Topological differences at infinity for nonlinear problems related to the fractional Laplacian

Abstract

In this paper, we are interested in the fractional Yamabe-type equation $A_s u= u^\frac{n+2s}{n-2s}$, ${u>0} \text{ in } \Omega \text{ and } u=0 \text{ on } \partial \Omega .$ Here $\Omega$ is a regular bounded domain of ${\mathbb {R}}^n, n\ge 2$ and $A_s, s\in (0, 1)$ represents the fractional Laplacian operator in $\Omega$ with zero Dirichlet boundary condition. Based on the theory of critical points at infinity of Bahri and the localization technique of Caffarelli and Silvestre, we compute the difference of topology induced by the critical points at infinity between the level sets of the variational functional associated to the problem. Our result can be seen as a nonlocal analog of the theorem of Bahri et al. (Cal. Var. Partial. Differ. Equ. 3 (1995) 67–94) on the classical Yamabe-type equation.

1 Introduction

In this paper, we consider the nonlinear fractional Yamabe-type problem

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle A_s u = \displaystyle u^{p},\\ \displaystyle u > 0 \;\hbox { in } \Omega ,\\ \displaystyle u = 0 \;\hbox { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(1.1)

where $\Omega \subset {\mathbb {R}}^n, n\ge 2$ is a regular bounded domain, $p= \frac{n+2s}{n-2s}, s\in (0, 1)$ and $A_s$ represents the fractional Dirichlet Laplacian operator $(-\Delta )^s$ in $\Omega$ defined by using the spectrum of the Laplacian $-\Delta$ in $\Omega$ with zero Dirichlet boundary condition. It can be viewed as the nonlocal version of the Yamabe-type equation

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle -\Delta u = \displaystyle u^\frac{n+2}{n-2},\\ \displaystyle u > 0 \;\hbox { in } \Omega ,\\ \displaystyle u = 0 \;\hbox { on } \partial \Omega . \end{array} \right. \end{aligned}$$

(1.2)

Fractional equations involving $(-\Delta )^s$ has attracted the attention of a lot of researchers as it naturally appears in many fields in various scientific areas. The nonlocal character of the fractional Laplacian makes it difficult to handle. After the paper of Caffarelli and Silvestre [15] who provided a local interpretation to the fractional Laplacian in one more dimension, a large number of studies have been developed. In [14], Cabré and Tan studied the subcritical cases; that is, equation (1.1) with subcritical nonlinearities ($p< \frac{n+2s}{n-2s}$) in the particular case $s=1/2$. They transform the equation in a local form as the Caffarelli-Silvestre extension and established the existence of positive solutions. For similar extensions, we refer to [13, 16, 28].

Motivated by the work of Pohozaev [22] on equation (1.2), Tan [27] proved that equation (1.1) has no solutions if $\Omega$ is a star-shaped domain and $s=1/2$. The resemblance between (1.1) and (1.2) led the authors in [1] to investigate the effect of the topology of $\Omega$ on the existence of solutions of (1.1). Such a result can be seen as the fractional counterpart of the famous result of Bahri and Coron [9]. For more recent results on (1.1) and related problems, we refer to [2,3,4,5,6,7,8,9,10, 18, 23] and the references therein.

Problem (1.1) is delicate from the variational viewpoint because the failure of the Palais-Smale condition (PS). This leads to the possibility of existence of non-compact gradient-flow lines along which the associated variational functional J is bounded and its gradient tends to zero, the so-called critical points at infinity, see [7].

Trying to prove the existence of solutions of (1.1) by studying the topological differences between the level sets of J, it will be useful to compute the topological contributions of the critical points at infinity between these level sets. The main purpose of the present paper is to characterize the critical points at infinity of problem (1.1) and evaluate its topological contributions. We shall prove a fractional analog of the theorem of Bahri et al. [11] on the classical Yamabe-type equation.

2 General framework and statement of results

We start this section by recalling some preliminaries related to the fractional Laplacian. Let $(e_k)_{k\in {\mathbb {N}}}$ be the basis of $L^2(\Omega )$ such that for any $k\in {\mathbb {N}}, \Vert e_k\Vert _{L^2(\Omega )}= 1, \langle e_k, e_\ell \rangle =0, \forall k\ne \ell$ and

$$\begin{aligned} \left\{ \begin{array} {lll} -\Delta e_k &{}=&{} \lambda _k e_k \hbox { in } \Omega , \\ e_k &{}=&{} 0 \;\; \; \; \; \; \hbox { on } \partial \Omega . \end{array} \right. \end{aligned}$$

So for any $k\in {\mathbb {N}}$, $\lambda _k>0$.

The fractional Laplacian $A_s, s\in (0, 1)$ is defined by

$$\begin{aligned} H^s_0(\Omega )\longrightarrow & {} H^{-s}_0(\Omega )\simeq H^s_0(\Omega ),\\ u= \sum _{k=1}^\infty b_k e_k\longmapsto & {} A_s(u)= \sum _{k=1}^\infty b_k \lambda _k^s e_k, \end{aligned}$$

where $H^s_0(\Omega ):= \big \{u= \sum _{k=1}^\infty b_k e_k\in L^2(\Omega ), \sum _{k=1}^\infty b_k^2 \lambda _k^s<\infty \big \}$ and $H^{-s}_0(\Omega )$ is the dual space of the Hilbert fractional Sobolev space $H^{s}_0(\Omega )$. Concerning the local equivalent problem to (1.1), we follow the results of [15] for $\Omega = {\mathbb {R}}^n$, and [14] for bounded domain $\Omega$, see also [13, 16, 25, 28]. Therefore, we consider the associated local problem on the half cylinder with base $\Omega$. Define

$$\begin{aligned} C= \Omega \times [0, \infty )= \{(x, t),\ \mathrm{s.t.} \ x\in \Omega \hbox { and } t\in [0, \infty )\} \end{aligned}$$

and

$$\begin{aligned} C_{0L}^\infty (C)=\{v\in C^\infty ({\bar{C}}),\ \mathrm{s.t.} \ v=0 \hbox { on } \partial _LC\}, \end{aligned}$$

where $\partial _LC$ denotes the lateral boundary of C, it is defined by $\partial \Omega \times [0, \infty )$. Let $H^s_{0L}(C)$ be the Hilbert Sobolev space defined by the closure of $C_{0L}^\infty (C)$ with respect to

$$\begin{aligned} |v| = \left( \int _{C}t^{1-2s} |\nabla v|^2\right) ^\frac{1}{2}, \end{aligned}$$

and equipped by the following inner product:

$$\begin{aligned} \langle v, w\rangle _{H^s_{0L}(C)} = \int _{C}t^{1-2s} \nabla v \nabla w, \; \forall \, v, w \in H^s_{0L}(C). \end{aligned}$$

Following [13, 28], we associate to any $u\in H^s_0(\Omega )$ the unique s-harmonic function denoted $s-h(u)$ in $H^s_{0L}(C)$, the unique solution of the following problem:

$$\begin{aligned} \left\{ \begin{array}{lll} \mathrm{div} (t^{1-2s} \nabla v) &{}= 0&{} \hbox { in } C,\\ v= 0&{}&{} \hbox { on } \partial _LC,\\ v= u&{} &{} \hbox { on } \Omega \times \{0\}. \end{array} \right. \end{aligned}$$

See [13, 28] for the explicit expression of $s-h(u)$. It follows that $A_s$ is expressed by the following map:

$$\begin{aligned} u=\sum _{k=1}^\infty b_k e_k \longmapsto A_s(u)= \partial ^s_\nu (s-h(u))/_{\Omega \times \{0\}}, \end{aligned}$$

where $\nu$ denotes the unit outward normal vector to C on $\Omega \times \{0\}$ and for any $v\in H^s_{0L}(C)$ and any $x\in \Omega$, we have

$$\begin{aligned} \partial ^s_\nu (v)(x, 0)=\displaystyle -c_s\lim _{t\rightarrow 0^+} t^{1-2s} \frac{\partial v}{\partial t}(x, t), \; \hbox { where } \; c_s:= \displaystyle \frac{\Gamma (s)}{2^{1-2s}\Gamma (1-s)}. \end{aligned}$$

In this way, problem (1.1) is equivalent to the following local problem

$$\begin{aligned} \left\{ \begin{array}{lllll} \mathrm{div}(t^{1-2s} \nabla v)=0&{}&{}&{} \hbox { in }&{} C,\\ v> 0&{}&{}&{} \hbox { in }&{} C,\\ v= 0&{}&{}&{} \hbox { on }&{} \partial _LC,\\ \partial ^s_\nu (v)= v^\frac{n+2s}{n-2s}&{}&{}&{} \hbox { on }&{} \Omega \times \{0\}. \end{array} \right. \end{aligned}$$

(2.1)

Therefore, if v satisfies (2.1), then $u(x)= v(x, 0):= tr (v)(x), \forall \, x\in \Omega$ is a solution of (1.1). Notice that

$$\begin{aligned} H_0^s(\Omega )=\{u=tr (v), v\in H^s_{0L}(C), \hbox { with } \mathrm{div}(t^{1-2s} \nabla v)= 0 \hbox { in } C\}. \end{aligned}$$

In order to present the variational structure associated to (1.1), we introduce the following Hilbert space constructed by all s-harmonic functions in $H^s_{0L}(C)$. More precisely, let

$$\begin{aligned} {\mathcal {H}}=\{v\in H^s_{0L}(C),\ \mathrm{s.t.}\ \mathrm{div}(t^{1-2s} \nabla v) = 0 \hbox { in } C\}. \end{aligned}$$

For all $v\in {\mathcal {H}}$, we denote

$$\begin{aligned} \Vert v\Vert ^2 := |v|^2= \int _{C}t^{1-2s} |\nabla v|^2 dx dt = c_s^{-1}\int _{\Omega \times \{0\}}\partial ^s_\nu v(x, 0) \cdot v(x, 0) dx, \end{aligned}$$

and for all $v, w \in {\mathcal {H}}$, we denote

$$\begin{aligned} \langle v, w\rangle =\langle v, w\rangle _{H^s_{0L}(C)}= c_s^{-1}\int _{\Omega \times \{0\}}\partial ^s_\nu v(x, 0) w(x, 0) dx. \end{aligned}$$

As mentioned above, problem (2.1) has a variational structure. The Euler–Lagrange functional is the following:

$$\begin{aligned} J(v)= c_s\frac{\Vert tr (v)\Vert ^2}{\left( \displaystyle \int _{\Omega }|v(x, 0|^\frac{2n}{n-2s}dx\right) ^\frac{n-2s}{n}} \end{aligned}$$

defined on

$$\begin{aligned} \displaystyle \Sigma = \{v\in {\mathcal {H}}, \Vert v\Vert = c_s^{-1/2}\}. \end{aligned}$$

Problem (2.1) is equivalent to finding the critical points of J subjected to the constraint $u\in \Sigma ^+:=\{u\in \Sigma , u>0\}$.

Since $p+1$ is the critical Sobolev exponent of the Sobolev trace embedding $v\in {\mathcal {H}}\mapsto tr (v) \in L^{p+1}(\Omega )$ which is continuous but not compact for $p= \frac{n+2s}{n-2s}$, the functional J does not satisfy the Palais–Smale condition. This means that there exist sequences along which J is bounded, its gradient goes to zero and which do not converge.

For $x, y\in \Omega , t>0$, let ${\tilde{G}}((x, t), y), x, y\in \Omega , t>0$ be the s-harmonic extension of the Green’s function of the fractional Dirichlet Laplacian $A_s$. It satisfies

$$\begin{aligned} \left\{ \begin{array}{lll} \mathrm{div} (t^{1-2s} \nabla {\tilde{G}}(\cdot , y)) &{}= 0&{} \hbox { in } C,\\ {\tilde{G}}(\cdot , y)= 0&{} &{} \hbox { on } \partial _LC,\\ \partial ^s_\nu {\tilde{G}}(\cdot , y)= \delta _y&{} &{} \hbox { on } \Omega \times \{0\}. \end{array} \right. \end{aligned}$$

We have

$$\begin{aligned} {\tilde{G}}((x, t), y)= \frac{\hat{c}}{\Vert (x-y, t)\Vert ^{n-2s}_{{\mathbb {R}}^{n+1}}}- {\tilde{H}}((x, t), y), \end{aligned}$$

where $\hat{c}$ is a fixed constant and ${\tilde{H}}$ is the regular part of ${\tilde{G}}$ (see [19], page 6542). It satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \mathrm{div} (t^{1-2s} \nabla {\tilde{H}}(\cdot , y))= 0 &{}\hbox { in } C,\\ {\tilde{H}}((x, t), y)= \displaystyle \frac{\hat{c}}{\Vert (x-y, t)\Vert ^{n-2s}_{{\mathbb {R}}^{n+1}}}&{} \hbox { on } \partial _LC,\\ \partial ^s_\nu {\tilde{H}}(\cdot , y)= 0 &{}\hbox { on } \Omega \times \{0\}. \end{array} \right. \end{aligned}$$

For any $a\in \Omega$ and $\lambda >0$, we set

$$\begin{aligned} \delta _{(a, \lambda )}(x)= \frac{\lambda ^\frac{n-2s}{2}}{(1+\lambda ^2|x-a|^2)^\frac{n-2s}{2}}, \; x\in {\mathbb {R}}^n. \end{aligned}$$

Following the classification results of [17, 20, 21], $\delta _{(a, \lambda )}(x), a\in \Omega , \lambda >0$ are the only solutions of

$$\begin{aligned} \left\{ \begin{array}{llll} A_s u = c_0 u^\frac{n+2s}{n-2s}&{} \hbox { in } {\mathbb {R}}^n,\\ u>0 &{}\hbox { in } {\mathbb {R}}^n,\\ \displaystyle \lim _{|x|\rightarrow \infty }u(x)= 0,&{} \end{array} \right. \end{aligned}$$

where $c_0$ is a fixed positive constant which depends only on n and s. Notice that in the case where $\Omega = {\mathbb {R}}^n$, the Sobolev space $H^s({\mathbb {R}}^n)$ is defined by

$$\begin{aligned} H^s({\mathbb {R}}^n)=\{u\in L^2({\mathbb {R}}^n),\int _{{\mathbb {R}}^n}(1+|2\pi x|^{2s}) |\hat{u}(x)|^2 dx<\infty \}. \end{aligned}$$

Here $\hat{u}$ denotes the Fourier transform of u. The fractional operator $A_s: H^s({\mathbb {R}}^n)\rightarrow H^{-s}({\mathbb {R}}^n)$ is defined by

$$\begin{aligned} \widehat{A_s(u)}(x)= |2\pi x|^{2s} \hat{u}(x). \end{aligned}$$

Let ${\hat{\delta }}_{(a, \lambda )}$ be the s-harmonic extension of $\delta _{(a, \lambda )}$ in ${\mathbb {R}}_+^{n+1}$ and let

$$\begin{aligned} {\hat{\gamma }}= c_s^\frac{-1}{2}\Vert {\hat{\delta }}_{(a, \lambda )}\Vert ^{-1}_{D^s({{\mathbb {R}}}^{n+1}_{+})}:= \left( c_s\int _{{\mathbb {R}}^{n+1}}t^{1-2s}|\nabla {\hat{\delta }}_{(a, \lambda )}|^2 dx dt\right) ^\frac{-1}{2}. \end{aligned}$$

(2.2)

It is more convenient in the next to work with ${\tilde{\delta }}_{(a, \lambda )}, a\in \Omega$ and $\lambda >0$ defined by

$$\begin{aligned} {\tilde{\delta }}_{(a, \lambda )}= {\hat{\gamma }}{\hat{\delta }}_{(a, \lambda )}. \end{aligned}$$

We have

$$\begin{aligned} \Vert {\tilde{\delta }}_{(a, \lambda )}\Vert _{D^s({{\mathbb {R}}}^{n+1}_{+})}= & {} c_s^\frac{-1}{2}, \end{aligned}$$

(2.3)

$$\begin{aligned} tr ({\tilde{\delta }}_{(a, \lambda )})= & {} {\hat{\gamma }} \delta _{(a, \lambda )} \hbox { on } {\mathbb {R}}^n \end{aligned}$$

(2.4)

and

$$\begin{aligned} \left\{ \begin{array}{lll} \mathrm{div}(t^{1-2s} \nabla {\tilde{\delta }}_{(a, \lambda )}) = 0 \;\; \hbox { in } {\mathbb {R}}_+^{n+1},\\ \partial ^s_\nu {\tilde{\delta }}_{(a, y)}= \gamma _0 {\tilde{\delta }}_{(a, y)}^\frac{n+2s}{n-2s} \qquad \,\,\hbox { on } {\mathbb {R}}^n\times \{0\}, \end{array} \right. \end{aligned}$$

where $\gamma _0= c_0 {\hat{\gamma }}^\frac{-4s}{n-2s}$.

For any $a\in \Omega$ and $\lambda >0$, we define the almost solutions $P {\delta }_{(a, \lambda )}$ of (2.1) as the unique solutions of the following problem:

$$\begin{aligned} \left\{ \begin{array}{llll} \mathrm{div} (t^{1-2s} \nabla P {\delta }_{(a, \lambda )}) = 0&{} \hbox { in } C,\\ P {\delta }_{(a, \lambda )}= 0&{} \hbox { on } \partial _LC,\\ \partial ^s_\nu P {\delta }_{(a, y)}= \partial ^s_\nu {\tilde{\delta }}_{(a, y)} = \gamma _0 {\tilde{\delta }}_{(a, y)}^\frac{n+2s}{n-2s}&{} \hbox { on } \Omega \times \{0\}. \end{array} \right. \end{aligned}$$

Next, we introduce the best constant of Sobolev. Let

$$\begin{aligned} \imath : H^s_{0L}(C)\longrightarrow & {} L^\frac{2n}{n-2s}(\Omega ),\\ v\longmapsto & {} tr (v) \end{aligned}$$

be the Sobolev trace embedding. The best constant of Sobolev is given by

$$\begin{aligned} S= \frac{\Vert tr \;{\tilde{\delta }}_{(a, \lambda )}\Vert _{L^\frac{2n}{n-2s}({\mathbb {R}}^n)}}{\Vert {\tilde{\delta }}_{(a, \lambda )}\Vert _{D^s({\mathbb {R}}^{n+1}_+)}}= c_s^\frac{1}{2}\Vert tr \;{\tilde{\delta }}_{(a, \lambda )}\Vert _{L^\frac{2n}{n-2s}({\mathbb {R}}^n)}, \end{aligned}$$

since $\Vert {\tilde{\delta }}_{(a, \lambda )}\Vert _{D^s({\mathbb {R}}^{n+1}_+)}= c_s^\frac{-1}{2}$. Notice that S is independent of a and $\lambda$ (see [29]). Observe that

$$\begin{aligned} \displaystyle \inf _{v\in \Sigma }J(v)= c_s^\frac{n}{n-2s} S^\frac{-2n}{n-2s}:= {\tilde{S}}= \gamma _0. \end{aligned}$$

Therefore,

$$\begin{aligned} {\tilde{S}}= \frac{1}{\Vert tr \;{\tilde{\delta }}_{(a, \lambda )}\Vert ^\frac{2n}{n-2s}_{L^\frac{2n}{n-2s}({\mathbb {R}}^n)}}=\displaystyle \frac{1}{\displaystyle \int _{{\mathbb {R}}^n}(tr \; {\tilde{\delta }}_{(a, \lambda )})^\frac{2n}{n-2s} dx}. \end{aligned}$$

(2.5)

Arguing as [26], the following proposition describes the Palais–Smale sequences of J.

PROPOSITION 2.1

Assume that (2.1) has no solution. Let $(v_k)_k$ be a sequence in $\Sigma ^+:= \{v\in \Sigma , v\ge 0\}$ such that $J(v_k)\rightarrow c$ and $\partial J(v_k)\rightarrow 0$. There exists $p\in {\mathbb {N}}^*$ and a subsequence of $(v_k)_k$ denoted again $(v_k)_k$ such that $v_k \in V(p, \varepsilon _k)$, where $\varepsilon _k\rightarrow 0$ as $k \rightarrow +\infty$ and

$$\begin{aligned} V(p, \varepsilon )= & {} \left\{ u\in \Sigma ^+,\ \mathrm{s.t.} \; \exists (a_1, \ldots , a_p)\in \Omega ^p, \; \exists (\lambda _1, \ldots , \lambda _p)\in \left[ \frac{1}{\varepsilon }, \infty \right) ^p \hbox { and }\right. \\&(\alpha _1, \ldots , \alpha _p) \in {\mathbb {R}}_+^p,\ \mathrm{s.t.}\ \left\| u - \frac{1}{\sqrt{c_s}}\frac{\sum _{i=1}^p \alpha _i P {\delta } _{(a_i, \lambda _i)}}{\Vert \sum _{i=1}^p \alpha _i P {\delta }_{ (a_i, \lambda _i)}\Vert }\right\|<\varepsilon \\&\hbox {with}~\left. (\lambda _id(a_i, \partial \Omega )>\frac{1}{\varepsilon } \hbox { and } \varepsilon _{ij}< \varepsilon , \forall \, i\ne j\right\} . \end{aligned}$$

Here $\varepsilon _{ij}= \frac{1}{\big (\frac{\lambda _i}{\lambda _j}+ \frac{\lambda _j}{\lambda _i}+ \lambda _i\lambda _j|a_i-a_j|^2\big )^\frac{n-2s}{2}}$.

The following proposition gives suitable parameters for $V(p, \varepsilon )$. The proof is similar to ([9], Proposition 7).

PROPOSITION 2.2

Let $p\in {\mathbb {N}}^*$. There exists $\varepsilon _p>0$ such that for any $0< \varepsilon < \varepsilon _p$ and $u\in V(p, \varepsilon )$, the following minimization problem:

$$\begin{aligned} \inf _{\alpha _i, a_i, \lambda _i}\left\| u- \sum _{i=1}^p \alpha _i P {\delta }_{(a_i, \lambda _i)}\right\| \end{aligned}$$

admits a unique solution $({\bar{\alpha }}, {\bar{a}}, {\bar{\lambda }})$ modulo a permutation on the indices set. Let $v=u- \sum _{i=1}^p {\bar{\alpha }}_i P {\delta }_{({\bar{a}}_i, {\bar{\lambda }}_i)}$. It satisfies

$$\begin{aligned} (V_{0}): \langle v ,\psi \rangle =0\ ~{ for} \ \psi \in \left\{ \displaystyle P\delta _{a_{i},\lambda _{i}}, \frac{\displaystyle \partial P\delta _{a_{i},\lambda _{i}}}{\displaystyle \partial \lambda _{i}}, \frac{\displaystyle \partial P\delta _{a_{i},\lambda _{i}}}{\displaystyle \partial a_{i}},\ i=1,\ldots ,p\right\} . \end{aligned}$$

For $q\in {\mathbb {N}}^*$ and $x=(x_1, \ldots , x_q)\in \Omega ^q$, such that $x_i \ne x_j$ for $i\ne j$, we denote $M(x)= (m_{ij})_{1\le i, j \le q}$ the matrix defined by

$$\begin{aligned} m_{ii}= H((x_i, 0), x_i), \qquad \ m_{ij}= -G((x_i, 0), x_j) \; \forall \, j\ne i. \end{aligned}$$

(2.6)

Let $\rho (x)$ be the least eigenvalue and by e(x) the eigenvector associated to $\rho (x)$ whose norm equals 1 and whose components are strictly positive.

Our main results are the following.

Theorem 2.3

Assume that zero is a regular value of $\rho$. For $\varepsilon >0$ sufficiently small, there exists a change of variables, such that for any $u = \sum _{i=1}^p \alpha _i P {\delta }_{(a_i, \lambda _i)} \in V(p, \varepsilon )$, $(a_i, \lambda _i , v)\rightarrow (a_i', \lambda _i', V)$ where V belongs to a neighborhood of zero in a fixed Hilbert space so that

$$\begin{aligned} J\left( \sum _{i=1}^p \alpha _i P {\delta }_{(a_i, \lambda _i)}+v\right) = J\left( \sum _{i=1}^p \alpha _i P {\delta }_{(a_i', \lambda _i')}\right) + \Vert V\Vert ^2. \end{aligned}$$

Furthermore, if each $a_i$ belongs to a neighborhood of $x_i$ such that $\rho (x_1, \ldots , x_p)>0$ and $\rho '(x_1, \ldots , x_p)=0,$ there exists another change of variables $(a_i, \lambda _i)\rightarrow (a_i', \lambda _i')$ such that

$$\begin{aligned} J\left( \sum _{i=1}^p \alpha _i P {\delta }_{(a_i, \lambda _i)}\right) = \displaystyle \frac{({\tilde{S}})^{2/n}(\sum _{i=1}^p \alpha _i^2)}{( \sum _{i=1}^p\alpha _i^{\frac{2n}{n-2s}})^{\frac{n-2s}{n}}} \left( 1 + \eta \rho (a') \sum _{i=1}^p \frac{1}{\lambda _i'^{ n-2s}}\right) , \end{aligned}$$

up to a multiplicative constant. Here $a'= (a_1', \ldots , a_p')$ and $\eta$ is a fixed positive constant.

The characterisation of the critical points at infinity is given in the following theorem.

Theorem 2.4

Assume that zero is a regular value of $\rho$. Then we have

(i) For $\varepsilon$ small enough, J does not have any critical point in $V(p, \varepsilon )$.

(ii) The only critical points at infinity of J in $V(p, \varepsilon )$ correspond to $\sum _{i=1}^p P {\delta }_{(x_i, +\infty )}$, where $p\in {\mathbb {N}}^*$ and the $x_i$’s satisfy

$$\begin{aligned} \rho (x_1, \ldots , x_p)>0\quad \hbox { and }\quad \rho '(x_1, \ldots , x_p)=0. \end{aligned}$$

(iii) There is $p_0\in {\mathbb {N}}^*$ such that J does not have any critical point at infinity in $V (p, \varepsilon )$ for each $p\ge p_0$.

The following result illustrates the usefulness of the above theorems. It computes the difference of topology between the level sets of the functional J. More precisely, it evaluates the contribution of the critical points at infinity to the relative homology between the sets $W_p$ and $W_{p-1}$, where

$$\begin{aligned} W_p=\{u\in \Sigma ^+, s.t, J(u)< ((p+1){\tilde{S}})^\frac{2}{n}\}. \end{aligned}$$

Theorem 2.5

Assume that J has no critical point in $\Sigma ^+$ and zero is a regular value of $\rho$. Then the relative homology $H_*(W_p,W_{p-1})$ between the sets $W_p$ and $W_{p-1}$ equals to

$$\begin{aligned} H_*(\Omega ^p\times _{\sigma _p}\Delta _{p-1}, \Omega ^p\times \partial \Delta _{p-1}\cup _{\sigma _p} I_p\times \Delta _{p-1}), \end{aligned}$$

where $I_p=\{x\in \Omega ^p, s.t. \,\rho (x)\le 0\}$, $\Delta _{p-1}= \{(\alpha _1, \ldots , \alpha _p), such\,that\, \alpha _i\ge 0, \; \sum _{i=1}^p \alpha _i=1\}$ and $\sigma _p$ is the permutation group.

The remainder of the present paper is organized as follow. Section 3 will be devoted to the expansion of J and its gradient. In Section 4, we will study the v-part of u. In Section 5, we will construct a suitable pseudo-gradient to characterize the critical points associated to problem (1.1). The proofs of Theorems 2.3, 2.4 and 2.5 are given in Section 6.

3 Expansion of the functional and its gradient near potential critical points at infinity

First, we deal with the asymptotic expansion of the functional J.

PROPOSITION 3.1

For $\varepsilon >0$ small enough and $u=\sum _{i=1}^p\alpha _i P {\delta }_{(a_i, \lambda _i)}+v \in V(p, \varepsilon )$, we have the following expansion:

$$\begin{aligned} J(u)&= \displaystyle \frac{(c_s^{-1})^{2/n}(\sum _{i=1}^p \alpha _i^2)}{( \sum _{i=1}^p\alpha _i^{\frac{2n}{n-2s}})^{\frac{n-2s}{n}}} \left[ 1 + \frac{c_1}{c_s^{-1}} \sum _{i=1}^p \frac{H((a_i, 0), a_i)}{\lambda _i^{n-2s}}\left( \frac{2 \alpha _i^{\frac{2n}{n-2s}}}{\sum _{j=1}^p\alpha _j^\frac{2n}{n-2s}}- \frac{\alpha _i^{2}}{\sum _{j=1}^p\alpha _j^2}\right) \right. \\&\left. \qquad - \frac{c_1}{c_s^{-1}} \sum _{i\ne j\ge 1} \left( \varepsilon _{ij} - \frac{H((a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\right) \left( \frac{2 \alpha _i^{\frac{n+2s}{n-2s}}\alpha _j}{\sum _{j=1}^p\alpha _j^\frac{2n}{n-2s}}- \frac{\alpha _i\alpha _j}{\sum _{j=1}^p\alpha _k^2}\right) \right. \\&\left. \qquad + \frac{Q(v, v)}{c_s^{-1}\sum _{j=1}^p\alpha _j^2 } - f(v)\right. \\&\left. \qquad + O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k=1}^p\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) + O(\Vert v \Vert ^{\inf (3, \frac{2n}{n-2s})})\right] , \end{aligned}$$

where $c_1$ is a positive constant, and

$$\begin{aligned} Q(v, v)&= \Vert v\Vert ^2 -\frac{(n+2s)}{(n-2s)} \frac{\sum _j\alpha _j^{2}}{\sum _{j}\alpha _j^{\frac{2n}{n-2s}}}\int \left( \sum _{i}\alpha _i P {\delta }_i\right) ^\frac{4s}{n-2s} v^2,\\ f(v)&= \frac{2 c_s}{\sum _{j}\alpha _j^{\frac{2n}{n-2s}}}\int \left( \sum _{i}\alpha _i P {\delta }_i\right) ^\frac{n+2s}{n-2s} v. \end{aligned}$$

Proof

Let us recall that

$$\begin{aligned} \displaystyle J(u) = \displaystyle \frac{\displaystyle \Vert u\Vert ^2}{\left( \displaystyle \int _{\Omega } u^\frac{2n}{n-2s} \right) ^\frac{n-2s}{n}} . \end{aligned}$$

We need to estimate

$$\begin{aligned} N(u) =\Vert u\Vert ^2 = \left\| \sum _{i=1}^p\alpha _i P {\delta }_i + v \right\| ^2 \quad \text { and }\quad D^\frac{n}{n-2s} = \int _\Omega u^{\frac{2n}{n-2s}}. \end{aligned}$$

Using the fact that v satisfies $(V_0)$, we have

$$\begin{aligned} N(u) = \sum _{i=1}^p \alpha _i^2\Vert P {\delta }_i\Vert ^2 + \sum _{i\ne j} \alpha _i\alpha _j \langle P {\delta }_i, P {\delta }_j\rangle + \Vert v\Vert ^2. \end{aligned}$$

(3.1)

A computation similar to the one performed in [1, 7] shows that, for $\lambda _i d_i$ large enough, we have the following estimates:

$$\begin{aligned} \Vert P {\delta }_i\Vert ^2= & {} c_s^{-1}- c_1\frac{H((a_i, 0), a_i)}{\lambda _i^{n-2s}}+ O\left( \frac{\log (\lambda _i d_i)}{(\lambda _i d_i)^{n+2-2s}}\right) , \end{aligned}$$

(3.2)

$$\begin{aligned} \langle P {\delta }_i, P {\delta }_j \rangle= & {} c_1\left( \varepsilon _{ij} - \frac{H((a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\right) \nonumber \\&+ O\left( \varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k= i, j}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) . \end{aligned}$$

(3.3)

Using (3.1), (3.2) and (3.3), we derive that

$$\begin{aligned} N(u)= & {} \sum _{i=1}^p\alpha _{i}^2\left( c_s^{-1}- c_1\frac{H((a_i, 0), a_i)}{\lambda _i^{n-2s}} \right) + \sum _{j\ne i}\alpha _i\alpha _j c_1\left( \varepsilon _{ij} - \frac{H((a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\right) \nonumber \\&+ \Vert v\Vert ^2+ R, \end{aligned}$$

(3.4)

where

$$\begin{aligned} R= O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k=1}^p\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) . \end{aligned}$$

For the denominator, we have

$$\begin{aligned} D^\frac{n}{n-2s}= & {} \int _{\Omega } \left( \sum _{i=1}^p\alpha _i P {\delta }_i + v \right) ^{\frac{2n}{n-2s}} = \int _{\Omega } \left( \sum _{i=1}^p\alpha _i P{\delta }_i \right) ^{\frac{2n}{n-2s}}\nonumber \\&\quad + \frac{2n}{n-2s} \int _{\Omega } \left( \sum _{i=1}^p\alpha _i P {\delta }_i \right) ^{ \frac{n+2s}{n-2s}} v \nonumber \\&\quad + \frac{n(n+2s)}{(n-2s)^2} \int _{\Omega } \left( \sum _{i=1}^p\alpha _i P {\delta }_i \right) ^{ \frac{4s}{n-2s}} v^2 + O( \Vert v\Vert ^{\inf (3, \frac{2n}{n-2s})}). \end{aligned}$$

(3.5)

Observe that

$$\begin{aligned} \begin{array}{ll} \displaystyle \int _{\Omega } \left( \sum _{i=1}^p\alpha _i P {\delta }_i \right) ^{\frac{2n}{n-2s}} \!=\!&{} \sum _{i=1}^p\alpha _i^{\frac{2n}{n-2s}}\int _{\Omega } \; P {\delta }_i^{ \frac{2n}{n-2s}}\! \! +\!\! \displaystyle \;\frac{2n}{n-2s} \sum _{i\ne j}\alpha _i^{\frac{n+2s}{n-2s}} \alpha _j\int _{\Omega } P {\delta }_i^{\frac{n+2s}{n-2s}} P {\delta }_j \\ &{}+ O\left( \sum _{i\ne j}\int P {\delta }_i^\frac{4s}{n-2s}\inf ( P{\delta }_i, P{\delta }_j)^2\right) . \end{array}\nonumber \\ \end{aligned}$$

(3.6)

A computation similar to the one performed in [1, 7] shows that

$$\begin{aligned}&\int P {\delta }_i^\frac{4s}{n-2s}\inf ( P {\delta }_i, P {\delta }_j)^2 = O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k= i, j}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) , \end{aligned}$$

(3.7)

$$\begin{aligned}&\int P {\delta }_i^{\frac{n+2s}{n-2s}} P{\delta }_j= c_1 \left( \varepsilon _{ij}-\frac{H((a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\right) \nonumber \\&+ O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k= i, j}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) , \end{aligned}$$

(3.8)

$$\begin{aligned}&\int P {\delta }_i ^{\frac{2n}{n-2s}} = c_s^{-1}- c_1\frac{2n}{n-2s}\frac{H((a_i, 0), a_i)}{\lambda _i^{n-2s}}+ O\left( \frac{\log (\lambda _i d_i)}{(\lambda _i d_i)^{n+2-2s}}\right) . \end{aligned}$$

(3.9)

Using (3.7), (3.8) and (3.9), we get

$$\begin{aligned}&\int \left( \sum _{i=1}^p\alpha _i P {\delta }_i \right) ^{\frac{2n}{n-2s}}= \sum _{i=1}^p\alpha _i^\frac{2n}{n-2s}\left( c_s^{-1}- c_1\frac{2n}{n-2s}\frac{H((a_i, 0), a_i)}{\lambda _i^{n-2s}}\right) \nonumber \\&\quad + \frac{2n}{n-2s} c_1 \sum _{i\ne j}\alpha _i^{\frac{n+2s}{n-2s}} \alpha _j \left( \varepsilon _{ij}-\frac{H((a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\right) \nonumber \\&\quad + O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k= i, j}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) . \end{aligned}$$

(3.10)

Combining (3.4), (3.5) and (3.10) and the fact that $J(u)^\frac{n}{n-2s}\alpha _i^\frac{4s}{n-2s}=1+o(1)$, for each i, the result follows.

PROPOSITION 3.2

For $u=\sum _{i=1}^p\alpha _i P {\delta }_{(a_i, \lambda _i)} \in V(p, \varepsilon )$, we have the following expansion:

$$\begin{aligned} \left\langle \partial J(u), \lambda _{i} \displaystyle \frac{\partial P \delta _{a_i, \lambda _i}}{\partial \lambda _{i}} \right\rangle= & {} 2c_1 J(u)\left[ -\frac{n-2s}{2} \alpha _i \frac{{H}((a_i, 0), a_i)}{\lambda _i^{n-2s}}(1+o(1))\right. \\&\left. - \displaystyle \sum _{j \ne i} \alpha _{j} \left( \lambda _{i}\displaystyle \frac{\partial \varepsilon _{i j}}{\partial \lambda _{i}} + \frac{n-2s}{2}\frac{{H}((a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\right) (1+o(1))+R\right] . \end{aligned}$$

Proof

For any $h\in {\mathcal {H}}$, we have

$$\begin{aligned} \langle \partial J(u), h\rangle = 2J(u)\left( \langle u, h\rangle - J(u)^\frac{n}{n-2s}\int _\Omega u^\frac{n+2s}{n-2s} h\right) .\end{aligned}$$

(3.11)

Thus

$$\begin{aligned} \left\langle \partial J(u), \lambda _i \frac{\partial P {\delta _i}}{\partial \lambda _i}\right\rangle&= 2J(u)\left[ \left\langle \sum _{j=1}^p\alpha _j {P {\delta }_j}, \lambda _i \frac{\partial P {\delta _i}}{\partial \lambda _i}\right\rangle \right. \\&\left. \quad -J(u)^\frac{n}{n-2s} \int _\Omega \left( \sum _{j=1}^p\alpha _j { P {\delta }}_j\right) ^\frac{n+2s}{n-2s}\lambda _i \frac{\partial P {\delta }_i}{\partial \lambda _i}\right] . \end{aligned}$$

Observe that

$$\begin{aligned}&\int _\Omega \left( \sum _{j=1}^p\alpha _j{P \delta }_j\right) ^\frac{n+2s}{n-2s}\lambda _i \frac{\partial P \delta _i}{\partial \lambda _i}= \sum _{j=1}^p\alpha _j^\frac{n+2s}{n-2s}\int _\Omega P {\delta }_j^\frac{n+2s}{n-2s}\lambda _i \frac{\partial P \delta _i}{\partial \lambda _i} \nonumber \\&\quad + \frac{n+2s}{n-2s}\sum _{i\ne j} \int _\Omega (\alpha _i P {\delta }_i)^\frac{4s}{n-2s}\lambda _i \frac{\partial P \delta _i}{\partial \lambda _i}( \alpha _j P {\delta }_j) + O\left( \sum _{j\ne i}\int _\Omega P{\delta }_j^\frac{4s}{n-2s}\inf ( {\delta }_j, {\delta }_i)^2\right) .\nonumber \\ \end{aligned}$$

(3.12)

A computation similar to the one performed in [7], shows that

$$\begin{aligned}&\left\langle P {\delta }_i, \lambda _i\frac{\partial P {\delta }_i}{\partial \lambda _i}\right\rangle = \frac{n-2s}{2}c_1 \frac{H((a_i, 0), a_i)}{\lambda _i^{n-2s}} + O\left( \frac{\log (\lambda _i d_i)}{(\lambda _i d_i)^{n+2-2s}}\right) , \end{aligned}$$

(3.13)

$$\begin{aligned}&\left\langle P {\delta }_j, \lambda _i\frac{\partial P {\delta }_i}{\partial \lambda _i}\right\rangle = c_1\left( \lambda _i\frac{\partial \varepsilon _{ij}}{\partial \lambda _i} + \frac{n-2s}{n} \frac{H((a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad +O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k= i, j}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) , \end{aligned}$$

(3.14)

$$\begin{aligned}&\int _\Omega P {\delta }_i^\frac{n+2s}{n-2s}\lambda _i \frac{\partial P \delta _i}{\partial \lambda _i}= (n-2s) c_1 \frac{H((a_i, 0), a_i)}{\lambda _i^{n-2s}} + O\left( \frac{\log (\lambda _i d_i)}{(\lambda _i d_i)^{n+2-2s}}\right) , \end{aligned}$$

(3.15)

$$\begin{aligned}&\int _\Omega P {\delta }_j^\frac{n+2s}{n-2s}\lambda _i \frac{\partial P \delta _i}{\partial \lambda _i}\!=\!\left\langle P {\delta }_j, \lambda _i\frac{\partial P {\delta }_i}{\partial \lambda _i}\right\rangle \!+\!O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} \!+\! \sum _{k= i, j}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) ,\nonumber \\ \end{aligned}$$

(3.16)

$$\begin{aligned}&\frac{n+2s}{n-2s}\int _\Omega P {\delta }_j P {\delta }_i^\frac{4s}{n-2s}\lambda _i \frac{\partial P \delta _i}{\partial \lambda _i}= \left\langle P {\delta }_j, \lambda _i\frac{\partial P {\delta }_i}{\partial \lambda _i}\right\rangle \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad + O\left( \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k= i, j}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\right) .\nonumber \\ \end{aligned}$$

(3.17)

Using (3.13)– (3.17) and the fact that $J(u)^\frac{n}{n-2s}\alpha _i^\frac{4s}{n-2s}=1+o(1)$, for each i, Proposition 3.2 follows.

PROPOSITION 3.3

For $u=\sum _{i=1}^p\alpha _i P {\delta }_{(a_i, \lambda _i)} \in V(p, \varepsilon )$, we have the following expansion:

Proof

Using (3.11), we have

$$\begin{aligned} \left\langle \partial J(u), \frac{1}{\lambda _i} \frac{\partial P {\delta _i}}{\partial a_i}\right\rangle&= 2J(u)\left[ \left\langle \sum _{j=1}^p\alpha _j {P {\delta }_j}, \frac{1}{\lambda _i} \frac{\partial P {\delta _i}}{\partial a_i}\right\rangle \right. \\&\quad \left. - J(u)^\frac{n}{n-2s} \int _\Omega \left( \sum _{j=1}^p\alpha _j { P {\delta }}_j\right) ^\frac{n+2s}{n-2s}\frac{1}{\lambda _i} \frac{\partial P {\delta }_i}{\partial a_i}\right] . \end{aligned}$$

Observe that

$$\begin{aligned} \int _\Omega \Big (\sum _{j=1}^p\alpha _j{P \delta }_j\Big )^\frac{n+2s}{n-2s}\frac{1}{\lambda _i} \frac{\partial P \delta _i}{\partial a_i}= & {} \sum _{j=1}^p\alpha _j^\frac{n+2s}{n-2s}\int _\Omega P {\delta }_j^\frac{n+2s}{n-2s}\frac{1}{\lambda _i} \frac{\partial P \delta _i}{\partial a_i} \nonumber \\&+\! \frac{n\!+\!2s}{n\!-\!2s}\sum _{i\ne j} \int _\Omega (\alpha _i P {\delta }_i)^\frac{4s}{n\!-\!2s}\frac{1}{\lambda _i} \frac{\partial P \delta _i}{\partial a_i}( \alpha _j P {\delta }_j) \nonumber \\&+ O\Bigg ( \sum _{j\ne i}\int _\Omega P{\delta }_j^\frac{4s}{n-2s}\inf ( {\delta }_j, {\delta }_i)^2\Bigg ).\nonumber \\ \end{aligned}$$

(3.18)

A computation similar to the one performed in [7], shows that

$$\begin{aligned}&\left\langle P {\delta }_i, \frac{1}{\lambda _i}\frac{\partial P {\delta }_i}{\partial a_i}\right\rangle = -\frac{1}{2}\frac{c_1 }{\lambda _i^{n+1-2s}}\frac{\partial H((a_i, 0), a_i)}{\partial a_i} + O\left( \frac{1}{(\lambda _i d_i)^{n+2-2s}}\right) , \end{aligned}$$

(3.19)

$$\begin{aligned}&\left\langle P {\delta }_j, \frac{1}{\lambda _i}\frac{\partial P {\delta }_i}{\partial a_i}\right\rangle = \frac{c_1}{\lambda _i}\left( \frac{\partial \varepsilon _{ij}}{\partial a_i} - \frac{1}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\frac{\partial H((a_i, 0), a_j)}{\partial a_i}\right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad + O\left( \sum _{k=i, j}\frac{1}{(\lambda _k d_k)^{n+2-2s}}+ \sum _{i\ne j} \lambda _i|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \right) , \end{aligned}$$

(3.20)

$$\begin{aligned}&\int _\Omega P {\delta }_i^\frac{n+2s}{n-2s}\frac{1}{\lambda _i} \frac{\partial P \delta _i}{\partial a_i}= -\frac{c_1 }{\lambda _i^{n+1-2s}}\frac{\partial H((a_i, 0), a_i)}{\partial a_i} + O\left( \frac{\log (\lambda _i d_i)}{(\lambda _i d_i)^{n+2-2s}}\right) , \end{aligned}$$

(3.21)

$$\begin{aligned}&\int _\Omega P {\delta }_j^\frac{n+2s}{n-2s}\frac{1}{\lambda _i} \frac{\partial P \delta _i}{\partial a_i}=\left\langle P {\delta }_j, \frac{1}{\lambda _i}\frac{\partial P {\delta }_i}{\partial a_i}\right\rangle + O\left( \sum _{k=i, j}\frac{1}{(\lambda _k d_k)^{n+2-2s}}+ \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} \right) ,\nonumber \\ \end{aligned}$$

(3.22)

$$\begin{aligned}&\frac{n+2s}{n-2s}\int _\Omega P {\delta }_j P {\delta }_i^\frac{4s}{n-2s}\frac{1}{\lambda _i} \frac{\partial P \delta _i}{\partial a_i}= \left\langle P {\delta }_j, \frac{1}{\lambda _i}\frac{\partial P {\delta }_i}{\partial a_i}\right\rangle \nonumber \\&+ O\left( \sum _{k=i, j}\frac{1}{(\lambda _k d_k)^{n+2-2s}}+ \sum _{j\ne i}\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1}\right) . \end{aligned}$$

(3.23)

Using (3.18)–(3.23) and the fact that $J(u)^\frac{n}{n-2s}\alpha _i^\frac{4s}{n-2s}=1+o(1)$, for each i, Proposition 3.3 follows.

4 The v-part of u

In this section, we deal with the v-part of u, in order to show that it is negligible with respect to the concentration phenomenon.

PROPOSITION 4.1

There is a ${\mathcal {C}}^{1}$-map which to each $(\alpha _{i},a_{i},\lambda _{i})$ such that $\sum _{i=1}^{p} \alpha _{i}P \delta _{(a_{i},\lambda _{i})}$ belongs to $V(p,\varepsilon )$ associates ${\bar{v}}={{\bar{v}}}(\alpha , a, \lambda )$ such that ${\bar{v}}$ is unique and satisfies

$$\begin{aligned} J\left( \sum _{i=1}^{p} \alpha _{i} P \delta _{(a_{i},\lambda _{i})}+{\bar{v}}\right) =\displaystyle \min _{v \in (V_0)}\left\{ J\left( \sum _{i=1}^{p} \alpha _{i} P \delta _{(a_{i},\lambda _{i})}+v\right) \right\} . \end{aligned}$$

Furthermore, we have the following estimate:

$$\begin{aligned} \Vert {\bar{v}}\Vert \le \; \left\{ \begin{array}{ll} \displaystyle \sum _{i=1}^p \frac{\log (\lambda _id_i)}{ (\lambda _id_i)^\frac{n+2s}{2}}+ \sum _{j \ne i} \varepsilon _{ij}^\frac{n+2s}{2(n-2s)}(\log \varepsilon _{ij}^{-1})^\frac{n+2s}{2n}, &{} \hbox { if } n \ge 6 \\ \displaystyle \sum _{i=1}^p \frac{1}{ (\lambda _id_i)^{n-2s}}+ \sum _{j \ne i} \varepsilon _{ij}(\log \varepsilon _{ij}^{-1})^\frac{n-2s}{n}, &{} \hbox { if } n \le 5 \\ \end{array} \right. \end{aligned}$$

Proof

Since $\frac{\alpha _i}{\alpha _j}=1 +o(1)$, then the quadratic form Q(v, v) defined in Proposition 3.1 is close to

$$\begin{aligned} \Vert v\Vert ^2 -\frac{(n+2s)}{(n-2s)}\sum _{i} \int P {\delta }_i^\frac{4s}{n-2s} v^2. \end{aligned}$$

(4.1)

Arguing as in [7], the existence of ${\bar{v}}$ follows, since Q(v, v) is definitive and positive. Thus

$$\begin{aligned} \exists \, \alpha >0,\ \mathrm{s.t.} \ \alpha \Vert {\bar{v}}\Vert ^2\le |(f, {\bar{v}})| \le \alpha |f| \Vert {\bar{v}}\Vert , \end{aligned}$$

(4.2)

where f is the linear form defined in Proposition 3.1. Thus, it is sufficient to estimate |f|. We have

$$\begin{aligned} f(v) = \frac{2 c_s}{\sum _{j}\alpha _j^{\frac{2n}{n-2s}}}\sum _{i=1}^p\alpha _i^\frac{n+2s}{n-2s} \int P {\delta }_i^\frac{n+2s}{n-2s} v + O\Bigg (\sum _{j\ne i} \int _{P{\delta }_j \le P{\delta }_i} P {\delta }_i^\frac{4s}{n-2s} P {\delta }_j |v| \Bigg ). \end{aligned}$$

(4.3)

Observe that

$$\begin{aligned} \int P {\delta }_i^\frac{n+2s}{n-2s} v = \int {\delta }_i^\frac{n+2s}{n-2s} v + O\Bigg (\int _{B_i\cup B_i^c} {\delta }_i^\frac{4s}{n-2s} \theta _i |v| \Bigg ), \end{aligned}$$

(4.4)

where $B_i =\{x, |x-a_i|< d_i\}$ and $\theta _i= {\delta }_i- P {\delta }_i$. Then, using the Holder’s inequality, we need to estimate

$$\begin{aligned}&\int _{ B_i^c} \big ({\delta }_i^\frac{4s}{n-2s} \theta _i\big )^\frac{2n}{n-2s}\le \int _{ B_i^c} {\delta }_i^\frac{2n}{n-2s} =O\Bigg (\frac{1}{(\lambda _i d_i)^{n+2-2s}}\Bigg ), \end{aligned}$$

(4.5)

$$\begin{aligned}&|\theta _i|_\infty \int _{ B_i} \big ({\delta }_i^\frac{8n}{(n+2s)(n-2s)} \big )^\frac{n+2s}{2n}\nonumber \\&\quad =O\left( \frac{1}{(\lambda _i d_i)^\frac{n+2s}{2}}+ (\hbox {if } n=6) \frac{\log (\lambda _i d_i)}{(\lambda _i d_i)^4} + (\hbox {if } n\le 5) \frac{1}{(\lambda _i d_i)^{n-2s}}\right) . \end{aligned}$$

(4.6)

Also, we have

$$\begin{aligned} \int _{P{\delta }_j \le P{\delta }_i} P {\delta }_i^\frac{4s}{n-2s} P {\delta }_j |v| \le |v|\Bigg [\int _{P{\delta }_j \le P{\delta }_i}\big ( P {\delta }_i^\frac{4s}{n-2s} P {\delta }_j\big )^\frac{2n}{n+2s}\Bigg ]^\frac{n+2s}{2n}. \end{aligned}$$

(4.7)

If $n\ge 6$, then $\frac{2n}{n+2s}\ge \frac{n}{n-2s}$. Therefore,

$$\begin{aligned} \int _{P{\delta }_j \le P{\delta }_i}\big ( P {\delta }_i^\frac{4s}{n-2s} P {\delta }_j\big )^\frac{2n}{n+2s}\le \int ( {\delta }_i {\delta }_j)^\frac{n}{n-2s}= O\Bigg (\varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1}\Bigg ). \end{aligned}$$

(4.8)

If $n\le 5$, then $1 < \frac{4s}{n-2s}$. In this case,

$$\begin{aligned} \int _{P{\delta }_j \le P{\delta }_i}\big ( P {\delta }_i^\frac{4s}{n-2s} P {\delta }_j\big )^\frac{2n}{n+2s}\le \bigg [\int ({\delta }_i {\delta }_j)^\frac{n}{n-2s}\bigg ]^\frac{2(n-2s)}{n+2s}= O\Bigg (\varepsilon _{ij}^\frac{2n}{n+2s}(\log \varepsilon _{ij}^{-1})^\frac{2(n-2s)}{n+2s}\Bigg ).\nonumber \\ \end{aligned}$$

(4.9)

This concludes the proof.

5 Construction of the pseudo-gradient

This section is devoted to the construction of a suitable pseudo-gradient of J for which the Palais–Smale condition is satisfied along the decreasing flow lines as long as these flow lines do not enter in some neighborhood of $\sum _{i=1}^p P {\delta }_{(x_i, +\infty )}$, $p\in {\mathbb {N}}^*$ such that

$$\begin{aligned} \rho (x_1, \ldots , x_p)>0\quad \hbox { and }\quad \rho '(x_1, \ldots , x_p)=0. \end{aligned}$$

Such a construction allows us to identify the critical points at infinity of the variational structure associated to (1.1).

Theorem 5.1

Assume that zero is a regular value of $\rho$. For any $p\ge 1$ and $\varepsilon >0$ small enough, there exists a pseudo-gradient W in ${V}(p, \varepsilon )$ satisfying the following:

There exists a constant $c>0$ such that for any $u= \sum _{i=1}^p\alpha _i P \delta _{a_i, \lambda _i} \in {V}(p, \varepsilon )$, we have

$$\begin{aligned}&(i)\,\, \langle \partial J(u), W(u)\rangle \le -c \bigg (\sum _{i=1}^p\Big (\frac{1}{(\lambda _id_i)^{n+1-2s}}\Big )+ \sum _{j\ne i}\varepsilon _{ij}^\frac{n+1-2s}{n-2s}\bigg ),\\&(ii)\,\, \Bigg \langle \partial J(u {+}{\bar{v}}), W(u) + \frac{\partial {\bar{v}}}{\partial (\alpha , a, \lambda )}(W(u))\Bigg \rangle {\le } -c \Bigg (\sum _{i=1}^p\Bigg (\frac{1}{(\lambda _id_i)^{n+1-2s}}\Bigg ){+} \sum _{j\ne i}\varepsilon _{ij}^\frac{n+1-2s}{n-2s}\Bigg ). \end{aligned}$$

|W| is bounded, the minimal distance to the boundary only increases if it is small enough and the only case where $\lambda _i(s), i=1, \ldots , p, s\ge 0,$ tend to $\infty$ is when $\rho (X)>0$ and $\rho '(X)=0$, where $X= (a_1, \ldots , a_p)$.

In order to construct the required pseudo-gradient, we need to introduce the following notations: For each $i\in \{1, \ldots , p\}$, let

$$\begin{aligned} I_1=\Bigg \{i,\ \mathrm{s.t.}\ \frac{1}{2^{p+1}}\sum _{k\ne i} \varepsilon _{ki}\le \sum _{j=1}^p\frac{H(a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\Bigg \}, \\ I_2=\Bigg \{i,\ \mathrm{s.t.}\ \frac{1}{2^{p+1}}\sum _{k\ne i} \varepsilon _{ki}> \sum _{j=1}^p\frac{H(a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\Bigg \}. \end{aligned}$$

Without loss of generality, we can assume that $\lambda _i d_i$: $\lambda _1 d_1\le \lambda _2 d_2\le \cdots \le \lambda _p d_p$. Let us define

$$\begin{aligned} I= \{1\}\cup \{i,\ \mathrm{s.t.}\ \forall \, k\le i, c_2\lambda _k d_k\le \lambda _{k-1} d_{k-1} \le \lambda _k d_k\}, \end{aligned}$$

where $c_2$ is a constant chosen small enough.

Case 1: $I\cap I_2\ne \emptyset$ and $I\ne \{1, \ldots , p\}$. We order all the concentrations $\lambda _i, i\in I_2$. Assume that

$$\begin{aligned} \lambda _{i_1}\le \lambda _{i_2}\le \cdots \le \lambda _{i_r}. \end{aligned}$$

Let

$$\begin{aligned} W_1= -\sum _{k=1}^r 2^k\alpha _{i_k}\lambda _{i_k}\frac{\partial P\delta _{i_k}}{\partial \lambda _{i_k}}. \end{aligned}$$

We claim that

$$\begin{aligned} \langle \partial J(u), W_1(u)\rangle \le -c \sum _{i\in I_2}\Bigg (\frac{1}{(\lambda _id_i)^{n-2s}}+ \sum _{j\ne i}\varepsilon _{ij}\Bigg ) + R. \end{aligned}$$

(5.1)

Indeed, using Proposition (3.2), we derive that

$$\begin{aligned} \langle \partial J(u), W_1(u)\rangle= & {} 2c_1 J(u)\sum _{k=1}^r \bigg [-\sum _{j\ne i_k}2^k \alpha _j\alpha _{i_k} \lambda _{i_k}\displaystyle \frac{\partial \varepsilon _{ji_k}}{\partial \lambda _{i_k}}(1+o(1))\nonumber \\&\quad - \frac{n-2s}{2}\sum _{j=1}^p 2^k\alpha _j\alpha _{i_k} \frac{{H}((a_j, 0), a_{i_k})}{(\lambda _j\lambda _{i_k})^\frac{n-2s}{2}}(1+o(1))+ R \bigg ]. \end{aligned}$$

(5.2)

Observe that

$$\begin{aligned} - \lambda _i \frac{\partial \varepsilon _{ij}}{\partial \lambda _i}= \frac{n-2s}{2}\varepsilon _{ij}\Bigg (1-2\frac{\lambda _j}{\lambda _i}\varepsilon _{ij}^\frac{2}{n-2s}\Bigg ). \end{aligned}$$

(5.3)

Thus for $\lambda _i\ge \lambda _j$,

$$\begin{aligned} -2\lambda _i \frac{\partial \varepsilon _{ij}}{\partial \lambda _i}- \lambda _j \frac{\partial \varepsilon _{ij}}{\partial \lambda _j}\ge -\lambda _i \frac{\partial \varepsilon _{ij}}{\partial \lambda _i}= \frac{n-2s}{2}\varepsilon _{ij}+O\Big (\varepsilon _{ij}^\frac{n+2-2s}{n-2s}\Big ). \end{aligned}$$

(5.4)

Furthermore, arguing as in [7, 24] and using the maximum principle, the regular part of the Green’s function satisfies

$$\begin{aligned} H((a_i, 0), a_j)\le \max (d_i, d_j)^{2s-n}. \end{aligned}$$

(5.5)

For $j\in I_1$ and $i\ne j$, if $d_j/2\le d_i\le 2 d_j$, using (5.5), we obtain

$$\begin{aligned} \frac{\lambda _j}{\lambda _i}\varepsilon _{ij}^\frac{2}{n-2s}= o(1). \end{aligned}$$

(5.6)

In the other case (i.e. $d_i\le d_j/2$ or $d_i\ge 2d_j$), we use the inequality $|a_i-a_j|\ge \frac{1}{2}\max (d_i, d_j)$ to obtain (5.6). Thus

$$\begin{aligned} \langle \partial J(u), W_1(u)\rangle\le & {} -(n-2s)c_1 J(u)\nonumber \\&\sum _{i\in I_2}\bigg [\sum _{i\ne j} \varepsilon _{ij}(1+o(1)) \!-\! 2^p\sum _{j=1}^p \frac{{H}((a_i, 0), a_{j})}{(\lambda _i\lambda _{j})^\frac{n\!-\!2s}{2}} \bigg ]\!+\! R. \end{aligned}$$

(5.7)

Since $i\in I_2$, we obtain

$$\begin{aligned} \langle \partial J(u), W_1(u) \rangle \le -c\sum _{i\in I_2}\bigg [\sum _{i\ne j} \varepsilon _{ij}+ 2^p\sum _{j=1}^p \frac{{H}((a_i, 0), a_{j})}{(\lambda _i\lambda _{j})^\frac{n-2s}{2}} \bigg ] + R. \end{aligned}$$

(5.8)

A similar computation as in the proof of (2.8) of [24] shows that

$$\begin{aligned} H((a_i, 0), a_i) = (2d_i)^{2s-n} + o(d_i^{2s-n}), \end{aligned}$$

(5.9)

for each point $a_i$ near the boundary. From another part, for each $a_i$ in a compact set K of $\Omega$, we have $H((a_i, 0) a_i)\ge c$. Thus

$$\begin{aligned} H((a_i, 0), a_i)\ge c (d_i)^{2s-n}, \end{aligned}$$

(5.10)

for each $a_i\in \Omega$. Using (5.10) and (5.8), claim (5.1) follows.

Since $\lambda _1 d_1\le \lambda _2 d_2\le \cdots \le \lambda _p d_p$, we can make appear the term $(\lambda _id_i)^{-(n-2s)}$ in the upper bound of (5.1). Therefore,

$$\begin{aligned} \langle \partial J(u), W_1(u)\rangle \le -c \Big (\sum _{i=1}^p\frac{1}{(\lambda _id_i)^{n-2s}}+ \sum _{i\in I_2, j\ne i}\varepsilon _{ij}\Big ) + R. \end{aligned}$$

(5.11)

For $i\in I_1$, we have

$$\begin{aligned} \frac{1}{2^{p+1}}\sum _{k\ne i} \varepsilon _{ki}\le \sum _{j=1}^p\frac{H(a_i, 0), a_j)}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\le \frac{p}{(\lambda _1d_1)^{n-2s}}. \end{aligned}$$

(5.12)

Therefore, from $(\lambda _1d_1)^{-(n-2s)}$ we can make appear the term $\sum _{i\in I_1, j\ne i}\varepsilon _{ij}$ in the upper bound of (5.11). Hence the estimates (i) of Theorem 5.1 follows in this case.

Case 2: $I\cap I_2\ne \emptyset$ and $I= \{1, \ldots , p\}$. Let $i_1= \min \{i,\ \mathrm{s.t.}\ i\in I_2\}$ and $I_{i_1}= \{j\not \in I_2,\ \mathrm{s.t.}\ \lambda _{i_1}d_{i_1}\le 2c_2 \lambda _j d_j\}.$ Let

$$\begin{aligned} W_2= -\sum _{i\in I_2\cup I_{i_1}}\lambda _i\frac{\partial P \delta _i}{\partial \lambda _i}. \end{aligned}$$

Using Proposition 3.2, we have

$$\begin{aligned} \langle \partial J(u), W_2(u) \rangle= & {} 2c_1 J(u)\sum _{i\in I_2\cup I_{i_1}} \bigg [-\sum _{j\ne i}2^i\alpha _j\alpha _{i} \lambda _{i}\displaystyle \frac{\partial \varepsilon _{ji}}{\partial \lambda _{i}}(1+o(1))\nonumber \\&- \frac{n-2s}{2}\sum _{j=1}^p 2^i\alpha _j\alpha _{i} \frac{{H}((a_j, 0), a_{i})}{(\lambda _j\lambda _{i})^\frac{n-2s}{2}}(1+o(1)) \bigg ]+R. \end{aligned}$$

(5.13)

Since $I= \{1, \ldots , p\}$, we have for each $i\ne j$,

$$\begin{aligned} \varepsilon _{ij}= & {} \Bigg (\frac{1}{\lambda _i\lambda _j|a_i-a_j|^2}\Bigg )^\frac{n-2s}{2}\Bigg (1+O\Bigg (\frac{1}{\lambda _i^2|a_i-a_j|^2}+ \frac{1}{\lambda _j^2|a_j-a_j|^2}\Bigg )\Bigg ) \nonumber \\= & {} \Bigg (\frac{1}{\lambda _i\lambda _j|a_i-a_j|^2}\Bigg )^\frac{n-2s}{2}+ O \Bigg (\frac{1}{(\lambda _1d_1)^{n+2-2s}}+ \varepsilon _{ij}^\frac{n}{n-2s}\Bigg ). \end{aligned}$$

(5.14)

Indeed, if $d_i\le d_j/2$ or $d_i\ge 2d_j$, we have $|a_i-a_j|\ge \frac{1}{2}\max (d_i, d_j)$ and the result follows. In the other case, if $d_j/2\le d_i\le 2 d_j$, using that $i, j\in I$, we derive that $\frac{\lambda _i}{\lambda _j}$ and $\frac{\lambda _j}{\lambda _i}$ are bounded. Therefore, $(\lambda _k|a_i-a_j|)^{-2}= O(\varepsilon _{ij}^\frac{2}{n-2s})$ for $k=i, j$. Thus, for $i\in I_{i_1}$, using (5.3) and (5.14), we get

$$\begin{aligned}&\sum _{j\ne i}\Bigg (-\lambda _i \frac{\partial \varepsilon _{ji}}{\partial \lambda _i} -\frac{n-2s}{2} \frac{{H}((a_i, 0), a_{j})}{(\lambda _i\lambda _{j})^\frac{n-2s}{2}}\Bigg ) \nonumber \\&\quad = \frac{n-2s}{2}\sum _{j\ne i} \frac{G((a_i, 0), a_{j})}{(\lambda _i\lambda _{j})^\frac{n-2s}{2}} + O \Bigg (\frac{1}{(\lambda _1d_1)^{n+2-2s}} + \varepsilon _{ij}^\frac{n}{n-2s}\Bigg ). \end{aligned}$$

(5.15)

Furthermore, for $i\in I_{i_1}$,

$$\begin{aligned} \frac{{H}((a_i, 0), a_{i})}{(\lambda _i)^{n-2s}}= O\Bigg (\frac{1}{(\lambda _i d_i)^{n-2s}}\Bigg )= O\Bigg (\frac{2c_2}{(\lambda _{i_1} d_{i_1})^{n-2s}}\Bigg )= o\Bigg (\frac{1}{(\lambda _{i_1} d_{i_1})^{n-2s}}\Bigg ). \end{aligned}$$

For $i\in I_2$, using (5.3) and (5.14), we get

$$\begin{aligned} \sum _{j\ne i}\lambda _i \frac{\partial \varepsilon _{ji}}{\partial \lambda _i} +\frac{n-2s}{2}\sum _{j=1}^p \frac{{H}((a_i, 0), a_{j})}{(\lambda _i\lambda _{j})^\frac{n-2s}{2}}\le -c \Bigg (\sum _{j\ne i} \varepsilon _{ij} + \frac{1}{(\lambda _id_i)^\frac{n}{n-2s}}\Bigg ). \end{aligned}$$

(5.16)

Therefore, using the fact that the Green’s function is positive, we derive

$$\begin{aligned} \langle \partial J(u), W_2(u) \rangle \le -c\sum _{i\in I_2}\Bigg (\sum _{j\ne i} \varepsilon _{ji}+ \frac{1}{(\lambda _i d_{i})^{n-2s}}\Bigg ) + R. \end{aligned}$$

(5.17)

Using the fact that $I\cap I_2\ne \emptyset$ and arguing as in the Case1, estimate (5.11) is valid. Therefore, we can make appear the term $\sum _{i\in I_1, j\ne i}\varepsilon _{ij}$ in the upper bound of (5.11). Thus estimate (i) of Theorem 5.1 follows in this case.

For $c_3$ a fixed small constant, let us define

$$\begin{aligned} L=\{j\in I_1,\ \mathrm{s.t.}\ \exists \, i\in I_1,\ \mathrm{s.t.}\ c_3 \max (d_i, d_{j})\ge |a_i-a_{j}|\}. \end{aligned}$$

For $i\in L$, let $i_0$ the index such that

$$\begin{aligned} c_3 \max (d_i, d_{i_0})\ge |a_i-a_{i_0}|. \end{aligned}$$

(5.18)

Case 3: $I\cap I_2= \emptyset$ and there exists $i, i_0\in I$ satisfying (5.18). Let

$$\begin{aligned} W_3= -\alpha _i\lambda _i\frac{\partial P\delta _i}{\partial \lambda _i}. \end{aligned}$$

We have

$$\begin{aligned} \langle \partial J(u), W_3(u) \rangle= & {} 2c_1 J(u) \bigg [-\sum _{j\ne i} \alpha _j\alpha _{i} \lambda _{i}\displaystyle \frac{\partial \varepsilon _{ji}}{\partial \lambda _{i}}(1+o(1))\nonumber \\&\quad - \frac{n-2s}{2}\sum _{j=1}^p \alpha _j\alpha _{i} \frac{{H}((a_j, 0), a_{i})}{(\lambda _j\lambda _{i})^\frac{n-2s}{2}}(1+o(1)) \bigg ]+R. \end{aligned}$$

(5.19)

Arguing as in Case 1, the terms with $j\in I_2$ can be seen like $O(\varepsilon _{ij})$. Next, we interest with the indices $j\in I_1$. Observe that for $i, k\in I_1$, we have (5.14). Indeed, if $d_i\le d_k/2$ or $d_i\ge 2d_k$), we have $|a_i-a_k|\ge \frac{1}{2}\max (d_i, d_k)$ and the result follows. In the other case, if $d_k/2\le d_i\le 2 d_k$, using that $i, k\in I_1$, we have as in (5.6)

$$\begin{aligned} \varepsilon _{ij}= o\Bigg (\Bigg (\frac{\lambda _i}{\lambda _k}\Bigg )^\frac{n-2s}{2}+ \Bigg (\frac{\lambda _k}{\lambda _i}\Bigg )^\frac{n-2s}{2}\Bigg ). \end{aligned}$$

(5.20)

Therefore,

$$\begin{aligned} \frac{1}{\lambda _r^2|a_i-a_k|^2}= & {} \frac{\lambda _i}{\lambda _k}\frac{1}{\lambda _i\lambda _k|a_i-a_k|^2}\le c\frac{\lambda _i}{\lambda _k} \varepsilon _{ij}^\frac{2}{n-2s}\le c\frac{\lambda _i}{\lambda _k} \frac{1}{(\lambda _id_i)(\lambda _1 d_1)}\\= & {} O\Bigg (\frac{1}{(\lambda _1 d_1)^2}\Bigg ), \forall \, r=i, k. \end{aligned}$$

Thus we obtain (5.15) with the indices $j\in I_1$. Using (5.10), and the fact that the Green’s function is positive, we derive that

$$\begin{aligned} \langle \partial J(u), W_3(u)\rangle \le -c \Bigg (\frac{1}{(\lambda _i d_i)^{n-2s}}+ \frac{G((a_i, 0), a_{i_0})}{(\lambda _i\lambda _{i_0})^\frac{n-2s}{2}} \Bigg ) +R + O \Bigg (\sum _{j\ne i, j\in I_2}\varepsilon _{ij}\Bigg ).\nonumber \\ \end{aligned}$$

(5.21)

Since $i, i_0\in I$ satisfying (5.18), we can assume that $\lambda _i\ge \lambda _{i_0}$ and thus

$$\begin{aligned} \frac{1}{(\lambda _i d_i)^{n-2s}} + \frac{H((a_i, 0), a_{i_0})}{(\lambda _i\lambda _{i_0})^\frac{n-2s}{2}}\le \Bigg (\frac{c_3^2 }{\lambda _i\lambda _{i_0}|a_i-a_{i_0}|^2}\Bigg )^\frac{n-2s}{2}. \end{aligned}$$

(5.22)

Using (5.14), we derive that

$$\begin{aligned} \langle \partial J(u), W_3(u) \rangle \le -c \Bigg (\varepsilon _{ii_0}+\frac{1}{(\lambda _i d_i)^{n-2s}}\Bigg ) +R + O \Bigg (\sum _{j\ne i, j\in I_2}\varepsilon _{ij}\Bigg ). \end{aligned}$$

(5.23)

Since $i\in I$ and the term $(\lambda _i d_i)^{2s-n}$ appears in the upper-bound of the above estimate, we argue as in the Case 1, we can make appear all the $(\lambda _k d_k)^{2s-n}$ and $\sum _{k\ne j, k\in I_1}\varepsilon _{kj}$ in this upper-bound. For $m_1$, a fixed large constant, the pseudo-gradient $W_3+m_1W_1$ satisfies estimate (i) of Theorem 5.1.

Case 4: $I\cap I_2= \emptyset$ and $\forall i, i_0\in I$, $c_3 \max (d_i, d_{i_0})< |a_i-a_{i_0}|$. Let $d_0$ be a fixed small positive constant. We introduce the following sets:

$$\begin{aligned} I'=\{i\in I, d_i< d_0\} \end{aligned}$$

and

$$\begin{aligned} L_i=\{j\in L,\ \mathrm{s.t.}\ i \hbox { and }j~\hbox {satisfy}~(5.18)\}. \end{aligned}$$

Case 4.1: If $I' \ne \emptyset$, let

$$\begin{aligned} W_4=\sum _{i\in I'}\alpha _i \frac{\partial P \delta _i}{\partial a_i}\Bigg (-\frac{\eta _i}{\lambda _{j_0}}\Bigg ), \end{aligned}$$

where $\lambda _{j_0}= \max \{\lambda _i, i\in I'\}$ and $\eta _i$ is the outward normal to $\partial \Omega _{d_i}=\{x\in \Omega ,\ \mathrm{s.t.}\ d(x, \partial \Omega )=d_i\}$ at $a_i$.

Using Proposition 3.3, we have

$$\begin{aligned}&\langle \partial J(u), \displaystyle W_4\rangle \\&\quad = \frac{J(u)c_1}{\lambda _{j_0}}\sum _{i\in I'} \Bigg [ \frac{\alpha _i^2}{\lambda _i^{n-2s}}\frac{\partial H((a_i, 0), a_i)}{\partial \eta _i}(1+o(1)) + O\Bigg (\lambda _i \sum _{i\ne j, j\in I_2\cup L_i} \varepsilon _{ij}\Bigg ) \nonumber \\&\qquad + \displaystyle 2\sum _{j \ne i} \alpha _i\alpha _{j} \Bigg (\displaystyle \frac{\partial \varepsilon _{i j}}{\partial \eta _{i}} - \frac{\frac{\partial {H}((a_i, 0), a_j)}{\partial \eta _i}}{ (\lambda _i\lambda _j)^\frac{n-2s}{2}}\Bigg ) \bigg (1- J(u)^\frac{n}{n-2s}(\alpha _j^\frac{4s}{n-2s}+ \alpha _j^\frac{4s}{n-2s})\bigg )\Bigg ]\nonumber \\&\qquad + O\Bigg (\lambda _i\Bigg (\sum _{i\ne j} \varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\Bigg )\Bigg ) \nonumber \\&\qquad + O\Bigg (\sum _{i\ne j} \lambda _i^2|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \Bigg ).\\ \end{aligned}$$

Observe that for $i\in I'$ and $j\in I_1\setminus (I\cup L_i)$, using (5.5), we have

$$\begin{aligned} \frac{\frac{\partial {H}((a_i, 0), a_j)}{\partial \eta _i}}{ (\lambda _i\lambda _j)^\frac{n-2s}{2}}\le \frac{ c H((a_i, 0), a_j)}{ (\lambda _i\lambda _j)^\frac{n-2s}{2} d_i}\le \frac{ c}{ d_i(\lambda _i d_i\lambda _j d_j)^\frac{n-2s}{2} }= o\Big (\frac{1}{ d_i(\lambda _i d_i)^{n-2s} }\Big ). \end{aligned}$$

For i and $j\in I'$, if $\frac{d_i}{d_j}$, $\frac{d_j}{d_i}$ and $\frac{|a_i-a_j|}{d_i}$ are bounded and arguing as in the Appendix of [11], we derive that $\frac{\partial {H}}{\partial \eta _i}((a_i, 0), a_j)>0$. In the other case, we have

$$\begin{aligned} \frac{\partial {H}}{\partial \eta _i}((a_i, 0), a_j)\le & {} \frac{ H((a_i, 0), a_j)}{ d_i}\le \frac{1}{ d_i \max (d_i, d_j, |a_i-a_j|)^{n-2s}}\nonumber \\= & {} o\Bigg (\frac{1}{(d_i d_j)^\frac{n+1-2s}{2}}\Bigg ). \end{aligned}$$

(5.24)

Thus

$$\begin{aligned} \frac{1}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\frac{\partial {H}}{\partial \eta _i}((a_i, 0), a_j) =o\Bigg (\frac{1}{d_i(\lambda _id_i)^{n-2s}}+ \frac{1}{d_j(\lambda _jd_j)^{n-2s}}\Bigg ). \end{aligned}$$

(5.25)

Observe that for each $i\in I'$, using (5.9) and arguing as [7, 24], we have

$$\begin{aligned} \frac{\partial {H}}{\partial \eta _i}\big (\big (a_i, 0\big ), a_i\big ) = \frac{n-2s}{2^{n-2s}}\frac{1}{d_i^{n+1-2s}}\big (1+o\big (1\big )\big ). \end{aligned}$$

(5.26)

Moreover, for $i, j \in I'$, we have $\eta _i-\eta _j= O(|a_i-a_j|)$. Therefore,

$$\begin{aligned} \frac{\partial \varepsilon _{ij}}{\partial a_i}\eta _i+ \frac{\partial \varepsilon _{ij}}{\partial a_j}\eta _j= \frac{n-2s}{2}\lambda _i \lambda _j(a_i-a_j)\varepsilon _{ij}^\frac{n+2-2s}{n-2s}(\eta _j-\eta _i)= O(\varepsilon _{ij}). \end{aligned}$$

(5.27)

Using the fact that $c_3\max (d_i, d_j)\le |a_i-a_j|$, for $i\in I'$ and $j\in I_1\setminus (I\cup L_i)$, we get

$$\begin{aligned} \Big |\frac{\partial \varepsilon _{ij}}{\partial a_i}\Big |= & {} \big (n-2s\big )\lambda _i \lambda _j|a_i-a_j|\varepsilon _{ij}^\frac{n+2-2s}{n-2s}\le \frac{c}{\big (\lambda _i \lambda _j\big )^\frac{n-2s}{2}}\frac{1}{|a_i-a_j|^{n+1-2s}}\nonumber \\= & {} O\Bigg ( \frac{1}{c_3^{n+1-2s}(\lambda _i d_i \lambda _j d_j)^\frac{n-2s}{2}d_i}\Bigg )\nonumber \\= & {} O\Bigg ( \frac{c_2^\frac{n-2s}{2}}{c_3^{n+1-2s}(\lambda _i d_i)^{n-2s}d_i}\Bigg ) \nonumber \\= & {} o\Bigg ( \frac{1}{(\lambda _i d_i)^{n-2s}d_i}\Bigg ), \end{aligned}$$

(5.28)

for $c_2$ and $c_3$ chosen such that $c_2^\frac{n-2s}{2}= o(c_3^{n+1-2s})$. For $i\in I'$ and $j\in I\setminus I'$, we claim that

$$\begin{aligned} \frac{\partial {H}}{\partial \eta _i}((a_i, 0), a_j)\frac{1}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}- \frac{\partial \varepsilon _{ij}}{\partial a_i} \eta _i= & {} -\frac{\partial G((a_i, 0), a_j)}{\partial \eta _i}\frac{1}{(\lambda _i\lambda _j)^\frac{n-2s}{2}}\nonumber \\&+ o\Bigg (\frac{\lambda _i}{(\lambda _1 d_1)^{n+3-2s}}\Bigg ). \end{aligned}$$

(5.29)

Indeed, since $I\cap I_2= \emptyset$ then i and j belong to $I_1$. Using (5.14) and the fact that (5.18) is not satisfied, we derive

$$\begin{aligned} \frac{\partial \varepsilon _{ij}}{\partial a_i}= & {} (n-2s)\lambda _i \lambda _j(a_i-a_j)\varepsilon _{ij}^\frac{n+2-2s}{n-2s}\\= & {} \frac{(n-2s)(a_i-a_j)}{(\lambda _i \lambda _j)^\frac{n-2s}{2}|a_i-a_j|^{n+2-2s}}\Bigg (1+ O\Bigg (\frac{1}{(\lambda _1 d_1)^2}\Bigg )\Bigg ). \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\partial \varepsilon _{ij}}{\partial a_i}= \frac{1}{(\lambda _i \lambda _j)^\frac{n-2s}{2}}\frac{\partial }{\partial a_i}\Big (\frac{1}{ |a_i-a_j|^{n-2s}}\Big ) + O\Big (\frac{\lambda _i}{c_3^{n+1-2s}(\lambda _1 d_1)^{n+3-2s}}\Big ). \end{aligned}$$

and our claim follows. Thus

$$\begin{aligned} \langle \partial J(u), \displaystyle W_4(u)\rangle\le & {} -\frac{c}{\lambda _{j_0}}\sum _{i\in I'} \Bigg [ \frac{1}{(\lambda _i d_i)^{n-2s} d_i}- \sum _{j\in I\setminus I'}\frac{1}{(\lambda _i\lambda _j)^\frac{n-2s}{2}} \frac{\partial G((a_i, 0), a_j)}{\partial \eta _i}\\&\quad + O\Bigg (\sum _{i\ne j, j\in I'} \varepsilon _{ij}\Bigg ) + O\Bigg ( \sum _{i\ne j, j\in I_2\cup L_i} \varepsilon _{ij}\Bigg )\\&\quad + O\Bigg (\sum _{i\ne j} \varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\Bigg )\\&\quad + O\Bigg (\sum _{i\ne j} \lambda _i|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \Bigg )\Bigg ]. \end{aligned}$$

Observe that $d_i\le d_j$ for $i\in I'$ and $j\in I\setminus I'$. Therefore, $- \frac{\partial G((a_i, 0), a_j)}{\partial \eta _i}>0$, see [7] and [24]. Now for $i, j \in I'$, we have

$$\begin{aligned} \varepsilon _{ij}= O\Big ( \frac{1}{(\lambda _i d_i)^{n-2s}}+ \frac{1}{(\lambda _jd_j)^{n-2s}}\Big ). \end{aligned}$$

Using the fact that $d_i$ and $d_j$ are small enough, we get

$$\begin{aligned} \varepsilon _{ij}= o\Big ( \frac{1}{d_i(\lambda _i d_i)^{n-2s}}+ \frac{1}{d_j(\lambda _jd_j)^{n-2s}}\Big ). \end{aligned}$$

Observe that $j_0\in I'$ and $\lambda _{j_0}d_{j_0}$ and $\lambda _{1}d_{1}$ are of the same order. Thus, we can make all the $(\lambda _i d_i)^{n+1-2s}$ for $i\in I_1$ appear in the upper bound of the last inequality. It follows that,

$$\begin{aligned}&\langle \partial J(u), \displaystyle W_4(u)\rangle \le -c\sum _{i=1}^p \frac{1}{(\lambda _i d_i)^{n+1-2s}} + O\Bigg (\sum _{i\ne j, j\in I_2\cup L_i} \varepsilon _{ij}\Bigg )\nonumber \\&\quad + O\Bigg (\sum _{i\ne j} \varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\Bigg ) + O\Bigg (\sum _{i\ne j} \lambda _i|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \Bigg ).\nonumber \\ \end{aligned}$$

(5.30)

Let

$$\begin{aligned} W_5(u)= -\sum _{i\in I'}\alpha _i\lambda _i\frac{\partial P\delta _i}{\partial \lambda _i} -\sum _{j\in L_i}\alpha _j\lambda _j\frac{\partial P\delta _j}{\partial \lambda _j}. \end{aligned}$$

Using Case 3, we get

$$\begin{aligned} \langle \partial J(u), W_5(u) \rangle \le -c \Bigg (\sum _{i\in I', j\in L_i}\varepsilon _{ij}\Bigg ) + O \Bigg (\sum _{j\ne i, i\in I', j\in I_2}\varepsilon _{ij}\Bigg ) +R. \end{aligned}$$

(5.31)

For $m_1$ and $m_2$ two fixed large constants, using (5.11) (5.30) and (5.31), we get

$$\begin{aligned} \langle \partial J(u), \displaystyle W_4+ m_1W_1+ m_2W_5\rangle\le & {} -c\Bigg (\sum _{i=1}^p \frac{1}{(\lambda _i d_i)^{n+1-2s}} + \sum _{i\ne j, j\in I_2} \varepsilon _{ij}\Bigg ) + R \nonumber \\&\quad + O\Bigg (\sum _{i\ne j} \lambda _i|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \Bigg ). \end{aligned}$$

(5.32)

As in the Case 1, we can make appear the term $\sum _{i\in I_1, j\ne i}\varepsilon _{ij}$ in the upper bound of (5.11). Hence the estimates (i) of Theorem 5.1 follows in this case.

Case 4.2: If $I'=\emptyset$, then $d_i\ge d_0$ for any $i\in I$. Let $M= (m_{ij})_{i, j\in I}$ be the matrix defined in (2.6). Let $\rho$ its least eigenvalue and e is the eigenvector associated to $\rho$. Fix $\eta >0$. We set

$$\begin{aligned} C(e, \eta )= \Big \{y\in ({\mathbb {R}}_+^*)^{r}, r= {card } I,\ \mathrm{s.t.} \ \Big | \frac{y}{|y|}-e\Big |< \eta \Big \}. \end{aligned}$$

For any $x\in C(e, \eta ), \eta$ small, we have

$$\begin{aligned} t_x M x - \rho |x|^2\le \frac{1}{2}|\rho ||x|^2 \end{aligned}$$

(5.33)

and

$$\begin{aligned} t_x \frac{\partial M}{\partial a_i} x= \Big (\frac{\partial \rho }{\partial a_i}+ o(1)\Big ) |x|^2. \end{aligned}$$

(5.34)

For any $x\in C(e, \eta )^c$, we have

$$\begin{aligned} t_x M x - \rho |x|^2 > c |x|^2. \end{aligned}$$

Denote $I= \{j_1, \ldots , j_r\}$ and $\Lambda = t_{\Bigg (\frac{1}{\lambda _{j_1}^\frac{n-2s}{2}}, \ldots , \frac{1}{\lambda _{j_r}^\frac{n-2s}{2}}\Bigg )}$.

If $\Lambda$ belongs to the set $C(e, \eta )^c$, then we move the vector $\Lambda$ to $C(e, \eta )$ as in [12] along

$$\begin{aligned} \Lambda (t)=|\Lambda |\frac{(1-t)\Lambda +t|\Lambda |e}{|(1-t)\Lambda +t|\Lambda |e|}. \end{aligned}$$

Using Proposition (3.2), we derive that there exists a pseudo-gradient $W_6$ such that

$$\begin{aligned} \langle \partial J(u), W_6 \rangle= & {} c\bigg [\frac{{d}}{dt}(t_\Lambda (t) M \Lambda (t) ) + o\Bigg (\displaystyle \sum _{j \ne i, i, j \in I} \varepsilon _{i j}+ \sum _{i\in I}\frac{1 }{\lambda _i^{n-2s}}\Bigg )\nonumber \\&\quad + O\Bigg (\sum _{i\ne j, i\in I, j\in I_2\cup I_1\setminus I} \varepsilon _{ij}\Bigg ) + R\Bigg ]. \end{aligned}$$

(5.35)

As in [12], we have

$$\begin{aligned} \frac{{d}}{dt}(t_\Lambda (t) M \Lambda (t) )<-c|\Lambda |^2=-c \sum _{i\in I}\frac{1 }{\lambda _i^{n-2s}} \end{aligned}$$

and

$$\begin{aligned} \varepsilon _{ij}= O\Bigg (\Bigg (\frac{1}{(\lambda _1 d_1) (\lambda _j d_j)}\Bigg )^\frac{n-2s}{2}\Bigg )= O\Bigg (\frac{c_2^\frac{n-2s}{2}}{d_0^{n-2s} \lambda _1^{n-2s}}\Bigg )= o\Bigg (\frac{1}{\lambda _1^{n-2s}}\Bigg ), \end{aligned}$$

(5.36)

for $i\in I, j\in I_1\setminus I$. Thus

$$\begin{aligned} \langle \partial J(u), W_6(u) \rangle \le -c \sum _{i\in I}\frac{1 }{\lambda _i^{n-2s}} + o\Bigg (\displaystyle \sum _{j \ne i, i, j \in I} \varepsilon _{i j}+ \sum _{i\in I}\frac{1 }{\lambda _i^{n-2s}}\Bigg )+ R. \end{aligned}$$

(5.37)

If $\Lambda$ belongs to $C(e, \eta )$, the construction of the vector-field $W_6$ depends on the value of $\rho$ and $|\rho '|$. Since zero is a regular value of $\rho$ then there exists a constant $\rho _0>0$ such that either $|\rho |>\rho _0$ or $|\rho '|>\rho _0$.

If $\rho < -\rho _0$, we decrease all the $\lambda _i$’s for $i\in I$. If we assume that $c_2^\frac{n-2s}{2}= o(\rho _0 d_0^{n-2s})$ then using Proposition 3.2, (5.3) (5.5) and (5.36), we obtain (5.37) in this case.

If $|\rho '|>\rho _0$ and $\rho > -\rho _0$, then we move the points $a_i$’s along $\lambda _{j_0} \dot{a_i}= -\frac{\partial \rho }{\partial a_i}$ for each $a_i\in I$ and $\lambda _{j_0}= \max \{\lambda _i, i\in I\}$. Using Proposition 3.3, we derive

$$\begin{aligned} \langle \partial J(u), W_6 \rangle= & {} \frac{1}{\lambda _{j_0}}\sum _{i\in I}\Bigg \langle \partial J(u), \displaystyle \frac{\partial P \delta _{a_i, \lambda _i}}{\partial a_{i}} \Bigg \rangle \Bigg (-\frac{\partial \rho }{\partial a_i}\Bigg )\nonumber \\= & {} -\frac{1}{\lambda _{j_0}}\sum _{i\in I}\Bigg [c \Bigg (\frac{\partial \rho }{\partial a_i}\Bigg )\Bigg (t_\Lambda \frac{\partial M}{\partial a_i}\Lambda \Bigg )+ O\Bigg (\sum _{j\in I_2\cup L_i}\lambda _i \varepsilon _{i j}\Bigg )\nonumber \\&\quad + O\Bigg (\sum _{j\in I_1\setminus (I\cup L_i)} \frac{\partial {H}((a_i, 0), a_j)}{\partial a_i}\frac{1}{ (\lambda _i\lambda _j)^\frac{n-2s}{2}} +\frac{\partial \varepsilon _{ij}}{\partial a_i}\Bigg )\nonumber \\&\quad + O\Bigg (\sum _{i\ne j} \varepsilon _{ij}^\frac{n}{n-2s}\log \varepsilon _{ij}^{-1} + \sum _{k}\frac{\log (\lambda _k d_k)}{(\lambda _k d_k)^{n+2-2s}}\Bigg )\nonumber \\&\quad + O\Bigg (\sum _{i\ne j} \lambda _i|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \Bigg )\Bigg ]. \end{aligned}$$

(5.38)

Observe that for $i\in I$ and $j\in I_1{\setminus }(I\cup L_i)$, (5.18) is not satisfied. Thus

$$\begin{aligned} \frac{\partial {H}((a_i, 0), a_j)}{\partial a_i}\frac{1}{ (\lambda _i\lambda _j)^\frac{n-2s}{2}}\le \frac{1}{d_i^{n+1-2s}}\frac{1}{ (\lambda _i\lambda _j)^\frac{n-2s}{2}}\le \frac{(c_2 D)^\frac{n-2s}{2}}{\lambda _1^{n-2s}d_0^\frac{3n+2-6s}{2}}, \end{aligned}$$

(5.39)

where D is the diameter of $\Omega$. Arguing as in (5.28), we obtain

$$\begin{aligned} \Big |\frac{\partial \varepsilon _{ij}}{\partial a_i}\Big |\le & {} \frac{c}{(\lambda _i \lambda _j)^\frac{n-2s}{2}}\frac{1}{|a_i-a_j|^{n+1-2s}}\nonumber \\\le & {} \frac{(c_2 D)^\frac{n-2s}{2}}{c_3^{n+1-2s}\lambda _1^{n-2s}d_0^\frac{3n+2-6s}{2}}. \end{aligned}$$

(5.40)

We can chose $c_2$ and $c_3$ such that

$$\begin{aligned} \frac{\partial {H}((a_i, 0), a_j)}{\partial a_i}\frac{1}{ (\lambda _i\lambda _j)^\frac{n-2s}{2}}+ \Big |\frac{\partial \varepsilon _{ij}}{\partial a_i}\Big |= o\Big (\frac{1}{\lambda _1^{n-2s}}\Big ). \end{aligned}$$

(5.41)

Since $\Lambda \in C(e, \eta )$, then using (5.33), (5.34), (5.38) and (5.41), we obtain

$$\begin{aligned} \langle \partial J(u), W_6 \rangle\le & {} -\frac{c}{2\lambda _{j_0}}|\rho '|^2 |\Lambda |^2 + O\Big (\sum _{i\in I, j\in I_2\cup L_i}\lambda _i \varepsilon _{i j}\Big )\nonumber \\&\quad + R + O\Big (\sum _{i\ne j} \lambda _i|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \Big )\nonumber \\\le & {} -\frac{c}{2\lambda _{j_0}}\rho _0^2 |\Lambda |^2 + O\Big (\sum _{i\in I, j\in I_2\cup L_i}\lambda _i \varepsilon _{i j}\Big )\nonumber \\&\quad + R + O\Big (\sum _{i\ne j} \lambda _i|a_i-a_j|\varepsilon _{ij}^\frac{n+3-2s}{n-2s} \Big ). \end{aligned}$$

(5.42)

Thus in both cases, the pseudo-gradient $W_6+ m_1 W_1+m_2W_5$ where $m_1$ and $m_2$ are two positive constants large enough, satisfies (5.32) and then the estimate (i) of Theorem 5.1 follows in this case.

The required vector field W required in Theorem 5.1 will be defined by a convex combination of $W_i, i=1, \ldots , 6$. It satisfies Claim (i) of Theorem 5.1. Moreover, by the argument of Corollary B.3 of [12], it satisfies Claim (ii). This completes the proof.

COROLLARY 5.2

Under the assumption of Theorem 5.1, the only critical points at infinity for J are $\sum _{i=1}^p P \delta _{(a_i, \infty )}, p\ge 1$, where $\rho (a_1, \ldots , a_p)>0$ and $\rho '(a_1, \ldots , a_p)=0$. Moreover, all concentration points of any critical point at infinity lie in a compact set K of $\Omega$.

6 Proof of the results

Proof of Theorem 2.3

It follows from the result of Theorem 5.1 and similar arguments of Appendix 2 of [8] (see also [12]).

Proof of Theorem 2.4

Claim (i) follows from the inequalities of Theorem 5.1. Claim (ii) corresponds to the result of Corollary 5.2. Concerning Claim (iii), we know that for a large integer $p_0$, there exists at least two points $a_i$ and $a_j$ in K, (where K is defined in Corollary 5.2), such that $|a_i - a_j|$ is very small. Therefore, any related $p_0\times p_0$-matrix, $M(a_1, \ldots , a_{p_0})$ is not positive definite. Claim (iii) follows from Corollary 5.2.

Proof of Theorem 2.5

Using the result of Corollary 5.2, the only critical points at infinity for J are $\sum _{i=1}^p P \delta _{(a_i, \infty )}$ with $\rho (a_1, \ldots , a_p)>0$ and $\rho '(a_1, \ldots , a_p)=0$. Near each critical point at infinity, the normal form of the expansion of J presented by Theorem 2.3, shows that the relative homology between $W_p$ and $W_{p-1}$ is given by the product of the homologies defined by each variables. This conclude the proof.