepsilon <\varepsilon_0$. Moreover, the family of solutions $u_\varepsilon$ blows-up and concentrates at two different points of $\Omega$ in the following sense: $$u_\varepsilon= \sum_{i=1}^2\alpha_i^\varepsilon P\delta _{x_i^\varepsilon, \lambda_i^\varepsilon} + v_\varepsilon,$$ where $\lambda_1^\varepsilon,\,\lambda_2^\varepsilon >0$, $x_1^\varepsilon,\, x_2^\varepsilon \in \Omega$ with $\lim_{\varepsilon\to 0}x_i^\varepsilon=x_i \in\Omega$, $x_1\ne x_2$, $\lambda_1^\varepsilon$ and $\lambda_2^\varepsilon$ are of order $\varepsilon^{-1/(n-8)}$, $v_\varepsilon \in E_{x_1^\varepsilon, \lambda_1^\varepsilon}\cap E_{x_2^\varepsilon,\lambda_2^\varepsilon}$ and as $\varepsilon\to 0,\,\, \alpha_i^\varepsilon \to 1,\; \| v_\varepsilon\| \to 0$. \end{thm} Note that the construction of solutions which concentrate around $k$ different points of $\Omega$, with $k \geq 2$ is related to suitable critical points of the function $\Psi_k:\mathbb{R}_+^k \times\Omega^k\to \mathbb{R}$ defined by $$\Psi_k (\Lambda ,x)=\frac{1}{2}(M(x)\Lambda,\Lambda)+\frac{1}{2}\sum_{i=1}^k \Lambda_i^{\frac{8}{n-4}},$$ where $\Lambda=^T(\Lambda_1,\dots ,\Lambda_k)$ and $M(x)=\left( m_{ij}(x)\right)_{1\leq i,j\leq k}$ is the matrix defined by $$\label{e:20} m_{ii}=H(x_i,x_i),\quad m_{ij}=-G(x_i,x_j) \quad \mbox{for } i\neq j.$$ Let $\rho (x)$ be the least eigenvalue of $M(x)$ and $e(x)$ the eigenvector corresponding to $\rho (x)$ whose norm is $1$ and whose components are all strictly positive (see Appendix A of \cite{BLR}). Now, we define the following subset of $\mathcal{H}(\Omega)$ \begin{align*} \mathcal{M}_\varepsilon=&\{ m=(\alpha,\lambda,x,v)\in \mathbb{R}^k \times(\mathbb{R}_+^*)^k\times\Omega_{d_0}^k\times\mathcal{H}(\Omega) :|\alpha_i-1|<\nu_0,\\ &\lambda_i>\frac{1}{\nu_0}\,\,\forall i,\frac{\lambda_i}{\lambda_j}d'_0\,\, \forall i\neq j,\,v \in E,\,\|v\|<\nu_0 \}. \end{align*} where $\nu_0,\,c_0,\,d_0,\,d'_0$ are some suitable positive constants, $\Omega_{d_0}=\{x\in\Omega : d(x,\partial \Omega)>d_0\}$ and $E=\bigcap_{i=1}^k E_{x_i,\lambda_i}$. Then, we have the following necessary condition. \begin{thm}\label{t:13} Assume that $u_\varepsilon$ is a solution of \eqref{Pe} of the form $$\label{e:21} u_\varepsilon = \sum_{i=1}^k \alpha_i^\varepsilon P\delta _{x_i^\varepsilon , \lambda_i^\varepsilon}+ v^\varepsilon,$$ where $(\alpha^\varepsilon,\lambda^\varepsilon,x^\varepsilon,v^\varepsilon) \in \mathcal{M}_\varepsilon$, then, when $\varepsilon\to 0$, $\alpha_i^\varepsilon \to 1,\,\,\, x_i^\varepsilon \to x_i$ for $i=1,\dots , k$ and we have either $\rho (x)=0$ and $\rho '(x)=0$ or $\rho (x)<0$ and $(\Lambda, x )$ is a critical point of $\Psi_k$, where $\Lambda_i=c \mu_i$, with $\mu_i= \lim_{\varepsilon \to 0} \varepsilon^{\frac{-1}{n-8}}\lambda_i^\varepsilon >0$ for $i=1,\dots , k$ and c is a positive constant. \end{thm} The proof of our results is inspired by the methods of \cite{B,BLR,BE,EH, MMP}. In Section 2, we develop the technical framework needed in the proofs of ours results. Section 3 is devoted to the proof of Theorems \ref{t:11} and \ref{t:13}, while Theorem \ref{t:12} is proved in Section 4. \section{The Technical Framework} First of all, let us introduce the general setting. For $\varepsilon>0$, we define on $\mathcal{H}(\Omega )$ the functional $$\label{e:201} J_\varepsilon (u) = \frac{1}{2}\int_\Omega |\Delta u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1}+\frac{\varepsilon}{2}\int_\Omega u^2.$$ If $u$ is a positive critical point of $J_\varepsilon$, $u$ satisfies on $\Omega$ the equation \eqref{Pe}. Conversely, we see that any solution of \eqref{Pe} is a critical point of $J_\varepsilon$. Let us define the functional $$\label{psi} K_\varepsilon : \mathcal{M}_\varepsilon \to \mathbb{R}, \,\, K_\varepsilon (\alpha,\lambda,x,v) = J_\varepsilon (\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i} +v).$$ Note that $(\alpha,\lambda,x,v)$ is a critical point of $K_\varepsilon$ if and only if $u=\sum_{i=1}^{k} \alpha_i P\delta _{x_i,\lambda_i} + v$ is a critical point of $J_\varepsilon$, i.e. if and only if there exist $A_i$, $B_i$, $C_{ij} \in \mathbb{R}$, $1\leq i\leq k$ and $1\leq j\leq n$, such that \begin{gather} \label{Eai} %(E_{\alpha_i}): \frac{\partial K_\varepsilon}{\partial \alpha_i} =0\quad \forall i,\\ \label{Eli} %(E_{\lambda_i}): \frac{\partial K_\varepsilon}{\partial \lambda_i} = B_i\big( \frac{\partial ^2P\delta _{x_i,\lambda_i}}{\partial\lambda_i^2}, v\big) + \sum_{j=1}^n C_{ij} \big( \frac{\partial ^2P\delta _{x_i,\lambda_i}}{\partial (x_i)_j\partial \lambda_i}, v\big)\quad\forall i, \\ \label{Exir} %(E_{(x_i)_r}): \frac{\partial K_\varepsilon}{\partial (x_i)_r} = B_i\big( \frac{\partial ^2P\delta _{x_i,\lambda_i}}{\partial\lambda_i\partial (x_i)_r}, v\big) + \sum_{j=1}^n C_{ij} \big( \frac{\partial ^2P\delta _{x_i,\lambda_i}}{\partial (x_i)_j \partial (x_i)_r}, v\big)\quad\forall r,\,\forall i,\\ \label{Ev} %(E_{v}):& \frac{\partial K_\varepsilon}{\partial v} = \sum_{i=1}^k \Big(A_i P\delta _{x_i,\lambda_i} + B_i\frac{\partial P\delta _{x_i,\lambda_i}}{\partial\lambda_i} + \sum_{j=1}^n C_{ij} \frac{\partial P\delta _{x_i,\lambda_i}}{\partial (x_i)_j}\Big), \end{gather} where the $(x_i)_r$ is the $r$-th component of $x_i$. As usual in this type of problems, we first deal with the $v$-part of $u$, in order to show that it is negligible with respect to the concentration phenomenon. Namely, we prove the following. \begin{pro}\label{p:21} There exists $\varepsilon_1>0$ such that, for $0<\varepsilon<\varepsilon_1$, there exists a $C^1$-map which to any $(\alpha,\lambda,x)$ with $(\alpha ,\lambda,x,0)\in \mathcal{M}_\varepsilon$, associates $v_\varepsilon= v_{(\varepsilon,\alpha,\lambda,x)}\in E$, $\|v_\varepsilon\| < \nu_0$, such that \eqref{Ev} is satisfied for some $(A,B,C) \in \mathbb{R}^k\times \mathbb{R}^k\times ( \mathbb{R}^n)^k$. Such a $v_\varepsilon$ is unique, minimizes $K_\varepsilon(\alpha,\lambda,x,v)$ with respect to $v$ in $E$ and we have the estimate $\|v_\varepsilon\| = O\Big(\sum_{i=1}^k \Big(\frac{(\log\lambda_i)^{\frac{n+4}{2n}}}{\lambda_i^{\frac{n+4}{2}}} +(\mbox{if } n\geq12) \frac{\varepsilon(\log\lambda_i)^{\frac{n+4}{2n}}}{\lambda_i^4} +(\mbox{if } n<12) (\frac{\varepsilon}{\lambda_i^{\frac{n-4}{2}}}+\frac{1}{\lambda_i^{n-4}}) \Big)\Big)$ \end{pro} \begin{proof} As in \cite{B} (see also \cite{R1}) we write \label{e:203} \begin{aligned} &K_\varepsilon (\alpha,\lambda, x,v)\\ &=J_\varepsilon(\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i}+ v)\\ &= \frac{1}{2}\|\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i} + v\|^2-\frac{1}{p+1}\int_\Omega |\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i} +v|^{p+1}\\ &\quad +\frac{\varepsilon}{2}\int_\Omega \Big(\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i}+v\Big)^2 \\ &=K_\varepsilon(\alpha,\lambda,x,0) -(f_\varepsilon,v) + \frac{1}{2}Q_\varepsilon(v,v) + O\big(\|v\|^{\min(3,p+1)}+\varepsilon \|v\|^2\big), \end{aligned} where \begin{gather*} (f_\varepsilon,v)= \int_\Omega |\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i}|^p v -\varepsilon \int_\Omega \Big(\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i}\Big) v,\\ Q_\varepsilon(v,v)= \|v\|^2 - p \int_\Omega (\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i})^{p-1}v^2= \|v\|^2 - p \sum_{i=1}^k \int_\Omega \delta _{x_i,\lambda_i}^{p-1}v^2 + o(\|v\|^2). \end{gather*} According to \cite{BE1}, there exists a positive constant $c$ such that $$\label{e:204} \|v\|^2 - p \sum_{i=1}^k \int_\Omega \delta _{x_i,\lambda_i}^{p-1}v^2 \geq c \|v\|^2,\quad \forall v\in E.$$ Now, we will estimate $(f_\varepsilon,v)$. Using the fact that $\left( P\delta _{x_i,\lambda_i},v\right)=0$, we obtain \label{e:205} \begin{aligned} &\int_\Omega |\sum_{i=1}^k \alpha_i P\delta _{x_i,\lambda_i}|^p v\\ &=O\Big(\sum_i\int_{B_i\cup B_i^c}\delta _{x_i,\lambda_i}^{p-1}\varphi_{x_i,\lambda_i}|v|+\sum_{j\neq i}\int_{B_i\cup B_i^c} \chi_{P\delta _j \leq P\delta _i}P\delta _{x_i,\lambda_i}^{p-1}P\delta _{x_j,\lambda_j}|v|\Big) \\ &=O\Big(\sum_{i,j}\Big(\frac{1}{\lambda_j^{(n-4)/2}}\int_{B_i} \delta _{x_i,\lambda_i}^{p-1}|v|+\int_{ B_i^c} \delta _{x_i,\lambda_i}^p|v|\Big)\Big), \end{aligned} where $B_i=\{y :|y-x_i|d'_0$ then \begin{align}\label{e:228}\lambda_i\frac{\partial\varepsilon_{ij}}{\partial\lambda_i} =-\frac{n-4}{2}\frac{1}{(\lambda_i \lambda_j |x_i-x_j|^2)^{(n-4)/2}}+ O\Big(\sum_{k=i,j} \frac{1}{\lambda_k^{n-2}}\Big). \end{align} The Claim $2$ follows from Proposition \ref{p:21} and \eqref{e:212}--\eqref{e:228}. Regarding Claim 3, its proof is similar to Claim $2$, so we will omit it. \end{proof} \begin{lem}\label{l:23} Assume that $(\alpha,\lambda,x,v)\in \mathcal{M}_\varepsilon$ and let $v:=v_\varepsilon$ be the function obtained in Proposition \ref{p:21}. Then the following expansion holds \begin{align*} &K_\varepsilon(\alpha,\lambda,x,v)\\ &=\frac{S_n}{2} \Big(\sum_{i=1}^k\alpha_i^2-\frac{n-4}{n}\sum_{i=1}^k \alpha_i^{p+1}\Big) +\frac{c_2}{2}\sum_{i=1}^k\alpha_i^2\left(2\alpha_i^{p-1}-1\right) \frac{H(x_i,x_i)}{\lambda_i^{n-4}}\\ &\quad +\frac{c_2}{2}\sum_{j\neq i}\alpha_i\alpha_j\left(1-2\alpha_i^{p-1}\right)\frac{G(x_i,x_i)}{(\lambda_j\lambda_i)^{(n-4)/2}} +\frac{\varepsilon}{2}\sum_{i=1}^k\alpha_i^2\frac{c_4}{\lambda_i^4}\\ &\quad +O\Big(\sum_{i=1}^k\Big(\frac{\varepsilon}{\lambda_i^{n-4}} +\frac{1}{\lambda_i^{n-2}}+(\mbox{if }n\geq12) \frac{\varepsilon^2(\log\lambda_i)^{\frac{n+4}{n}}}{\lambda_i^8} +(\mbox{if } n<12)\frac{\varepsilon^2}{\lambda_i^{n-4}}\Big)\Big). \end{align*} \end{lem} \begin{proof} Using \eqref{e:203} and Proposition \ref{p:21}, this lemma follows from \eqref{e:209}--\eqref{e:211}. \end{proof} Let $\mathcal{M}_\varepsilon^1=\{(\lambda,x)\in (\mathbb{R}_+^*)^k\times\Omega_{d_0}^k : \lambda_i>\frac{1}{\nu_0}\,\,\forall i, \frac{\lambda_i}{\lambda_j}d'_0\,\,\, \forall i\neq j\}.$ For $(\lambda,x)\in\mathcal{M}_\varepsilon^1$, our aim is to study the $\alpha$-part of $u$. Namely, we prove the following result. \begin{pro}\label{p:22} There exists $\varepsilon_1>0$ such that, for $0<\varepsilon<\varepsilon_1$, there exists a $C^1$-map which to any $(\lambda,x)\in \mathcal{M}_\varepsilon^1$, associates $\alpha:= \alpha_{(\varepsilon,\lambda,x)}$, which satisfies \eqref{Eai} for each $i$ and we have the following estimate $$|\alpha_i-1|=O\Big(\frac{\varepsilon}{\lambda_i^4}+\frac{1}{\lambda_i^{n-4}}\Big).$$ \end{pro} \begin{proof} Let $\beta_i=1-\alpha_i$. By Lemma \ref{l:21}, we have $$\frac{8}{n-4}\beta_i S_n+O(\beta_i^2)=O\Big(\frac{\varepsilon}{\lambda_i^4} +\frac{1}{\lambda_i^{n-4}}\Big),$$ then $\beta_i=O\big(\frac{\varepsilon}{\lambda_i^4}+\frac{1}{\lambda_i^{n-4}}\big)$. On the other hand, we have $\frac{\partial^2 K_\varepsilon}{\partial \alpha_i \partial \alpha_j}(\alpha,\lambda,x,v)=(1-p)S_n \delta _i^j+o(1),$ with $\delta _i^j$ the Kronecker symbol and $o(1)$ tends to zero when $\varepsilon\to 0$, where we have used \eqref{e:209}, \eqref{e:2091}, Proposition \ref{p:21}, the fact that $\partial v/\partial \alpha_i \in E$ and $\|\partial v/\partial \alpha_i\|=o(1)$. Using the implicit function theorem the proposition follows. \end{proof} \section{Proof of Theorems \ref{t:11} and \ref{t:13}} \begin{proof}[Proof of Theorem \ref{t:13}] Assume that $u_\varepsilon$ is a family of solutions of \eqref{Pe} which has the form \eqref{e:21} where $(\alpha^\varepsilon,\lambda^\varepsilon,x^\varepsilon,v^\varepsilon)\in \mathcal{M_\varepsilon}$. The result of the theorem will be obtained through a careful analysis of \eqref{Eai}, \eqref{Eli}, \eqref{Exir} and \eqref{Ev}. From Proposition \ref{p:21}, there exists $v^\varepsilon$ satisfying \eqref{Ev}. We estimate now the corresponding numbers $A_i,\,B_i,\,C_{ij}$ by taking the scalar product of \eqref{Ev} with $P\delta _i$, $\partial P\delta _i/\partial\lambda_i$ and $\partial P\delta _i/\partial (x_i)_r$ for $i=1,\dots ,k$ and $r=1,\dots ,n$. Thus from the right side we get a quasi-diagonal system whose coefficients are given by \begin{gather*} \left(P\delta _i,P\delta _j\right) =S_n \delta _i^j+O\big( \frac{1}{{\lambda_i^{n-4}}}\big), \big(\frac{\partial P\delta _j}{\partial \lambda_j}, P\delta _i \big) =O\Big(\sum_{k=i,j}\frac{1}{{\lambda_k^{n-3}}}\Big), \\ \big(\frac{\partial P\delta _j}{\partial (x_j)_r}, P\delta _i\big)=O\Big(\sum_{k=i,j}\frac{1}{{\lambda_k^{n-4}}}\Big), \big(\frac{\partial P\delta _j}{\partial\lambda_j}, \frac{\partial P\delta _i}{\partial\lambda_i}\big) =\frac{n+4}{n-4}\frac{C_n}{\lambda_i^2} \delta _i^j+O\Big(\sum_{k=i,j}\frac{1}{{\lambda_k^{n-2}}}\Big), \\ \big(\frac{\partial P\delta _j}{\partial\lambda_j}, \frac{\partial P\delta _i}{\partial (x_i)_r}\big)=O\Big(\sum_{k=i,j}\frac{1}{{\lambda_k^{n-3}}}\Big), \big(\frac{\partial P\delta _i}{\partial x_i}, \frac{\partial P\delta _j}{\partial x_j}\big)=O\Big(\sum_{k=i,j}\frac{1}{{\lambda_k^{n-5}}}\Big)\mbox { for } i\neq j, \\ \big(\frac{\partial P\delta _i}{\partial (x_i)_r}, \frac{\partial P\delta _i}{\partial (x_i)_j}\big)=\frac{n+4}{n-4}C'_n \lambda_i^2 \delta _r^j+ O\big(\frac{1}{{\lambda_i^{n-5}}}\big), \end{gather*} where $\delta _i^j$ is the Kronecker symbol, $S_n$ is defined in Lemma \ref{l:21}, $C_n=\frac{(n-4)^2}{4}\int_{\mathbb{R}^n} \frac{(1-|y|^2)^2}{(1+|y|^2)^{n+2}} dy\quad{and}\quad C'_n=\frac{(n-4)^2}{4n}\int_{\mathbb{R}^n} \frac{|y|^2}{(1+|y|^2)^{n+2}} dy\,.$ The left side is given by $$\big(\frac{\partial K_\varepsilon}{\partial v}, P\delta _i\big) =\frac{\partial K_\varepsilon}{\partial \alpha_i}, \quad \big(\frac{\partial K_\varepsilon}{\partial v}, \frac{\partial P\delta _i}{\partial \lambda_i}\big)= \frac{1}{\alpha_i}\frac{\partial K_\varepsilon}{\partial \lambda_i},\quad \big(\frac{\partial K_\varepsilon}{\partial v}, \frac{\partial P\delta _i}{\partial (x_i)_r}\big)= \frac{1}{\alpha_i}\frac{\partial K_\varepsilon}{\partial (x_i)_r}.$$ Let $\beta_i=1-\alpha_i$. By Lemma \ref{l:21}, we have \begin{gather*} \frac{\partial K_\varepsilon}{\partial \alpha_i}(\alpha,x,\lambda,v^\varepsilon) =O\Big(|\beta_i|+\frac{\varepsilon}{\lambda_i^4}+\frac{1}{\lambda_i^{n-4}}\Big),\\ \frac{\partial K_\varepsilon}{\partial \lambda_i}(\alpha,x,\lambda,v^\varepsilon) =O\Big(\frac{\varepsilon}{\lambda_i^5}+\frac{1}{ \lambda_i^{n-3}}\Big), \end{gather*} \begin{align*} &\frac{\partial K_\varepsilon}{\partial (x_i)_r}(\alpha,x,\lambda,v^\varepsilon)\\ &=O\Big(\frac{1}{\lambda_i^{n-4}}+\frac{\varepsilon}{\lambda_i^{n-4}}+ (\mbox{if } n\geq12)\frac{\varepsilon^2(\log\lambda_i)^{{(n+4)}/{n}}}{\lambda_i^7} +(\mbox{if } n<12)\frac{\varepsilon^2}{\lambda_i^{n-5}} \Big). \end{align*} The solution of the system in $A_i, B_i, C_{ij}$ shows that \begin{gather*} A_i = O\big(|\beta_i|+\frac{\varepsilon}{\lambda_i^4}+\frac{1}{\lambda_i^{n-4}} \big) \\ B_i =O\big(\frac{\varepsilon}{\lambda_i^3}+\frac{1}{ \lambda_i^{n-5}}\big)\\ C_{ij}=O\big(\frac{1}{\lambda_i^{n-2}}+\frac{\varepsilon }{ \lambda_i^{n-2}}+ (\mbox {if } n\geq12)\frac{\varepsilon^2(\log\lambda_i)^{\frac{n+4}{n}}}{\lambda_i^9}+(\mbox{if } n<12)\frac{\varepsilon^2}{\lambda_i^{n-3}}\big). \end{gather*} This allows us to evaluate the right hand side in the equations \eqref{Eai} and \eqref{Exir}, namely \label{e:25} \begin{aligned} &B_i\big( \frac{\partial ^2P\delta _i}{\partial\lambda_i\partial (x_i)_r}, v^\varepsilon\big) + \sum_{j=1}^n C_{ij} \big( \frac{\partial ^2P\delta _i}{\partial (x_i)_j\partial (x_i)_r}, v^\varepsilon\big) \\ &=O\big(|B_i\||v^\varepsilon\|+\sum_{j=1}^n \lambda_i^2|C_{ij}\||v^\varepsilon\|\big) =O\big(\|v^\varepsilon\|(\frac{\varepsilon}{\lambda_i^3}+ \frac{1}{ \lambda_i^{n-5}}) \big) =O\big(\frac{1}{\lambda_i^{n-3}}+\frac{\varepsilon^2}{\lambda_i^5}\big). \end{aligned} In the same manner, we obtain $$\label{e:26} B_i\big( \frac{\partial ^2P\delta _i}{\partial\lambda_i^2}, v^\varepsilon\big)+ \sum_{j=1}^n C_{ij} \big( \frac{\partial ^2P\delta _i}{\partial (x_i)_j\partial \lambda_i}, v^\varepsilon\big) =O\big(\frac{1}{\lambda_i^{n-1}}+\frac{\varepsilon^2}{\lambda_i^{5}}\big).$$ From Proposition \ref{p:22} , there exists $\alpha^\varepsilon$ satisfying \eqref{Eai} for each $i$, and we have $$\label{e:27} 1-\alpha_i^\varepsilon=\beta_i^\varepsilon =O\Big(\frac{\varepsilon}{(\lambda_i^\varepsilon)^4} +\frac{1}{(\lambda_i^\varepsilon)^{n-4}}\Big).$$ Using \eqref{e:25}, \eqref{e:26}, \eqref{e:27} and Lemma \ref{l:21}, we deduce that \eqref{Eli} and \eqref{Exir} are equivalent to \label{e:28} \begin{aligned} &-2c_4\frac{\varepsilon}{(\lambda_i^\varepsilon)^4} -\frac{c_2(n-4)H(x_i^\varepsilon,x_i^\varepsilon)}{2(\lambda_i^\varepsilon)^{n-4}} +c_2\sum_{j\neq i}\frac{(n-4)G(x_i^\varepsilon,x_j^\varepsilon)} {2(\lambda_i^\varepsilon\lambda_j^\varepsilon)^{(n-4)/2}} \\ &=O\Big(\frac{\varepsilon}{(\lambda_i^\varepsilon)^{n-4}} +\frac{1}{(\lambda_i^\varepsilon)^{n-2}} +\frac{\varepsilon^2}{(\lambda_i^\varepsilon)^4}\Big), \end{aligned} \label{e:29} \begin{aligned} &\quad-\frac{c_2(n-4)}{2(\lambda_i^\varepsilon)^{n-4}}\frac{\partial H(x_i^\varepsilon,x_i^\varepsilon)}{\partial x_i} +c_2\sum_{j\neq i}\frac{(n-4)}{2(\lambda_i^\varepsilon \lambda_j^\varepsilon)^{(n-4)/2}}\frac{\partial G(x_i^\varepsilon,x_j^\varepsilon)}{\partial x_i}\\ &=O \Big(\frac{\varepsilon}{(\lambda_i^\varepsilon)^{n-4}} +\frac{1}{(\lambda_i^\varepsilon)^{n-3}} +\frac{\varepsilon^2}{(\lambda_i^\varepsilon)^4}\Big). \end{aligned} Let us perform the change of variables \begin{align}\label{e:30} \lambda_i^\varepsilon=(\Lambda_i^\varepsilon)^{\frac{-2}{n-4}} \varepsilon^{\frac{-1}{n-8}} \big(\frac{c_4}{c_2}\big)^{\frac{-1}{n-8}}. \end{align} Note that \label{e:m1} \Lambda_i^\varepsilon\varepsilon^{\frac{n-4}{2(n-8)}}\to 0 \quad \mbox{as } \varepsilon \to 0,\quad \frac{\Lambda_i^\varepsilon}{ \Lambda_j^\varepsilon}d'_0\}$, we get $$\frac{\partial M(x^\varepsilon)}{\partial x_i}e'(x^\varepsilon)=O(|e'(x^\varepsilon)|)=o(\xi^\varepsilon).$$ The scalar product of \eqref{e:37} with$e(x^\varepsilon)gives \begin{align}\label{e:38} ^Te(x^\varepsilon)\frac{\partial M(x^\varepsilon)}{\partial x_i}e(x^\varepsilon)=o(1). \end{align} Since|e(x^\varepsilon)|^2=1$and$e(x^\varepsilon).\frac{\partial e(x^\varepsilon)}{\partial x_i}=0, therefore \begin{align}\label{e:39} ^Te(x^\varepsilon)\frac{\partial M(x^\varepsilon)}{\partial x_i}e(x^\varepsilon)=\frac{\partial \rho}{\partial x_i}(x^\varepsilon). \end{align} Passing to the limit in \eqref{e:38} and \eqref{e:39}, we obtain $$\frac{\partial \rho}{\partial x_i}(x)=0.$$ This concludes the proof of Theorem \ref{t:13}. \end{proof} \begin{proof}[Proof of Theorem \ref{t:11}] Arguing by contradiction, suppose that \eqref{Pe} has a solution of the form \eqref{e:18} and satisfying \eqref{e:19}. Multiplying \eqref{Pe} byv_\varepsilon$and integrating over$\Omega, we obtain \begin{align}\label{e:40} \|v_\varepsilon\|^2&= \int_\Omega |\alpha_\varepsilon P\delta _{x_\varepsilon,\lambda_\varepsilon} + v_\varepsilon|^p v_\varepsilon -\varepsilon \int_\Omega \left(\alpha_\varepsilon P\delta _{x_\varepsilon,\lambda_\varepsilon} + v_\varepsilon\right)v_\varepsilon \\ &=\alpha_\varepsilon^p\int_\Omega P\delta _{x_\varepsilon,\lambda_\varepsilon}^p v_\varepsilon +p\alpha_\varepsilon^{p-1}\int_\Omega P\delta _{x_\varepsilon,\lambda_\varepsilon}^{p-1}v_\varepsilon ^2 +o(\|v_\varepsilon\|^2)+ O\Big(\varepsilon \int_\Omega \delta _{x_\varepsilon,\lambda_\varepsilon} |v_\varepsilon|\Big). \end{align} From Proposition 3.4 of \cite{BE1} and the fact that\alpha_\varepsilon\to1$, there exists a positive constant$c, such that \begin{align}\label{e:41} \|v_\varepsilon\|^2- p\alpha_\varepsilon^{p-1} \int_\Omega P\delta _{x_\varepsilon,\lambda_\varepsilon}^{p-1}v_\varepsilon^2 =\|v_\varepsilon\|^2- p\int_\Omega \delta _{x_\varepsilon,\lambda_\varepsilon}^{p-1}v_\varepsilon^2 + o\left(\|v_\varepsilon\|^2\right)\geq c \|v_\varepsilon\|^2. \end{align} On the other hand, using the fact thatv_\varepsilon \in E_{x_\varepsilon,\lambda_\varepsilon}, we obtain \begin{align*} \int_\Omega P\delta _{x_\varepsilon,\lambda_\varepsilon}^p v_\varepsilon =O\Big(\int_{B\cup B^c} |\varphi_{x_\varepsilon,\lambda_\varepsilon}| \delta _{x_\varepsilon,\lambda_\varepsilon}^{p-1}|v_\varepsilon|\Big), \end{align*} whereB=\{y :|y-x_\varepsilon| d'_0$and$\lambda_i^\varepsilon$satisfies$\lambda_i^\varepsilon=(\Lambda_i^\varepsilon)^{\frac{-2}{n-4}} \varepsilon^{\frac{-1}{n-8}}(\frac{c_4}{c_2})^{\frac{-1}{n-8}}. For the rest of this article, we will consider the set \begin{equation*} \mathcal{M}_\varepsilon^2=\{ (\Lambda,x)\in (\mathbb{R}_+^*)^2\times\Omega_{d_0}^2 : c<\Lambda_i<\frac{1}{c}\, \forall i,\,|x_1-x_2|>d'_0 \}. \end{equation*} Let us define the functional $$K_\varepsilon^2(\Lambda,x)=J_\varepsilon(u_\varepsilon).$$ \begin{lem}\label{l:29} We have the expansion \begin{align*} K_\varepsilon^2(\Lambda,x) &=\frac{4S_n}{n }+\varepsilon^{\frac{n-4}{n-8}}c_4^{\frac{n-4}{n-8}} c_2^{\frac{-4}{n-8}}\Big[\frac{1}{ 2} H(x_1,x_1)\Lambda_1^2 +\frac{1}{ 2}H(x_2,x_2)\Lambda_2^2\\ &\quad -G(x_1,x_2)\Lambda_1\Lambda_2 +\frac{1}{ 2}\big(\Lambda_1^{\frac{8}{n-4}}+\Lambda_2^{\frac{8}{ n-4}}\big)\Big]+o(\varepsilon^{\frac{n-4}{n-8}}), \end{align*} in theC^1$-norm with respect to$(\Lambda,x) \in \mathcal{M}_\varepsilon^2$, where$c_2 \mbox {and }\,c_4$are defined in Lemma \ref{l:21}. \end{lem} The proof of this lemma follows from Propositions \ref{p:21}, \ref{p:22} and Lemmas \eqref{l:21}, \eqref{l:23}. To find a solution of \eqref{Pe} with two blow-up points in$\Omega$, it is enough to find sufficiently stable'' critical point of the function$\Psidefined by \begin{align*} \Psi&:=\Psi_2(\Lambda,x)\\ &=\frac{1}{ 2}\left( H(x_1,x_1)\Lambda_1^2+H(x_2,x_2)\Lambda_2^2-2G(x_1,x_2)\Lambda_1\Lambda_2\right) +\frac{1}{2}\big(\Lambda_1^{\frac{8}{n-4}}+\Lambda_2^{\frac{8}{n-4}}\big). \end{align*} Here we follow the ideas of \cite{MMP}, \cite{DFM1}. LetD$be a bounded domain in$\mathbb{R}^n$with smooth boundary which contains the origin 0. The following result holds (see Corollary 2.1 of \cite{DFM1} which is analogue corollary for the Laplacian). \begin{cor}\label{c:11} For any sufficiently small$\sigma >0$there exists$r_0>0$such that if$0< r0$and$\rho>0$the following manifold $$W_\rho^\delta =\{ x\in \Omega_\delta ^2 : \rho(x)<-\rho\}.$$ Let$\rho_0=-\max_{x\in \mathcal{S}^2}\rho(x)$and$\delta _0=\mathop{\rm dist}(\mathcal{S},\partial \Omega)$. It holds for any$0<\rho<\rho_0$and$0<\delta <\delta _0$that$\mathcal{S}^2\subset W_\rho^\delta $. Since$\frac{8}{n-4}<2$, there exist$R_0 >0$such that $$\label{m} b=\max_{x\in \mathcal{S}^2,0\leq R\leq R_0} \Psi(Re(x),x)>0 \quad\mbox{ and }\quad \max_{x\in \mathcal{S}^2, R=0,R_0} \Psi(Re(x),x)=0.$$ Next we let$\Gamma$be the class of continuous function$\gamma:[0,R_0]\times\mathcal{S}^2\times [0,1]\to \mathbb{R}_+^2\times W_\rho^\delta $, such that \begin{enumerate} \item$\gamma(0,x,t)=(0,x)$and$\gamma(R_0,x,t)=(R_0e(x),x)$for all$x \in \mathcal{S}^2,t\in [0,1]$. \item$\gamma(R,x,0)=(R e(x),x)$for all$(R,x)\in [0,R_0]\times\mathcal{S}^2$. \end{enumerate} For every$(R,x,t)\in [0,R_0]\times\mathcal{S}^2\times [0,1]$we denote$\gamma(R,x,t)=(\tilde{\Lambda}(R,x,t),\tilde{x}(R,x,t))$and, for$\tau>0$, we define the set $$\mathcal{I_\tau}=\{(R,x)\in [0,R_0]\times\mathcal{S}^2 : \tilde{\Lambda}_1(R,x,1)\tilde{\Lambda}_2(R,x,1)=\tau\}.$$ In the following we prove that$\Psi$has a critical level between$a$and$b$where$b$is defined in \eqref{m} and$a$will be defined in Corollary \ref{c:12}. The first step in this direction is the following topological result which is similar to \cite[Lemma 7.1]{DFM1}. \begin{lem}\label{l:30} For every open neighborhood$\mathcal{U}$of$\mathcal{I_\tau}$in$\mathbb{R}_+^2\times\mathcal{S}^2$, the projection$g:\mathcal{U}\to \mathcal{S}^2$induces a monomorphism in cohomology, that is$g^*:H^*(\mathcal{S}^2)\to H^*(\mathcal{U})$is a monomorphism. \end{lem} \begin{cor}\label{c:12} For$\tau>0$small, there exist$a=a(\tau)>0$, such that $$\sup_{x\in\mathcal{S}^2,0 \leq R\leq R_0}\Psi(\gamma(R,x,1))\geq a \mbox{ for all } \gamma \in \Gamma.$$ \end{cor} \begin{proof} Since$\Omega$is smooth, there is$c_0>0$such that if$x_1,x_2\in \Omega_\delta $and$|x_1-x_2|0$so that$G(x_1,x_2)\geq K$implies$|x_1-x_2|0$, there exists$\gamma \in \Gamma$such that $$\Psi(\gamma(R,x,1))\delta \}.$$ To prove that the function$\Psi$constrained to$\mathbb{R}_+^2\times(W_\rho^\delta \cap V_\delta )$has a critical level between$a$and$b$we need to care about the fact that the domain$\mathbb{R}_+^2\times(W_\rho^\delta \cap V_\delta )$is not necessarily closed for the gradient flow of$\Psi$. The following lemma, is the first step in this direction. \begin{lem}\label{l:31} There exists$\delta '_0>0$such that for any$\delta \in (0,\delta '_0)$and for any$(\Lambda,x) \in \mathbb{R}_+^2\times (W_\rho^\delta \cap V_\delta )$with$\Psi(\Lambda,x)\in[a,b],\,\,\nabla_\Lambda\Psi(\Lambda,x)=0$and$x=(x_1,x_2)\in \partial V_\delta $, then there exists a vector$T$tangent to$\mathbb{R}_+^2\times \partial V_\delta $at the point$(\Lambda,x)$such that$\nabla\Psi(\Lambda,x).T \neq 0$. \end{lem} \begin{proof} The proof will be given in two steps.\\ {\it Step1}. We argue by contradiction. Let$(\Lambda_\delta ,x_\delta )\in\mathbb{R}_+^2\times \Omega^2$be such that$\Psi(\Lambda_\delta ,x_\delta )\in[a,b]$,$\nabla_\Lambda\Psi(\Lambda_\delta ,x_\delta )=0,\,\rho(x_\delta) <-\rho,\,\, \mathop{\rm dist}(x_{1_\delta },\partial \Omega)=\delta ,\,\,\mathop{\rm dist}(x_{2_\delta },\partial \Omega)\geq\delta $,$|x_{1_\delta }-x_{2_\delta }|\geq\delta $and for any vector$T$tangent to$\mathbb{R}_+^2\times \partial V_\delta $at the point$(\Lambda_\delta ,x_\delta )$it holds $$\nabla\Psi(\Lambda_\delta ,x_\delta ).T=0.$$ Set$\tilde{\Omega}_\delta =\frac{\Omega-\tilde{x}_{1_\delta}}{\delta }$,$y=\frac{x-\tilde{x}_{1_\delta}}{\delta }$and$ \mu_\delta =\delta ^{\frac{-(n-4)^2}{2(n-8)}}\Lambda_\delta $, where$\tilde{x}_{1_\delta}\in \partial \Omega $satisfies$|x_{1_\delta}-\tilde{x}_{1_\delta}|=\delta $. Then$\mathop{\rm dist}(y_{1_\delta },\partial \tilde{\Omega}_\delta )=1$,$\mathop{\rm dist}(y_{2_\delta },\partial \tilde{\Omega}_\delta )\geq 1$and$|y_{1_\delta }-y_{2_\delta }|\geq 1$. After a rotation and translation we may assume without loss of generality that$y_{1_\delta }\to (0,1)\in \mathbb{R}^{n-1}\times \mathbb{R}$as$\delta $tends to$0$and the domain$\tilde{\Omega}_\delta $becomes the half-space$\pi=\{(y',y^n)\in \mathbb{R}^{n-1}\times\mathbb{R}:y^n>0\}$. We observe that if$\tilde{G}_\delta $and$\tilde{H}_\delta $are the Green's function and its regular part associated to the domain$\tilde{\Omega}_\delta $then $$\tilde{G}_\delta (y_1,y_2)=\delta ^{n-4}G(\delta y_1,\delta y_2),\quad \tilde{H}_\delta (y_1,y_2)=\delta ^{n-4}H(\delta y_1,\delta y_2).$$ Recall that $$\label{53} \lim_\delta \tilde{H}_\delta (y_1,y_2)=H_\pi(y_1,y_2) =\frac{1}{|y_1-\bar{y}_2|^{n-4}} \quad C^1 \mbox{-uniformly on compact sets of } \pi^2,$$ and $$\label{54} \lim_\delta \tilde{G}_\delta (y_1,y_2)=G_\pi(y_1,y_2)=\frac{1}{| y_1-y_2|^{n-4}}-\frac{1}{| y_1-\bar{y}_2|^{n-4}} ,$$$ C^1$-uniformly on compact sets of$\{(y_1,y_2)\in\pi^2 : y_1\neq y_2\}$. Here for$y=(y',y^n)$, we denote$\bar{y}=(y',-y^n)$. Moreover,$\tilde{\Psi}_\delta $denotes by $$\tilde{\Psi}_{\delta }(\mu,y)=\frac{1}{ 2}\left( \tilde{H}_\delta (y_1,y_1)\mu_1^2+\tilde{H}_\delta (y_2,y_2)\mu_2^2-2\tilde{G}_{\delta }(y_1,y_2)\mu_1\mu_2\right)+\frac{1}{ 2}\big( \mu_1^{\frac{8}{n-4}}+\mu_2^{\frac{8}{n-4}}\big),$$ then $$\tilde{\Psi}_{\delta }(\mu,y)=\delta ^{\frac{-4(n-4)}{n-8}}\Psi(\Lambda,x)\,.$$ From \cite[appendix A]{MMP}, we have $$\nabla\Psi(\Lambda,x)=0 \mbox{ if and only if }\nabla\tilde{\Psi}_{\delta }(\mu,y)=0.$$ First of all, we claim that \label{55} 01$ and $$0=\nabla_{y_2^n}\Psi_\pi(\hat{\mu},\hat{y})=(n-4)\hat{\mu}_2 \Big(\Gamma_{n-3}(\beta)\hat{\mu}_1- \frac{1}{(2\beta)^{n-3}}\hat{\mu}_2\Big),$$ where $$\Gamma_{n-3}(\beta)=\frac{1}{(\beta-1)^{n-3}}-\frac{1}{(\beta+1)^{n-3}}>0.$$ We deduce that $$\label{m2} \hat{\mu}_2=(2\beta)^{n-3}\Gamma_{n-3}(\beta)\hat{\mu}_1.$$ On the other hand, by the condition $M_\pi(\hat{y})\hat{\mu}=0$, we get $$\label{m3} \begin{gathered} \frac{1}{2^{n-4}}\hat{\mu}_1-\Gamma_{n-4}(\beta)\hat{\mu}_2=0, \\ -\Gamma_{n-4}(\beta)\hat{\mu}_1 +\frac{1}{(2\beta)^{n-4}}\hat{\mu}_2=0, \end{gathered}$$ where $$\Gamma_{n-4}(\beta)=\frac{1}{(\beta-1)^{n-4}}-\frac{1}{(\beta+1)^{n-4}}.$$ Equations \eqref{m2} and \eqref{m3} imply $$\left(2\beta\Gamma_{n-3}(\beta)-\Gamma_{n-4}(\beta)\right)\hat{\mu}_1=0$$ and a contradiction arises since $2\beta\Gamma_{n-3}(\beta)-\Gamma_{n-4}(\beta)>0$. \noindent {\it Step2}. as in step 1, we prove the following: for any $(\Lambda,x) \in \mathbb{R}_+^2\times \Omega^2$ with $\Psi(\Lambda,x)\in [a,b]$, $\nabla_\Lambda\Psi( \Lambda,x)=0,\,\,\rho(x)<-\rho,\,\,\mathop{\rm dist}(x_1,\partial\Omega)\geq \delta ,\,\,\mathop{\rm dist}(x_2,\partial\Omega)\geq \delta$ and $|x_1-x_2|=\delta$, then there exist a vector $T$ tangent to $\mathbb{R}_+^2\times \partial V_\delta$ at the point $(\Lambda,x)$ such that $$\nabla\Psi(\Lambda,x).T\neq 0.$$ The lemma follows. \end{proof} \begin{lem}\label{l:32} There exist $\delta '_0>0$ and $\rho'_0>0$ such that for any $\delta \in (0,\delta '_0)$ and $\rho\in (0,\rho'_0)$ the function $\Psi$ satisfies the following property: For any sequence $(\Lambda_n,x_n)\in \mathbb{R}_+^2\times (W_\rho^\delta \cap V_\delta )$ such that $\lim_n(\Lambda_n,x_n)=(\Lambda,x)\in\partial (\mathbb{R}_+^2\times (W_\rho^\delta \cap V_\delta ))$ and $\Psi(\Lambda_n,x_n)\in[a,b]$ there exists a vector $T$ tangent to $\partial (\mathbb{R}_+^2\times (W_\rho^\delta \cap V_\delta ))$ at the point $(\Lambda,x)$ such that $$\nabla\Psi(\Lambda,x).T \neq 0.$$ \end{lem} \begin{proof} First, it is easy to check that $00$ such that $$\label{59} \rho(x)<-\rho'_0.$$ In fact, since $\nabla_\Lambda \Psi(\Lambda,x)=0$, we have $$\Psi(\Lambda,x)=\frac{n-8}{2(n-4)}\big(\Lambda_1^{\frac{8}{n-4}} +\Lambda_2^{\frac{8}{n-4}}\big)=\frac{8-n}{4}(M(x)\Lambda,\Lambda),$$ and since $\Psi(\Lambda,x)\in [a,b]$ we deduce that $$|\Lambda|^2\leq \big(\frac{2(n-4)}{n-8}\big)^{\frac{n-4}{4}} b^{ \frac{n-4}{4}} \mbox{ and } (M(x)\Lambda,\Lambda)\leq \frac{4}{8-n}a,$$ which implies \eqref{59} because $(M(x)\Lambda,\Lambda)\geq \rho(x)|\Lambda|^2$. Therefore we have that $x\in \partial V_\delta$ (if we choose $\rho<\rho'_0$ ) and we can apply Lemma \ref{l:31} to conclude the proof. \end{proof} \begin{lem}\label{l:33} The function $\Psi$ constrained to $\mathbb{R}^2_+\times (W_\rho^\delta \cap V_\delta )$ satisfies the Palais-Smal condition in $[a,b]$. \end{lem} \begin{proof} Let $(\Lambda_n,x_n) \in \mathbb{R}_+^2\times (W_\rho^\delta \cap V_\delta )$ be such that $\lim_n\Psi(\Lambda_n,x_n)=c$ and $\lim_n \nabla \Psi(\Lambda_n,x_n)=0$. Arguing as in the proof of Lemma \ref{l:31} it can be shown that $\Lambda_n$ remains bounded component-wise from above and below by a positive constant. As in Lemma \ref{l:32}, $\Lambda \in (\mathbb{R}_+^*)^2$ and by Lemma \ref{l:31}, $x\in (W_\rho^\delta \cap V_\delta )$. \end{proof} \begin{pro}\label{p:23} There exists a critical level for $\Psi$ between $a$ and $b$. \end{pro} \begin{proof} Assume by contradiction that there are no critical levels in the interval $[a,b]$. By Lemmas \ref{l:31} and \ref{l:32}, We can define an appropriate negative flow that will remain in $\mathcal{A}:=\mathbb{R}_+^2 \times (W_\rho^\delta \cap V_\delta )$ at any level $c\in [a,b]$. Moreover the Palais-Smale condition holds for $\Psi_{|\mathcal{A}}$ in $[a,b]$ (see Lemma \ref{l:33} ). Hence there exists a continuous deformation \begin{equation*} \eta : [0,1] \times \Psi_{|\mathcal{A}} ^b \to \Psi_{|\mathcal{A}}^b, \end{equation*} such that for some $a' \in (0,a)$ \begin{gather*} \eta (0,u)=u \quad \forall u \in \Psi_{|\mathcal{A}} ^b \\ \eta (t,u)=u \quad \forall u \in \Psi_{|\mathcal{A}} ^{a'}\\ \eta (1,u) \in \Psi _{|\mathcal{A}}^{a'}. \end{gather*} Then there exist a continuous function $\gamma \in \Gamma$ such that $$\gamma_{/[0,R_0]\times(\mathcal{S}^2\backslash T_\delta )}=\eta_{/[0,R_0]\times(\mathcal{S}^2\backslash T_\delta )}$$ and using \eqref{e}, we obtain \$\Psi(\gamma(R,x,1))