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Research article

Spherical fuzzy rough Hamacher aggregation operators and their application in decision making problem

  • Aggregation operators are the most effective mathematical tools for aggregating many variables into a single result. The aggregation operators operate to bring together all of the different assessment values offered in a common manner, and they are highly helpful for assessing the options offered in the decision-making process. The spherical fuzzy sets (SFSs) and rough sets are common mathematical tools that are capable of handling incomplete and ambiguous information. We also establish the concepts of spherical fuzzy rough Hamacher averaging and spherical fuzzy rough Hamacher geometric operators. The key characteristics of the suggested operators are thoroughly described. We create an algorithm for a multi-criteria group decision making (MCGDM) problem to cope with the ambiguity and uncertainty. A numerical example of the developed models is shown in the final section. The results show that the specified models are more efficient and advantageous than the other existing approaches when the offered models are contrasted with specific present methods.

    Citation: Muhammad Naeem, Muhammad Qiyas, Lazim Abdullah, Neelam Khan, Salman Khan. Spherical fuzzy rough Hamacher aggregation operators and their application in decision making problem[J]. AIMS Mathematics, 2023, 8(7): 17112-17141. doi: 10.3934/math.2023874

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  • Aggregation operators are the most effective mathematical tools for aggregating many variables into a single result. The aggregation operators operate to bring together all of the different assessment values offered in a common manner, and they are highly helpful for assessing the options offered in the decision-making process. The spherical fuzzy sets (SFSs) and rough sets are common mathematical tools that are capable of handling incomplete and ambiguous information. We also establish the concepts of spherical fuzzy rough Hamacher averaging and spherical fuzzy rough Hamacher geometric operators. The key characteristics of the suggested operators are thoroughly described. We create an algorithm for a multi-criteria group decision making (MCGDM) problem to cope with the ambiguity and uncertainty. A numerical example of the developed models is shown in the final section. The results show that the specified models are more efficient and advantageous than the other existing approaches when the offered models are contrasted with specific present methods.



    The study of solitary waves Φi=exp(ιωit)ui of the nonlinear Schrödinger system

    ιtΦi=ΔΦi+Φimj=1βij|Φj|2, Φi:ΩC, i=1,,m,

    where Ω is a smooth domain in RN naturally leads to study the elliptic system

    Δui+ωiui=mj=1βiju2jui, ui:ΩR, i=1,,m. (1.1)

    Here ωi and βij=βji are real numbers and βii>0. This type of systems arises in many physical models such as incoherent wave packets in Kerr medium in nonlinear optics (see [1]) and in Bose–Einstein condensates for multi–species condensates (see [25]). The coefficient βij represents the interaction force between components ui and uj. The sign of βij determines whether the interactions between components are repulsive (or competitive), i.e., βij<0, or attractive (or cooperative), i.e., βij>0. In particular, one usually assumes βii>0. We observe that system (1.1) has always the trivial solution, namely when all the components vanish. If one or more components are identically zero, then system (1.1) reduces to a system with a smaller number of components. Therefore, we are interested in finding solutions whose all components are not trivial. These are called fully nontrivial solutions.

    In low dimensions 1N4, problem (1.1) has a variational structure: solutions to (1.1) are critical points of the energy J:HR defined by

    J(u):=12mi=1Ω(|ui|2+ωiu2i)14mi,j=1βijΩu2iu2j,

    where the space H is either H1(Ω) or H10(Ω), depending on the boundary conditions associated to ui in (1.1) in the case of not empty Ω. Therefore, the existence and multiplicity of solutions can be obtained using classical methods in critical point theory. However, there is an important difference between the dimensions 1N3 and the dimension N=4. Actually, in dimension N=4 the nonlinear part of J has a critical growth and the lack of compactness of the Sobolev embedding H1(Ω)L4(Ω) makes difficult the search for critical points. On the other hand, in dimensions 1N3 the problem has a subcritical regime and the variational tools can be successfully applied to get a wide number of results. We refer to the introduction of the most recent paper [6] for an overview on the topic and for a complete list of references. Up to our knowledge, the higher dimensional case N5 is completely open, because the problem does not have a variational structure and new ideas are needed.

    In this paper, we will focus on problem (1.1) when Ω is a smooth bounded domain in R4 with Dirichlet boundary condition. We shall rewrite (1.1) in the form

    Δui=mj=1βiju2jui+λiui in Ω, ui=0 on Ω, i=1,,m, (1.2)

    where λi are real numbers, as this way it can be seen as a generalization of the celebrated Brezis–Nirenberg problem [5]

    Δu=u3+λu in Ω, u=0 on Ω. (1.3)

    It is worthwhile to remind that the existence of solutions to (1.3) strongly depends on the geometry of Ω. In particular, if Ω is a starshaped domain, then Pohozaev's identity ensures that (1.3) has no solution when λ0. On the other hand, Brezis and Nirenberg [5] proved that (1.3) has a positive solution if and only if λ(0,Λ1(Ω)) where Λ1 is the first eigenvalue of Δ with homogeneous Dirichlet condition on Ω. These solutions are often referred to as least energy solutions, as they can be obtained also by minimizing the functional

    12Ω(|u|2λ|u|2)14Ω|u|4

    restricted to the associated Nehari manifold. Later, Han [12] and Rey [18] studied the asymptotic behaviour of this solution as λ0 and proved that it blows–up at a point ξ0Ω which is a critical point of the Robin's function, whereas far away from ξ0 his shape resembles the bubble

    Uδ,ξ(x):=αδδ2+|xξ|2, α=22. (1.4)

    Recall that it is well known (see [2,23]) that {Uδ,ξ: δ>0, ξR4} is the set of all the positive solutions to the critical problem

    ΔU=U3 in R4. (1.5)

    Let us also remind that the Robin's function is defined by r(x):=H(x,x), xΩ, where H(x,y) is the regular part of the Green function of Δ in Ω with Dirichlet boundary condition.

    Successively, relying on the profile of the bubble as a first order approximation, the Ljapunov–Schmidt procedure has been fruitfully used to build both positive and sign–changing solutions to (1.3) blowing–up at different points in Ω as the parameter λ approaches zero (see for example Rey [18] and Musso and Pistoia [14]).

    As far as we know, few results are available about existence and multiplicity of solutions to the critical system (1.2). The first result is due to Chen and Zou [9], who considered (1.2) with 2 components only

    {Δu1=μ1u31+βu1u22+λ1u1in ΩΔu2=μ2u32+βu21u2+λ2u2in Ωu1=u2=0on Ω. (1.6)

    When 0<λ1,λ2<Λ1(Ω), they proved the existence of a least energy positive solution in the competitive case (i.e., β<0) and in the cooperative case (i.e., β>0) if β(0,β_][¯β,+), for some ¯βmax{μ1,μ2}>min{μ1,μ2}β_>0. In the cooperative case, when λ1=λ2 the least energy solution is synchronized, i.e., (u1,u2)=(c1u,c2u) where u is the least energy positive solution of the Eq (1.3) and (c1,c2) is a positive solution to the algebraic system

    {1=μ1c21+βc221=μ2c22+βc21.

    In the competitive case, the authors studied also the limit profile of the components of the least energy solution and proved that the following alternative occurs: either one of the components vanishes and the other one converges to a least energy positive solution of the Eq (1.3), or both components survive and their limits separate in different regions of the domain Ω, i.e., a phase separation phenomenon takes place. In the subcritical regime such a phenomenon has been studied by Noris, Tavares, Terracini and Verzini [15].

    Afterwards, Chen and Lin [8] studied the asymptotic behavior of the least energy solution of (1.6) in the cooperative case as max{λ1,λ2}0 and found that both components blow–up at the same critical point of the Robin's function, in the same spirit of the result by Han and Rey for the single equation (1.3).

    The existence of blowing–up solutions for system (1.2) with an arbitrary number of components has been studied by Pistoia and Tavares [17]. Using a Ljapunov–Schmidt procedure, they built solutions to (1.2) whose m components blow–up at m different non–degenerate critical points of the Robin's function as λ:=max{λ1,,λm}0, provided the interaction forces are either negative or not too large, namely β:=maxijβij¯β for some ¯β>0. For example, their result holds in dumbbell shaped domains which are obtained by connecting m mutually disjoint connected domains D1,,Dm by thin handles. In this case the Robin's function has m distinct local minimum points which are non–degenerate for a generic choice of the domain as proved by Micheletti and Pistoia [13]. Moreover, if, as λ0, we let β:=maxi,jβij approach with a sufficiently low velocity (depending on λ), then it is still possible to show that all the components blow–up at different points and a segregation phenomen occurs.

    To conclude the state of the art, we would like to mention some recent results obtained by exploiting a variational point of view. Guo, Luo and Zou [11] proved the existence of a least energy solution to (1.2) in the purely cooperative regime (i.e., minijβij0) when λ1==λm and showed that such a solution is synchronized under some additional technical conditions on the coupling coefficients. Tavares and You [24] generalized the previous result to a mixed competitive/weakly cooperative regime (i.e., maxijβij not too large). Clapp and Szulkin [10] found a least energy solution in the purely competitive regime (i.e., maxijβij<0), which is not synchronized when the coupling terms βij diverge to .

    Now, let us go back to the result obtained by Pistoia and Tavares [17] concerning the existence of solutions to (1.2) with all the components blowing–up around different points in Ω when all the mixed forces are repulsive or weakly attractive. It is natural to ask what happens for more general mixed repulsive and attractive forces. Our idea is to assemble the components ui in groups so that all the interaction forces βij among components of the same group are attractive, while forces among components of different groups are repulsive or weakly attractive. In this setting, we address the following question:

    (Q) is it possible to find solutions such that each component within a given group concentrates around the same point and different groups concentrate around different points?

    Taking the notation introduced in [16], given 1qm, let us introduce a qdecomposition of m, namely a vector (l0,,lq)Nq+1 such that

    0=l0<l1<<lq1<lq=m.

    Given a q–decomposition of m, we set, for h=1,,q,

    Ih:={i{1,,m}:lh1<ilh}.

    In this way, we have partitioned the set {1,,m} into q groups I1,,Iq, and we can consequently split the components of our system into q groups {ui: iIh}. Notice that if lhlh1=1, then Ih reduces to the singleton {i}, for some i{1,,m}. We will assume that for every h=1,,q.

    (A1) the algebraic system

    1=jIhβijc2j,iIh, (1.7)

    has a solution ch=(ci)iIh with ci>0 for every iIh;

    (A2) the matrix (βij)i,jIh is invertible and all the entries are positive.

    We observe that (A1) is satisfied for instance if for every ij (see [3])

    βij=:β>maxiIhβii for every iIh.

    Remark 1.1. Assumptions (A1) and (A2) are necessary to build the solutions to (1.2) using the classical Ljapunov-Schmidt procedure, namely (A1) allows to find a good ansatz which is non-degenerate because of (A2). Let us be more precise.

    From a PDE point of view, assumption (A1) is equivalent to require that the nonlinear PDE sub-system

    ΔWi=WijIhβijW2jin Rn, iIh, (1.8)

    has a synchronized solution Wi=ciU, iIh, where the positive function

    U(x):=α11+|x|2, α=22,

    solves the critical Eq (1.5). The first key point in the reduction procedure is done: the main order term of the components ui with iIh is nothing but the syncronized solution of the sub-system (1.8).

    Assumption (A2) ensures that such a synchronized solution of (1.8) is non–degenerate (see [16, Proposition 1.4]), in the sense that the linear system (obtained by linearizing system (1.8) around the synchronized solution)

    Δvi=U2[(3βiic2i+jIhjiβijc2j)vi+2jIhjiβijcicjvj] inRn,iIh, (1.9)

    has a 5–dimensional set of solutions

    (v1,,v|Ih|)span{ehψ | =0,1,,4}(H10(Ω))|Ih| (1.10)

    where ehR|Ih| is a suitable vector (see [16, Lemma 6.1]) and the functions

    ψ0(y)=1|y|2(1+|y|2)2andψ(y)=y(1+|y|2)2, =1,,4,

    solve the linear equation

    Δψ=3U2ψin R4.

    The non-degeneracy of the building block is ensured and the second key point in the reduction procedure is also done.

    We are now in position to state our main result.

    Theorem 1.2. Assume (A1) and (A2). Assume furthermore that the Robin's function has q distint non–degenerate critical points ξ01,,ξ0q. There exist ¯β>0 and λ0>0 such that, if β:=max(i,j)Ih×Ikhkβij<¯β then, for every (λi)mi=1 with λi(0,λ0), i=1,,m, there exists a solution (u1,,um) to (1.2) such that, for every h=1,,q, each group of components {ui : iIh} blows–up at ξ0h as λ:=maxi=1,,mλi0.

    Moreover, if, as λ0, β approaches slowly enough (depending on λ), i.e., |β|=O(edλ) forsomedsufficientlysmall, then all the components belonging to different groups blow–up at different points and segregate, while the components belonging to the same group blow–up at the same point and aggregate.

    Remark 1.3. We remind that in dumbbell shaped domains which are obtained by connecting q mutually disjoint connected domains by thin handles, the Robin's function has q distinct critical points and in a domain with holes the Robin's function has at least 2 critical points (see Pistoia and Tavares [17, Examples 1.5 and 1.6]). All these critical points are non–degenerate for a generic choice of the domain as proved by Micheletti and Pistoia [13].

    Remark 1.4. Theorem 1.2 deals with systems with mixed aggregating and segregating forces (i.e., some βij's are positive, and some others are negative). This is particularly interesting since there are few results about systems with mixed terms. The subcritical regime has been recently investigated by Byeon, Kwon and Seok [6], Byeon, Sato and Wang [7], Sato and Wang [19,20], Soave and Tavares [22], Soave [21] and Wei and Wu [26]. As far as we know, there are only a couple of results concerning the critical regime. The first one has been obtained by Pistoia and Soave in [16], where the authors studied system (1.2) when all the λi's are zero and the domain has some holes whose size approaches zero. The second one is due to Tavares and You, who in [24] found a least energy solution to system (1.2) provided all the parameters λi are equal. It would be interesting to compare this least energy solution with the blowing-up solutions found in the present paper.

    Remark 1.5. We strongly believe that the solutions found in Theorem 1.2 are positive, because they are constructed as the superposition of positive function and small perturbation term. This is true for sure if the attractive forces βij are small, as proved in [17]. In the general case, the proof does not work and some refined Lestimates of the small terms are needed. We will not afford this issue in the present paper, because the study of the invertibility of the linear operator naturally associated to the problem (see Proposition 3.1) should be performed in spaces equipped with different norms (i.e., L–weighted norms) that may deserve further investigations.

    The proof of Theorem 1.2 relies on the well known Ljapunov–Schmidt reduction. The main steps are described in Section 3, where the details of the proof are omitted whenever it can be obtained, up to minor modifications, by combining the arguments in Pistoia and Tavares [17] and in Pistoia and Soave [16]. Here we limit ourselves to give a detailed proof of the first step of the scheme, as it suggests how to adapt the ideas of [16,17] to the present setting. The technical details of this part are developed in the Appendix. Before getting to this, in Section 2 we recall some well known results that are needed in the following.

    We denote the standard inner product and norm in H10(Ω) by

    u,vH10(Ω):=Ωuv,uH10(Ω):=(u,uH10(Ω))12,

    and the Lq-norm (q1) by ||Lq(Ω). Whenever the domain of integration Ω is out of question, we also make use of the shorthand notation u for uH10(Ω) and |u|q for |u|Lq(Ω).

    Let i:H10(Ω)L4(Ω) be the canonical Sobolev embedding. We consider the adjoint operator (Δ)1:L43(Ω)H10(Ω) characterized by

    (Δ)1(u)=v{Δv=uinΩvH10(Ω)

    It is well known that (Δ)1 is a continuous operator, and relying on it we can rewrite (1.2) as

    ui=(Δ)1(mj=1βiju2jui+λiui),  i=1,,m. (2.1)

    From now on, we will focus on problem (2.1).

    We are going to build a solution u=(u1,,um) to (2.1), whose main term, as the parameters λi approach zero, is defined in terms of the bubbles Uδ,ξ given in (1.4). More precisely, let us consider the projection PUδ,ξ of Uδ,ξ into H10(Ω), i.e., the unique solution to

    Δ(PUδ,ξ)=ΔUδ,ξ=U3δ,ξ in Ω, PUδ,ξ=0 on Ω.

    We shall use many times the fact that 0PUδ,ξUδ,ξ, which is a simple consequence of the maximum principle. Moreover it is well known that

    PUδ,ξ(x)=Uδ,ξ(x)αδH(x,ξ)+O(δ3).

    Here G(x,y) is the Green function of Δ with Dirichlet boundary condition in Ω and H(x,y) is its regular part.

    Now, we search for a solution u:=(u1,,um) to (2.1) as

    \begin{equation} {\bf{u}} = {\bf{W}}+\boldsymbol{\phi},\ \hbox{where}\ {\bf{W}}: = \left(\mathfrak c_1 PU_{\delta_1,\xi_1},\dots,\mathfrak c_qPU_{\delta_q,\xi_q}\right)\in (H_0^1(\Omega ))^{|I_{1}|}\times\dots\times (H_0^1(\Omega ))^{|I_{q}|}, \end{equation} (2.2)

    where each vector \mathfrak c_h\in\mathbb R^{|I_h|} is defined in (1.7), the concentration parameters {\delta}_h = e^{-{d_h\over\lambda^*_h}} with {\lambda}^*_h: = \max_{i\in I_h}\lambda_i , and the concentration points \xi_h\in \Omega are such that ({\bf{{d}}}, \boldsymbol{{\xi}}) = (d_1, \dots, d_q, \xi_1, \dots, \xi_q)\in X_{\eta} , with

    \begin{equation} X_{\eta}: = \left\{ ({\bf{{d}}},\boldsymbol{{\xi}}) \in {\mathbb{R}}^q \times \Omega^q: \ \eta < d_h < \eta^{-1},\ {\rm{dist}}(\xi_h,\partial\Omega)\ge\eta, \ |\xi_h-\xi_k|\ge\eta\ \ \hbox{if}\ h\not = k\right\}, \end{equation} (2.3)

    for some \eta \in (0, 1). Recall that |I_1|+\dots+|I_q| = m. The higher order term \boldsymbol{\phi} = (\phi_1, \dots, \phi_m) \in \left(H_0^1(\Omega)\right)^m belongs to the space \boldsymbol{{K}} ^{\perp} whose definition involves the solutions of the linear equation

    \begin{equation} -\Delta \psi = 3 U_{{\delta},{\xi}}^{2} \psi \quad {\rm{in}}\; {\mathbb{R}}^4 , \quad \psi \in \mathcal{D}^{1,2}({\mathbb{R}}^4). \end{equation} (2.4)

    More precisely, we know that the set of solutions to (2.4) is a 5- dimensional space, which is generated by (see [4])

    \begin{aligned} \psi_{{\delta},{\xi}}^0 &: = \frac{ \partial U_{{\delta},{\xi}}}{ \partial {\delta}} = \alpha \frac{|x-\xi|^2 -\delta^2}{\left( {\delta}^2 + |x-{\xi}|^2 \right)^{2}} \\ \psi_{{\delta},{\xi}}^\ell&: = \frac{ \partial U_{{\delta},{\xi}}}{ \partial {\xi}_\ell} = 2\alpha_N {\delta} \frac{x_\ell-\xi_\ell}{\left( {\delta}^2 + |x-{\xi}|^2 \right)^{ {2}}}, \quad \ell = 1,\dots,4. \end{aligned}

    It is necessary to introduce the projections P \psi_{{\delta}, \xi}^\ell of \psi_{{\delta}, \xi}^\ell ( \ell = 0, \dots, N ) into H_0^1(\Omega) , i.e.,

    \begin{equation} -\Delta (P\psi_{{\delta},\xi}^\ell) = -\Delta \psi_{{\delta},{\xi}}^\ell = 3 U_{{\delta},{\xi}}^{2} \psi_{{\delta},{\xi}}^\ell \hbox{ in}\ \Omega,\quad P \psi_{{\delta},{\xi}}^\ell = 0\ \hbox{ on}\ \partial\Omega, \end{equation} (2.5)

    and it is useful to recall that

    \begin{aligned}P\psi_{\delta,\xi}^0(x)& = \psi^0_{\delta,\xi}(x)-\alpha H(x,\xi)+\mathcal O\left(\delta^{2}\right), \\ P\psi_{\delta,\xi}^\ell(x)& = \psi^\ell_{\delta,\xi}(x)-\alpha\delta \partial_\ell H(x,\xi)+\mathcal O\left(\delta^{3}\right), \quad \ell = 1,\dots,4. \end{aligned}

    Now, we define the space \boldsymbol{{K}} ^{\perp} as

    \begin{equation} {\bf{{K}}} : = K_1 \times \cdots \times K_q\ \hbox{ and }\ \boldsymbol{{K}} ^{\perp} = K_1^\perp \times \cdots \times K_q^\perp, \end{equation} (2.6)

    where (see (1.10))

    \begin{equation} K_h: = {\rm{span}}\left\{\mathfrak e_h P \psi_{{\delta}_h,\xi_h}^\ell: \ \ell = 0,\dots,4\right\}\subset \left(H_0^1(\Omega)\right)^{|I_{h}|}, \quad h = 1,\dots,q. \end{equation} (2.7)

    The unknowns in (2.2) are the rates of the concentration parameters d_h 's, the concentration points \xi_h 's and the remainder terms \phi_i 's. To identify them, we will use a Ljapunov–Schmidt reduction method. First, we rewrite system (2.1) as a couple of systems. Let us introduce the orthogonal projections

    \boldsymbol{{\Pi}}: = (\Pi_1,\dots,\Pi_q):(H_0^1(\Omega ))^{|I_{1}|}\times\dots\times H_0^1(\Omega ))^{|I_{q}|} \to {\bf{{K}}}

    and

    \ \boldsymbol{{\Pi}}^\perp: = (\Pi_1^\perp,\dots,\Pi_q^\perp):(H_0^1(\Omega ))^{|I_{1}|}\times\dots\times H_0^1(\Omega ))^{|I_{q}|} \to {\bf{{K}}} ^\perp,

    where \Pi_h :\left(H_0^1(\Omega)\right)^{|I_{h}|}\to K_h and \Pi_h^{\perp} :\left(H_0^1(\Omega)\right)^{|I_{h}|} \to K_h^{\perp} denote the orthogonal projections, for every h = 1, \dots, q .

    It is not difficult to check that (2.1) is equivalent to the couple of systems

    \begin{equation} \boldsymbol{{\Pi}}^\perp\left[\boldsymbol{{\mathcal L}}(\boldsymbol{\phi})+\boldsymbol{{\mathcal N}}(\boldsymbol{\phi})+\boldsymbol{{\mathcal E}}\right] = 0 \end{equation} (2.8)

    and

    \begin{equation} \boldsymbol{{\Pi}} \left[\boldsymbol{{\mathcal L}}(\boldsymbol{\phi})+\boldsymbol{{\mathcal N}}(\boldsymbol{\phi})+\boldsymbol{{\mathcal E}}\right] = 0, \end{equation} (2.9)

    where the linear operator {\boldsymbol{{\mathcal L}} (\boldsymbol{{\phi}})} = (\mathcal L^1 (\boldsymbol{{\phi}}), \dots, \mathcal L^m (\boldsymbol{{\phi}})):\left(H^1_0(\Omega)\right)^m\to\left(H^1_0(\Omega)\right)^m is defined for every i\in I_h and h = 1, \dots, q as

    \begin{equation} \begin{aligned} \mathcal L^i (\boldsymbol{{\phi}}): = & \phi_i -(-\Delta)^{-1} \left\{\left[\left(3\beta_{ii}{c _i}^2+\sum\limits_{j \in I_h\atop j\neq i} \beta_{ij} {c _j}^{2}\right) \phi_i+2\sum\limits_{j \in I_h\atop j\neq i} \beta_{ij} c _j c _i\phi_j\right](P U_{{\delta}_h,{\xi}_h})^2\right. \\ & \left. \qquad+\sum\limits_{k\not = h}\sum\limits_{j \in I_k }\beta_{ij} \left((c _j P U_{{\delta}_k,{\xi}_hk} )^2\phi_i +2c _i c _j P U_{{\delta}_h,{\xi}_h} P U_{{\delta}_k,{\xi}_hk}\phi_j\right)+\lambda_i\phi_i \right\} ,\end{aligned} \end{equation} (2.10)

    the nonlinear term {\boldsymbol{{\mathcal N}} (\boldsymbol{{\phi}})} = (\mathcal N^1 (\boldsymbol{{\phi}}), \dots, \mathcal N^m (\boldsymbol{{\phi}}))\in \left(L^\frac43(\Omega)\right)^m is defined for every i\in I_h and h = 1, \dots, q as

    \begin{equation} \begin{aligned} \mathcal N^i (\boldsymbol{{\phi}}): = & - (-\Delta)^{-1}\left\{\sum\limits_{j \in I_h\atop j\neq i} \beta_{ij} (c _i P U_{{\delta}_h,{\xi}_h} \phi_j ^{2}+2c _j P U_{{\delta}_h,{\xi}_h}\phi_j\phi_i+\phi_j^2\phi_i) \right.\\ & \left. +\sum\limits_{k\not = h}\sum\limits_{j \in I_k }\beta_{ij} (c _i P U_{{\delta}_h,{\xi}_h} \phi_j ^{2}+2c _j P U_{{\delta}_k,{\xi}_k}\phi_j\phi_i+\phi_j^2\phi_i) +\beta_{ii}\left(3 c _i P U_{{\delta}_h,{\xi}_h} \phi_i^2 + \phi_i^3\right)\right\}, \end{aligned} \end{equation} (2.11)

    and the error term \boldsymbol{{\mathcal E}} = (\mathcal E^1, \dots, \mathcal E^m)\in \left(L^\frac43(\Omega)\right)^m is defined for every i\in I_h and h = 1, \dots, q as

    \begin{equation} \begin{aligned} \mathcal E^i : = & -(-\Delta)^{-1}\left\{ \left( \sum\limits_{j \in I_h } \beta_{ij} {c _j}^{2} c _i \right) \left[ (P U_{{\delta}_h,{\xi}_h})^3-( U_{{\delta}_h,{\xi}_h})^3 \right]\right. \\ & \left. + \sum\limits_{k\not = h}\sum\limits_{j \in I_k } \beta_{ij} (c _j P U_{{\delta}_k,{\xi}_k} )^{2} ( c _i P U_{{\delta}_h,{\xi}_h} )+\lambda_i c _i P U_{{\delta}_h,{\xi}_h} \right\} .\end{aligned} \end{equation} (2.12)

    In the above computation, we used (1.7) and (1.8), so that for every i\in I_h and h = 1, \dots, q

    c _i P U_{{\delta}_h,{\xi}_h} = (-\Delta)^{-1}\left[\left( \sum\limits_{j \in I_h } \beta_{ij} {c _j}^2c _i \right)( U_{{\delta}_h,{\xi}_h} )^3\right].

    The proof of our main result consists of two main steps. First, for fixed {\bf{d}} = (d_1, \dots, d_q) , and \boldsymbol{\xi} = (\xi_1, \dots, \xi_q) we solve the system (2.8), finding \boldsymbol{{\phi}} = \boldsymbol{{\phi}}({\bf{d}}, \boldsymbol{\xi})\in {\bf{{K}}}^\perp . Plugging this choice of \boldsymbol{{\phi}} into the second system (2.9), we obtain a finite dimensional problem in the unknowns {\bf{{d}}} and \boldsymbol{{\xi}} , whose solution is identified as a critical point of a suitable function.

    We briefly sketch the main steps of the proof.

    As a first step, it is important to understand the solvability of the linear problem naturally associated to (2.8), i.e., given \boldsymbol{{\mathcal L}} as in (2.10)

    \boldsymbol{{\mathcal L}}(\boldsymbol{{\phi}}) = {\bf h}, \quad {\rm{with}} \quad {\bf h} \in K ^\perp.

    Proposition 3.1. For every \eta > 0 small enough there exist \bar \beta > 0 , \lambda_0 > 0 and C > 0 , such that if \lambda_i \in (0, \lambda_0) , for every i = 1, \dots, m , and

    \begin{equation} \beta^*: = \max\limits_{(i,j)\in I_h\times I_k \atop h\not = k}\beta_{ij}\leq \overline{\beta}\,, \end{equation} (3.1)

    then

    \begin{equation} \|\boldsymbol{{\mathcal L}} (\boldsymbol{{\phi}})\|_{(H_0^1(\Omega))^m} \ge C \|\boldsymbol{{\phi}}\|_{(H_0^1(\Omega))^m} \qquad \forall \boldsymbol{{\phi}} \in \bf K ^\perp\,, \end{equation} (3.2)

    for every ({\bf{{d}}}, \boldsymbol{{\xi}}) \in X_\eta . Moreover, \boldsymbol{{\mathcal L}} is invertible in {\bf{{K}}} ^\perp with continuous inverse.

    Proof. It is postponed to Appendix.

    We need to estimate the error term \boldsymbol{{\mathcal E}} defined in (2.12).

    Lemma 3.2. For every \eta > 0 small enough there exist \lambda_0 > 0 and C > 0 such that, if \lambda_i\in(0, \lambda_0) for every i = 1, \dots, m , then

    \begin{equation} \|\boldsymbol{{\mathcal E}} \|_{(H_0^1(\Omega))^m} \le C \sum\limits_{h = 1}^q\left(O(\delta_{h}^2)+ O(\lambda^*_{h}\delta_{h})+\sum\limits_{k\neq h}O(|\beta^*|\delta_{h}\delta_{k})\right) \end{equation} (3.3)

    for every ({\bf{{d}}}, \boldsymbol{{\xi}}) \in X_\eta , where \lambda^*_{h}: = \max\limits_{i\in I_h}\lambda_i and \beta^*: = \max\limits_{(i, j)\in I_h\times I_k\atop i\not = k}\beta_{ij}.

    Proof. We argue as in [17, Lemma A.1–A.3]. Note first that, by the continuity of (-\Delta)^{-1} , for every i\in I_h

    \begin{split} \|\mathcal E ^i\|\leq & C\left( \sum\limits_{j \in I_h } |\beta_{ij}| {c _j}^{2} c _i \right)\left|(P U_{{\delta}_h,{\xi}_h})^3-( U_{{\delta}_h,{\xi}_h})^3\right|_{\frac{4}{3}}\\ +&C\sum\limits_{k\not = h}\sum\limits_{j \in I_k } |\beta_{ij}| c _i c _j^2\left|P U_{{\delta}_h,{\xi}_h}(P U_{{\delta}_k,{\xi}_k} )^{2}\right|_\frac{4}{3}+\lambda_i c _i\left|P U_{{\delta}_h,{\xi}_h} \right|_\frac{4}{3}. \end{split}

    Moreover,

    \left|(P U_{{\delta}_h,{\xi}_h})^3-( U_{{\delta}_h,{\xi}_h})^3\right|_{\frac{4}{3}} = O(\delta_{h}^2),
    \left|P U_{{\delta}_h,{\xi}_h}P U_{{\delta}_k,{\xi}_k} ^{2}\right|_\frac{4}{3} = O(\delta_h\delta_k)

    and

    \left|P U_{{\delta}_h,{\xi}_h} \right|_\frac{4}{3} = O(\delta_{h})\,.

    Then the claim follows.

    We combine all the previous results and a standard contraction mapping argument and we prove the solvability of the system (2.8).

    Proposition 3.3. For every \eta > 0 small enough there exist \bar \beta > 0 , \lambda_0 > 0 and C > 0 such that, if \lambda_i \in (0, \lambda_0) for every i = 1, \dots, m and (3.1) holds, then for every ({\bf{{d}}}, \boldsymbol{{\xi}}) \in X_\eta there exists a unique function \boldsymbol{{\phi}} = \boldsymbol{{\phi}}({\bf{{d}}}, \boldsymbol{{\xi}}) \in \boldsymbol{{K}} ^\perp solving system (2.8). Moreover,

    \begin{equation} \|\boldsymbol{{\phi}} \|_{\left(H_0^1(\Omega )\right)^m} \le C\sum\limits_{h = 1}^q\left(O(\delta_{h}^2)+O(\lambda^*_{h}\delta_{h})+\sum\limits_{k\neq h}O(|\beta^*|\delta_{h}\delta_{k})\right) \end{equation} (3.4)

    and ({\bf{{d}}}, \boldsymbol{{\xi}}) \mapsto \boldsymbol{{\phi}}({\bf{{d}}}, \boldsymbol{{\xi}}) is a C^1- function.

    Proof. The claim follows by Proposition 3.1 and Lemma 3.2 arguing exactly as in [17, Proposition 3.2 and Lemma 3.3], noting that the nonlinear part \boldsymbol{{\mathcal N}} given in (2.11) has a quadratic growth in \boldsymbol{\phi} . In particular (3.4) follows by (3.3).

    Once the first system (2.8) has been solved, we have to find a solution to the second system (2.9) and so a solution to system (1.2).

    Proposition 3.4. For any \eta > 0 small enough there exist \overline\beta > 0 and \lambda_0 > 0 such that, if \lambda_i\in(0, \lambda_0) for every i = 1, \dots, m and (3.1) holds, then {\bf{u}} = {\bf{W}}({\bf{d}}, \boldsymbol{\xi})+\boldsymbol{\phi}({\bf{d}}, \boldsymbol{\xi}) defined in (2.2) solves system (1.2), i.e., it is a critical point of the energy

    J({\bf{u}}): = \frac12\sum\limits_{i = 1}^m\int\limits_{\Omega} |\nabla u_i|^2- \frac14\sum\limits_{i,j = 1 }^m\beta_{ij} \int\limits_{\Omega}u_i^2u_j^2-\frac12\sum\limits_{i = 1}^m\int\limits_{\Omega}\lambda_iu_i^2

    if and only if ({\bf{d}}, \boldsymbol{\xi})\in X_\eta is a critical point of the reduced energy

    \tilde J({\bf{d}},\boldsymbol{\xi}): = J\left({\bf{W}}+\boldsymbol{\phi}\right).

    Moreover, the following expansion holds true

    \begin{equation} \tilde J(\boldsymbol{\delta},\boldsymbol{\xi}) = \sum\limits_{h = 1}^q\left(\sum\limits_{i\in I_h}c _i^2\right)\left( A_0+ A_1\delta_h^2\mathtt r(\xi_h)+ A_2\lambda_h^*\delta_h^2|\ln\delta_h| +o(\delta_h^2)\right) \end{equation} (3.5)

    C^1- uniformly in X_\eta. Here the A_i 's are positive constants, \mathtt r is the Robin's function and \lambda_h^* = \max\limits_{i\in I_h}\lambda_i.

    Proof. The proof follows by combining the arguments in [17, Section 3 and Section 5] and [16, Section 5]. We remark that in this case the fact that \boldsymbol{\phi}({\bf{d}}, \boldsymbol{\xi}) solves (2.8) is equivalent to claim that it solves the system

    \boldsymbol{{\mathcal L}}(\boldsymbol{\phi}) -\boldsymbol{{\mathcal N}}(\boldsymbol{\phi})- \boldsymbol{{\mathcal E}} = \left(\sum\limits_{\ell = 0}^4a_1^\ell\mathfrak e_1 P \psi_{{\delta}_1,\xi_1}^\ell,\dots,\sum\limits_{\ell = 0}^4a_h^\ell\mathfrak e_h P \psi_{{\delta}_h,\xi_h}^\ell\right),

    for some real numbers a_i^\ell. Therefore, {\bf{W}}({\bf{d}}, \boldsymbol{\xi})+\boldsymbol{\phi}({\bf{d}}, \boldsymbol{\xi}) solves system (2.9) if and only if all the a_i^\ell 's are zero. We also point out that it is quite standard to prove that J\left({\bf{W}}+\boldsymbol{\phi}\right)\approx J\left({\bf{W}} \right) and moreover by (1.7) we deduce

    \begin{aligned}J\left({\bf{W}} \right)& = \sum\limits_{h = 1}^q \underbrace{\left(\sum\limits_{i\in I_h}\frac12 c_i^2\int\limits_\Omega| \nabla PU_{\delta_h,\xi_h}|^2-\frac14 \sum\limits_{i,j\in I_h }\beta_{ij} (c_i c_j)^2\int\limits_\Omega(PU_{\delta_h,\xi_h})^4 \right) }_{ = \left(\sum\limits_{i\in I_h} c_i^2\right)\left(\frac12\int\limits_\Omega| \nabla PU_{\delta_h,\xi_h}|^2-\frac14 \int\limits_\Omega(PU_{\delta_h,\xi_h})^4\right)}\\ &-\frac12\sum\limits_{h,k = 1\atop h\not = k}\beta_{ij}\int\limits_\Omega ( c_iPU_{\delta_h,\xi_h})^2( c_j PU_{\delta_k,\xi_k})^2- \sum\limits_{h = 1}^q\sum\limits_{i\in I_h}\frac12\lambda_i \int\limits_\Omega ( c_iPU_{\delta_h,\xi_h})^2 \end{aligned}

    so that the claim follows just arguing as in [17].

    Arguing exactly as in [17, Proof of Theorem 1.3, p. 437], we prove that the reduced energy (3.5) has a critical point ({\bf{d}}_{\boldsymbol{\lambda}}, \boldsymbol{\xi}_{\boldsymbol{\lambda}}) provided \boldsymbol{\lambda} = (\lambda_1, \dots, \lambda_m) is small enough and \boldsymbol{\xi}_{\boldsymbol{\lambda}}\to (\xi^0_1, \dots, \xi_q^0) as \lambda^* = \max_i\lambda_i\to0. Theorem 1.2 immediately follows by Proposition 3.4. Moreover, if the \beta_{ij} 's depend on the \lambda_i 's and \beta^* satisfies |\beta^*| = O\left(e^{d^*\over\lambda^*}\right) with d^* < \min _{h = 1, \dots, q}d_h, then for every h = 1, \dots, m

    |\beta^*|\delta_h\lesssim e^{{d^* \over\lambda^*}-{d_h \over\lambda^*_h}}\lesssim e^{{d^* -d_h \over\lambda^*}} = o(1)

    and by estimate (3.4) we can still conclude the validity of (3.5), and so the last part of Theorem 1.2 follows (see also [17, Section 5.3, p.438].

    The authors declare no conflict of interest.

    Proof of Proposition 3.1

    We argue combining ideas of [17, Lemma 3.2] and [16, Lemma 5.4]. We first prove (3.2) by contradiction. Assume thus that there exist \{({\bf d}_n, \boldsymbol{\xi}_n)\}_n\subset X_\eta so that \boldsymbol{\xi}_n\to\boldsymbol{\xi} as n\to+\infty , \boldsymbol{\lambda}_n: = (\lambda_{1, n}, \, \dots, \, \lambda_{m, n})\to0 as n\to+\infty , and \boldsymbol{\phi}^n: = (\phi_1^n, \, \dots, \, \phi_m^n)\in{\bf K}^\perp so that \|\boldsymbol{\phi}^n\| = 1 for every n\in\mathbb{N} and

    \|\boldsymbol{{\mathcal L}} (\boldsymbol{\phi}^n)\|\to 0\qquad {\rm{as }}\;n\to+\infty\,.

    We recall that the spaces introduced in (2.6) and (2.7) depend on \bf d_n and \boldsymbol{\xi}_n, so for the sake of clarity, let us introduce the following notation. For every h = 1, \, \dots, \, q , let

    \begin{split} K_h^n&: = K_{d_{h,n},\xi_{h,n} },\quad\quad (K_h^n)^\perp: = K_{d_{h,n},\xi_{h,n} }^\perp,\\ U_h^n&: = U_{\delta_{h,n},\xi_{h,n}},\qquad\quad PU_h^n: = PU_{\delta_{h,n},\xi_{h,n}},\\ \psi_{h,n}^l&: = \psi_{\delta_{h,n},\xi_{h,n}}^l,\quad\quad\,\, P\psi_{h,n}^l: = P\psi_{\delta_{h,n},\xi_{h,n}}^l,\qquad l = 0,\,\dots,\,4\,, \end{split}

    where \delta_{h, n}: = e^{-\frac{d_{h, n}}{ {\lambda}^*_{h, n}}} and {\lambda}^*_{h, n}: = \max_{i\in I_h}\lambda_{i, n} . Moreover, set {\bf h}_n: = \boldsymbol{{\mathcal L}} (\boldsymbol{\phi}^n) .

    By definition of \boldsymbol{{\mathcal L}} and the fact that \boldsymbol{\phi}^n\in {\bf K} ^\perp , we have, for every h = 1, \, \dots, \, q and i\in I_h

    \begin{equation} \begin{split} \phi_i^n = &(-\Delta)^{-1}\left\{\left[\left(3\mu_i c _i^2+\sum\limits_{j\in I_h\atop j\neq i}\beta_{ij} c _j^2\right)\phi_i^n+2\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _i c _j\phi_j^n\right]\left(PU_h^n\right)^2\right.\\ &\left.+\sum\limits_{k\neq h}\sum\limits_{j \in I_k}\beta_{ij}\left[ c _j^2(PU_k^n)^2\phi_i^n+2 c _i c _jPU_k^n PU_h^n\phi_j^n\right]+\lambda_{i,n}\phi_i^n\right\}+h_i^n-w_i^n\,, \end{split} \end{equation} (A.1)

    for a suitable {\bf w}_n: = (w_i^n)_{i\in I_h}\in K_h^n . Here \mu_i: = \beta_{ii}.

    Step 1: \|{\bf w}_n\|\to0 as n\to+\infty . Multiplying (A.1) by \delta_{h, n}^2w_i^n and recalling the definition of (-\Delta)^{-1} yields

    \begin{split} \delta_{h,n}^2\|w_i^n\|^2 = &\delta_{h,n}^2\langle h_i^n-\phi_i^n,w_i^n\rangle +\delta_{h,n}^2\left(3\mu_i c _i^2+\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _j^2\right)\int (PU_h^n)^2\phi_i^n w_i^n\\ +&2\delta_{h,n}^2\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _i c _j\int(PU_h^n)^2\phi_j^n w_i^n+\delta_{h,n}^2\sum\limits_{k\neq h}\sum\limits_{j \in I_k }\beta_{ij} c _j^2\int(PU_k^n)^2\phi_i^n w_i^n\\ +&2\delta_{h,n}^2 c _i\sum\limits_{k\neq h}\sum\limits_{j \in I_k }\beta_{ij} c _j\int PU_k^n PU_h^n\phi_j^n w_i^n +\delta_{h,n}^2\lambda_{i,n}\int \phi_i^n w_i^n\,, \end{split}

    so that, summing over i\in I_h and making use of (\phi_i^n)_{i\in I_h}, (h_i^n)_{i\in I_h}\in (K_h^n)^\perp ,

    \begin{equation} \begin{split} &\underbrace{\delta_{h,n}^2\sum\limits_{i\in I_h}\|w_i^n\|^2}_{I}\\ = &\delta_{h,n}^2\underbrace{\sum\limits_{i\in I_h}\left[\left(3\mu_i c _i^2+\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _j^2\right)\int (PU_h^n)^2\phi_i^n w_i^n+2\delta_{h,n}^2\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _i c _j\int(PU_h^n)^2\phi_j^n w_i^n\right]}_{II}\\ +&\delta_{h,n}^2\underbrace{\sum\limits_{i\in I_h}\sum\limits_{k\neq h}\sum\limits_{j \in I_k }\beta_{ij} c _j^2\int(PU_k^n)^2\phi_i^n w_i^n}_{III}+2\delta_{h,n}^2\underbrace{\sum\limits_{i\in I_h} c _i\sum\limits_{k\neq h}\sum\limits_{j \in I_k }\beta_{ij} c _j\int PU_k^n PU_h^n\phi_j^n w_i^n}_{IV}\\ +&\delta_{h,n}^2\underbrace{\sum\limits_{i\in I_h}\lambda_{i,n}\int \phi_i^n w_i^n}_{V}\,. \end{split} \end{equation} (A.2)

    Note first that, since (w_i^n)_{i\in I_h}\in K_h^n , for l = 0, \, \dots, \, 4 there are a_{h, n}^l\in\mathbb{R} for which it holds (see (2.6))

    (w_i^n)_{i\in I_h} = \sum\limits_{l = 0}^n a_{h,n}^l\mathfrak{e}_h P\psi_{h,n}^l\,,

    so that arguing as in [17, p. 417] and for sufficiently large n we can write

    \begin{equation} I = \delta_{h,n}^2\sum\limits_{i\in I_h}\sum\limits_{l,p = 0}^na_{h,n}^la_{h,n}^p|\mathfrak{e}_{i,h}|^2\int\nabla P\psi_{h,n}^l\cdot\nabla P\psi_{h,n}^p = \sum\limits_{l = 0}^n(a_{h,n}^l)^2\sigma_{ll}+o(1)\sum\limits_{l = 0}^na_{h,n}^l a_{h,n}^p\,, \end{equation} (A.3)

    for suitable positive constants \sigma_{ll} , l = 0, \, \dots, \, 4 .

    Let us thus estimate terms III and IV in (A.2). On the one hand, for every h, k = 1, \, \dots, q , k\neq h , i\in I_h and l = 0, \, \dots, 4 ,

    \begin{equation} \begin{split} \left|\int (PU_k^n)^2\phi_i^n P\psi_{h,n}^l\right|\leq& \left|\int(PU_k^n)^2\phi_i^n\psi_{h,n}^l\right|+\left|\int(PU_k^n)^2\phi_i^n(P\psi_{h,n}^l-\psi_{h,n}^l)\right|\\ \leq&\|\phi_i^n\|\left|(PU_n^k)^2\psi_{h,n}^l\right|_{\frac{4}{3}}+\|\phi_i^n\|\left|(PU_k^n)^2(P\psi_{h,n}^l-\psi_{h,n}^l)\right|_{\frac{4}{3}}\,, \end{split} \end{equation} (A.4)

    where we made use of Hölder and Sobolev inequality. Then, by [17, Lemma A.1] we get

    \begin{equation} \left|(PU_k^n)^2(P\psi_{h,n}^0-\psi_{h,n}^0)\right|_{\frac{4}{3}}\leq C(\delta_{h,n}+o(\delta_{h,n})\left(\int(PU_k^n)^{\frac{8}{3}}H(\cdot,\xi_{k,n})^{\frac{4}{3}}\right)^{\frac{3}{4}}\leq C'\delta_{h,n}+o(\delta_{h,n}) \end{equation} (A.5)

    and

    \begin{equation} \left|(PU_k^n)^2(P\psi_{h,n}^l-\psi_{h,n}^l)\right|_{\frac{4}{3}}\leq C(\delta_{h,n}^2+o(\delta_{h,n}^2))\left(\int(PU_k^n)^{\frac{8}{3}}\frac{\partial H}{\partial\xi}(\cdot,\xi_{k,n})^{\frac{4}{3}}\right)\leq C'\delta_{h,n}^2+o(\delta_{h,n}^2) \end{equation} (A.6)

    for every l = 1, \, \dots, \, 4 . Moreover, since direct calculations show

    \begin{split} &|\psi_{\delta,\xi}^0|\leq\frac{C}{\delta}U_{\delta,\xi}\\ &|\psi_{\delta,\xi}^l|\leq\frac{C}{\delta}U_{\delta,\xi}^2|x_l-\xi_l|,\quad l = 1,\,\dots,\,4\,, \end{split}

    recalling that 0\leq PU_k^n\leq U_k^n by the maximum principle and making use of [17, Lemma A.2–A.4], we also have

    \begin{split} \left|(PU_k^n)\psi_{h,n}^0\right|_{\frac{4}{3}}\leq&\frac{C}{\delta_{h,n}}\left|(PU_k^n)^2 U_h^n\right|_{\frac{4}{3}}\leq\frac{C}{\delta_{h,n}}\left|(U_k^n)^2 U_h^n\right|_{\frac{4}{3}}\\ \leq&\frac{C'}{\delta_{h,n}}\left(O(\delta_{h,n}\delta_{k,n})+O(\delta_{k,n}^2\delta_{h,n})\right) = O(\delta_{k,n}) \end{split}

    and, for l = 1, \, \dots, \, 4 ,

    \begin{split} \left|(PU_k^n)\psi_{h,n}^l\right|_{\frac{4}{3}}\leq\frac{C}{\delta_{h,n}}\left|(U_k^n)^2(U_h^n)^2\right|_{\frac{4}{3}}\leq\frac{C'}{\delta_{h,n}}\left(O(\delta_{k,n}^2\delta_{h,n})+O(\delta_{k,n}\delta_{h,n}^2)\right) = o(\delta_{k,n})\,. \end{split}

    Combining the previous estimates with (A.4) and the fact that \|\boldsymbol{\phi}_n\| = 1 thus leads to

    \begin{equation} \delta_{h,n}^2 |III|\leq o(\delta_{h,n}^2)\sum\limits_{l = 0}^n |a_{h,n}^l|\,. \end{equation} (A.7)

    Similarly, by Hölder and Sobolev inequality, \|\boldsymbol{\phi}_n\| = 1 and [17, Lemma A.5],

    \left|\int PU_k^n PU_h^n \phi_j^n P\psi_{h,n}^l\right|\leq C\left|PU_k^n PU_h^n P\psi_{h,n}^l\right|_{\frac{4}{3}} = O(\delta_{k,n}\delta_{h,n})\,,

    for every k, h = 1, \, \dots, \, q , k\neq h , j\in I_k and l = 0, \, \dots, \, 4 , so that

    \begin{equation} \delta_{h,n}^2|IV|\leq o(\delta_{h,n}^2)\sum\limits_{l = 0}^n |a_{h,n}^l|\,. \end{equation} (A.8)

    Furthermore, by Hölder inequality and recalling that \boldsymbol{\lambda}_n\to0 as n\to+\infty and \|\boldsymbol{\phi}_n\| = 1 , we also have

    \begin{equation} \delta_{h,n}^2|V|\leq o(\delta_{h,n}^2)\|(w_i^n)_{i\in I_h}\|\,. \end{equation} (A.9)

    We are thus left to estimate the term II in (A.2). To this purpose, we set, for every i, j\in I_h ,

    \alpha_{ii}: = 3\mu_i c _i^2+\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _j^2,\qquad\alpha_{ij}: = 2\beta_{ij} c _i c _j\,,

    so that

    \begin{split} II = &\sum\limits_{i\in I_h}\int\left(\alpha_{ii}\phi_i^n+\sum\limits_{j \in I_h\atop j\neq i}\alpha_{ij}\phi_j^n\right)(PU_h^n)^2 w_i^n\\ = &\underbrace{\sum\limits_{i\in I_h}\int\left(\alpha_{ii}\phi_i^n+\sum\limits_{j \in I_h\atop j\neq i}\alpha_{ij}\phi_j^n\right)\left((PU_h^n)^2-(U_h^n)^2\right)w_i^n}_{II.1}\\+&\underbrace{\sum\limits_{i\in I_h}\int\left(\alpha_{ii}\phi_i^n+\sum\limits_{j \in I_h\atop j\neq i}\alpha_{ij}\phi_j^n\right)(U_h^n)^2 w_i^n}_{II.2}\,. \end{split}

    As for II.1 , we have, for every i, j\in I_h

    \left|\int \left((PU_h^n)^2-(U_h^n)^2\right)\phi_j^n w_i^n\right|\leq\left|(PU_h^n)^2-(U_h^n)^2\right|_2|\phi_j^n|_4|w_i^n|_4\leq \left|(PU_h^n)^2-(U_h^n)^2\right|_2\|w_i^n\|

    by Hölder and Sobolev inequality and \|\boldsymbol{\phi}_n\| = 1 . Furthermore, by [17, Lemma A.3] and 0\leq PU_h^n\leq U_h^n

    \left|(PU_h^n)^2-(U_h^n)^2\right|_2\leq C\left(\left|U_h^n(PU_h^n-U_h^n)\right|_2+\left|(PU_h^n-U_h^n)^2\right|_2\right)\leq C'\left|U_h^n(PU_h^n-U_h^n)\right|_2,

    and by [17, Lemma A.1–A.2]

    \begin{split} \left|U_h^n(PU_h^n-U_h^n)\right|_2 = &\left(\int (U_h^n)^2(PU_h^n-U_h^n)^2\right)^{\frac{1}{2}} = \left(\int (U_h^n)^2(\delta_{h,n}AH(\cdot,\xi_{h,n})+o(\delta_{h,n}))^2\right)^{\frac{1}{2}}\\ \leq&(C\delta_{h,n}+o(\delta_{h,n}))\left(\int (U_h^n)^2\right)^\frac{1}{2}\leq C'\delta_{h,n}^2|\ln\delta_{h,n}|^{\frac{1}{2}}+o(\delta_{h,n}^2)\,, \end{split}

    in turn yielding

    \begin{equation} |II.1|\leq C\sum\limits_{i,j\in I_h}\left|\int \left((PU_h^n)^2-(U_h^n)^2\right)\phi_j^n w_i^n\right|\leq C'(\delta_{h,n}^2|\ln\delta_{h,n}|+o(\delta_{h,n}^2))\|(w_i^n)_{i\in I_h}\|\,. \end{equation} (A.10)

    To estimate II.2 , note first that

    \begin{equation} \begin{split} II.2 = &\sum\limits_{i\in I_h}\int\left(\alpha_{ii}\phi_i^n+\sum\limits_{j \in I_h\atop j\neq i}\alpha_{ij}\phi_j^n\right)(U_h^n)^2\sum\limits_{l = 0}^n a_{h,n}^l\mathfrak{e}_{i,h}P\psi_{h,n}^l\\ = &\sum\limits_{l = 0}^n a_{h,n}^l\int (U_h^n)^2P\psi_{h,n}^l\sum\limits_{i\in I_h}\left(\alpha_{ii}\mathfrak{e}_{i,h}\phi_i^n+\mathfrak{e}_{i,h}\sum\limits_{j \in I_h\atop j\neq i}\alpha_{ij}\phi_j^n\right)\\ = &\sum\limits_{l = 0}^n a_{h,n}^l\int (U_h^n)^2P\psi_{h,n}^l\sum\limits_{i\in I_h}\phi_i^n\left(\alpha_{ii}\mathfrak{e}_{i,h}+\sum\limits_{j \in I_h\atop j\neq i}\alpha_{ij}\mathfrak{e}_{j,h}\right)\\ = &3\sum\limits_{l = 0}^n a_{h,n}^l \int (U_h^n)^2P\psi_{h,n}^l\sum\limits_{i\in I_h}\phi_i^n\mathfrak{e}_{i,h}\,, \end{split} \end{equation} (A.11)

    since by construction \mathfrak{e}_h is an eigenvector of the matrix (\alpha_{ij})_{i, j\in I_h} corresponding to the eigenvalue \Lambda_1 = 3 (see [16, Lemma 6.1]). Recalling that (\phi_i^n)_{i\in I_h}\in (K_h^n)^\perp , so that by (2.5) and (2.6)

    0 = \sum\limits_{i\in I_h}\int \mathfrak{e}_{i,h}\nabla P\psi_{h,n}^l\cdot\nabla\phi_i^n = 3\sum\limits_{i\in I_h}\int (U_h^n)^2\mathfrak{e}_{i,h}\psi_{h,n}^l\phi_i^n

    for every l = 0, \, \dots, \, 4 , we can then rewrite (A.11) as

    II.2 = 3\sum\limits_{l = 0}^n\sum\limits_{i\in I_h} a_{h,n}^l \mathfrak{e}_{i,h} \int (U_h^n)^2(P\psi_{h,n}^l-\psi_{h,n}^l)\phi_i^n\,.

    Arguing as in (A.4)–(A.5)–(A.6) above we get

    \left|\int (U_h^n)^2(P\psi_{h,n}^l-\psi_{h,n}^l)\phi_i^n\right|\leq C\delta_{h,n}+o(\delta_{h,n})

    for every l = 0, \dots, 4 , h = 1, \dots, q and i\in I_h , thus implying

    \begin{equation} |II.2|\leq (C\delta_{h,n}+o(\delta_{h,n}))\sum\limits_{l = 0}^n|a_{h,n}^l|\,. \end{equation} (A.12)

    Coupling (A.10)–(A.12) then gives

    |II|\leq C'(\delta_{h,n}^2|\ln\delta_{h,n}|+o(\delta_{h,n}^2))\|{\bf w}_n\|+(C\delta_{h,n}+o(\delta_{h,n}))\sum\limits_{l = 0}^n |a_{h,n}^l|\,,

    and combining with (A.2), (A.7), (A.8), (A.9) we finally obtain

    \delta_{h,n}^2\|(w_i^n)\|_{i\in I_h}^2\leq o(\delta_{h,n}^2)\|(w_i^n)\|_{i\in I_h}+o(\delta_{h,n}^2)\sum\limits_{l = 0}^n|a_{h,n}^l|\,.

    Together with (A.3), this ensures that \|(w_i^n)\|_{i\in I_h}\|\to0 as n\to+\infty , and repeating the argument for every h = 1, \dots, q , gives \|{\bf w}_n\|\to0 as desired.

    Step 2. For every h = 1, \dots, q and i\in I_h , we set

    \widetilde{\phi}_i^n(y): = \begin{cases} \delta_{h,n}\phi_i^n\left(\xi_{h,n}+\delta_{h,n}y\right) & \;{\rm{if }}\;y\in\widetilde{\Omega}_{h,n}: = \frac{\Omega-\xi_{h,n}}{\delta_{h,n}}\\ 0 & \;{\rm{if }}\;y\in\mathbb{R}^n\setminus\widetilde{\Omega}_{h,n}\,. \end{cases}

    By definition, \|\widetilde{\phi}_i^n\|_{H_0^1(\mathbb{R}^n)} = \|\phi_i^n\|_{H_0^1(\Omega)} , so that \widetilde{\phi}_i^n\rightharpoonup\widetilde{\phi}_i in \mathcal{D}^{1, 2}(\mathbb{R}^n) as n\to+\infty , for some \widetilde{\phi}_i . Let us thus show that \widetilde{\phi}_i\equiv0 for every i = 1, \dots, m . To this aim, note first that (A.1) can be rewritten as

    \begin{split} &\int_{\widetilde{\Omega}_{h,n}}\nabla\widetilde{\phi}_i^n\cdot\nabla\varphi\\ = &\underbrace{\delta_{h,n}^2\int_{\widetilde{\Omega}_{h,n}}\left[\left(3\mu_i c _i^2+\sum\limits_{j\in I_h\atop j\neq i}\beta_{ij} c _j^2\right)\widetilde{\phi}_i^n+2\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _i c _j\widetilde{\phi}_j^n\right]\left(PU_h^n\right)^2(\xi_{h,n}+\delta_{h,n}y)\varphi}_{A_n}\\ +&\underbrace{\delta_{h,n}^2\int_{\widetilde{\Omega}_{h,n}}\sum\limits_{k\neq h}\sum\limits_{j \in I_k}\beta_{ij} c _j^2(PU_k^n)^2(\xi_{h,n}+\delta_{h,n}y)\widetilde{\phi}_i^n\varphi}_{B_n}\\ +&\underbrace{2\delta_{h,n}^3\int_{\widetilde{\Omega}_{h,n}}\sum\limits_{k\neq h}\sum\limits_{j \in I_k} c _i c _jPU_k^n(\xi_{h,n}+\delta_{h,n}y) PU_h^n(\xi_{h,n}+\delta_{h,n}y)\phi_j^n(\xi_{h,n}+\delta_{h,n}y)\varphi}_{C_n}\\ +&\delta_{h,n}^2\int_{\widetilde{\Omega}_{h,n}}\lambda_{i,n}\widetilde{\phi}_i^n\varphi+\int_{\widetilde{\Omega}_{h,n}}\nabla(\widetilde{h}_i^n-\widetilde{w}_i^n)\cdot\nabla\varphi, \end{split}

    for every \varphi\in C_C^\infty(\mathbb{R}^n) , where

    \begin{split} \widetilde{h}_i^n(y): = &\begin{cases} \delta_{h,n}h_i^n(\xi_{h,n}+\delta_{h,n}y) & \;{\rm{if }}\;y\in\widetilde{\Omega}_{h,n}\\ 0 & \;{\rm{if }}\;y\in\mathbb{R}^n\setminus\widetilde{\Omega}_{h,n} \end{cases}\\ \widetilde{w}_i^n(y): = &\begin{cases} \delta_{h,n}w_i^n(\xi_{h,n}+\delta_{h,n}y) & \;{\rm{if }}\;y\in\widetilde{\Omega}_{h,n}\\ 0 & \;{\rm{if }}\;y\in\mathbb{R}^n\setminus\widetilde{\Omega}_{h,n}\,. \end{cases} \end{split}

    Let now \varphi be so that K_\varphi: = {\rm{supp}}\varphi\subset\widetilde{\Omega}_{h, n} , which is always true for any given \varphi\in C_c^\infty(\mathbb{R}^n) and n large enough. On the one hand, it is readily seen that

    \begin{split} &\delta_{h,n}^2\int_{\widetilde{\Omega}_{h,n}}\lambda_{i,n}\widetilde{\phi}_i^n\varphi\to0\\ &\int_{\widetilde{\Omega}_{h,n}}\nabla(\widetilde{h}_i^n-\widetilde{w}_i^n)\cdot\nabla\varphi\to0 \end{split}

    as n\to+\infty , since \boldsymbol{\lambda}_n\to0 , \|{\bf h}_n\|\to0 and \|{\bf w}_n\|\to0 .

    On the other hand, for every i\in I_h , j\in I_k , h\neq k

    \begin{split} \int_{\widetilde{\Omega}_{h,n}}(PU_k^n)^2(\xi_{h,n}+&\delta_{h,n}y)\widetilde{\phi}_i^n(y)\varphi(y) = \int_{\widetilde{\Omega}_{h,n}}(U_k^n)^2(\xi_{h,n}+\delta_{h,n}y)\widetilde{\phi}_i^n(y)\varphi(y)+o(1)\\ = &\delta_{k,n}^2\int_{\widetilde{\Omega}_{h,n}\cap K_\varphi}\frac{\alpha_4^2}{(\delta_{k,n}^2+|\delta_{h,n}y+\xi_{h,n}-\xi_{k,n}|^2)^2}\widetilde{\phi}_i^n(y)\varphi(y)+o(1)\to0 \end{split}

    and

    \begin{split} \delta_{h,n}^3\int_{\widetilde{\Omega}_{h,n}}&PU_k^n(\xi_{h,n}+\delta_{h,n}y) PU_h^n(\xi_{h,n}+\delta_{h,n}y)\phi_j^n(\xi_{h,n}+\delta_{h,n}y)\varphi(y)\\ = &\delta_{h,n}^2\int_{\widetilde{\Omega}_{h,n}\cap K_\varphi}\frac{\varphi(y)}{1+|y|^2}\frac{\alpha_4\delta_{k,n}}{\delta_{k,n}^2+|\delta_{h,n}y+\xi_{h,n}-\xi_{k,n}|^2}\phi_j^n(\xi_{h,n}+\delta_{h,n}y)+o(1)\to0 \end{split}

    as y\in K_\varphi , which is fixed and bounded, and |\xi_{h, n}-\xi_{k, n}|\geq\eta by assumption. Hence,

    B_n\to0,\quad C_n\to0\quad {\rm{as }}\;n\to+\infty\,.

    Furthermore, for every j\in I_h ,

    \begin{split} \delta_{h,n}^2\int_{\widetilde{\Omega}_{h,n}}\left(PU_h^n\right)^2(\xi_{h,n}+&\delta_{h,n}y)\widetilde{\phi}_j^n(y)\varphi(y)\\ = &\delta_{h,n}^2\int_{\widetilde{\Omega}_{h,n}}\left(U_h^n\right)^2(\xi_{h,n}+\delta_{h,n}y)\widetilde{\phi}_j^n(y)\varphi(y)+o(1)\\ = &\int_{\widetilde{\Omega}_{h,n}\cap K_\varphi}\frac{\alpha_4^2}{(1+|y|^2)^2}\widetilde{\phi}_j^n(y)\varphi(y)+o(1)\to\int_{\mathbb{R}^n}\frac{\alpha_4^2}{(1+|y|^2)^2}\widetilde{\phi}_j(y)\varphi(y)\\ = &\int_{\mathbb{R}^n}U_{1,0}^2(y)\widetilde{\phi}_j(y)\varphi(y) \end{split}

    since \widetilde{\phi}_j^n\rightharpoonup\widetilde{\phi}_j in \mathcal{D}^{1, 2}(\mathbb{R}^n) and U_{1, 0}\in L^n(\mathbb{R}^n) . Therefore,

    A_n\to\int_{\mathbb{R}^n}\left[\left(3\mu_i c _i^2+\sum\limits_{j\in I_h\atop j\neq i}\beta_{ij} c _j^2\right)\widetilde{\phi}_i+2\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _i c _j\widetilde{\phi}_j\right]\left(U_{1,0}\right)^2\varphi\qquad {\rm{as }}\;n\to+\infty

    that is, for every i\in I_h

    -\Delta\widetilde{\phi}_i = \left[\left(3\mu_i c _i^2+\sum\limits_{j\in I_h\atop j\neq i}\beta_{ij} c _j^2\right)\widetilde{\phi}_i+2\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _i c _j\widetilde{\phi}_j\right]\left(U_{1,0}\right)^2\quad\;{\rm{in }}\;\mathbb{R}^n,\quad\widetilde{\phi}_i\in\mathcal{D}^{1,2}(\mathbb{R}^n).

    Therefore, the weak limit (\widetilde{\phi}_i)_{i\in I_h} solves the linearized system (1.9) for every h = 1, \dots, q . Thus, (\widetilde{\phi}_i)_{i\in I_h}\in{\rm{span}}\{\mathfrak{e}_h\psi_{1, 0}^l\, :\, l = 0, \dots, 4\} . However, since (\phi_i^n)_{i\in I_h}\in(K_h^n)^\perp for every n , then it follows

    \begin{split} 0 = &\delta_{h,n}\langle(\phi_i^n)_{i\in I_h},\mathfrak{e}_hP\psi_{h,n}^0\rangle = 3\sum\limits_{i\in I_h}\delta_{h,n}\int_{\Omega} (U_h^n)^2\mathfrak{e}_{i,h}\psi_{h,n}^0\phi_i^n\\ = &3\sum\limits_{i\in I_h}\int_{\widetilde{\Omega}_{h,n}}\mathfrak{e}_{i,h}\alpha_4^3\frac{|y|^2-1}{(1+|y|^2)^n}\widetilde{\phi}_{i}^n = 3\sum\limits_{i\in I_h}\int_{\widetilde{\Omega}_{h,n}}U_{1,0}^2\psi_{1,0}^0\mathfrak{e}_{i,h}\widetilde{\phi}_i^n \end{split}

    and, for every l = 1, \dots, 4 ,

    \begin{split} 0 = &\delta_{h,n}\langle(\phi_i^n)_{i\in I_h},\mathfrak{e}_hP\psi_{h,n}^l\rangle = 3\sum\limits_{i\in I_h}\delta_{h,n}\int_{\Omega} (U_h^n)^2\mathfrak{e}_{i,h}\psi_{h,n}^l\phi_i^n\\ = &3\sum\limits_{i\in I_h}\int_{\widetilde{\Omega}_{h,n}}\mathfrak{e}_{i,h}2\alpha_4^3\frac{y_l}{(1+|y|^2)^n}\widetilde{\phi}_i^n = 3\sum\limits_{i\in I_h}\int_{\widetilde{\Omega}_{h,n}}U_{1,0}^2\psi_{1,0}^l\mathfrak{e}_{i,h}\widetilde{\phi}_i^n. \end{split}

    Passing to the limit as n\to+\infty and making use of \widetilde{\phi}_i^n\rightharpoonup\widetilde{\phi}_i , we obtain

    3\sum\limits_{i\in I_h}\int_{\mathbb{R}^n}U_{1,0}^2\psi_{1,0}^l\mathfrak{e}_{i,h}\widetilde{\phi}_i = 0,\qquad l = 0,\dots,4\,.

    This shows that (\widetilde{\phi}_i)_{i\in I_h}\in({\rm{span}}\{\mathfrak{e}_h\psi_{1, 0}^l\, :\, l = 0, \dots, 4\})^\perp , thus implying \widetilde{\phi}_i\equiv0 for every i\in I_h and concluding Step 2.

    Step 3. We now prove that \phi_{i}^n\to0 strongly in H_0^1(\Omega) for every i = 1, \dots, m , which in turn concludes the proof of (3.2) as it contradicts the assumption \|\boldsymbol{\phi}_n\| = 1 for every n .

    To this aim, let us test (A.1) with \phi_{i}^n , so to have

    \begin{equation} \begin{split} \|\phi_i^n\|^2 = &\underbrace{\left(3\mu_i c _i^2+\sum\limits_{j\in I_h\atop j\neq i}\beta_{ij} c _j^2\right)\int_{\Omega}\left(PU_h^n\right)^2(\phi_i^n)^2}_{I}+2\underbrace{\sum\limits_{j \in I_h\atop j\neq i}\beta_{ij} c _i c _j\int_{\Omega}\left(PU_h^n\right)^2\phi_j^n\phi_i^n}_{II}\\ +&\underbrace{\sum\limits_{k\neq h}\sum\limits_{j \in I_k}\beta_{ij} c _j^2\int_{\Omega}(PU_k^n)^2(\phi_i^n)^2}_{III}+\underbrace{\sum\limits_{k\neq h}\sum\limits_{j \in I_k }2\beta_{ij} c _i c _j\int_{\Omega}PU_k^n PU_h^n\phi_j^n\phi_i^n}_{IV}\\ +&\lambda_{i,n}\|\phi_i^n\|^2+\langle h_i^n-w_i^n,\phi_i^n\rangle\,. \end{split} \end{equation} (A.13)

    Since \boldsymbol{\lambda}_n\to0 , \|{\bf h}_n\|\to0 , \|{\bf w}_n\|\to0 and \boldsymbol{\phi}_n is bounded in H_0^1(\Omega) uniformly on n ,

    \begin{equation} \lambda_{i,n}\|\phi_i^n\|^2+\langle h_i^n-w_i^n,\phi_i^n\rangle\to0\qquad{\rm{as }}\;n\to+\infty\,. \end{equation} (A.14)

    Moreover, recalling that 0\leq PU_h^n\leq U_h^n for every h = 1, \dots, q , we have

    \begin{split} \int_{\Omega}\left(PU_h^n\right)^2(\phi_i^n)^2\leq& \int_{\Omega}\left(U_h^n\right)^2(\phi_i^n)^2 = \int_{\widetilde{\Omega}_{h,n}}U_{1,0}^2(\widetilde{\phi}_i^n)^2\to0\\ \int_{\Omega}\left(PU_h^n\right)^2\phi_j^n\phi_i^n = &\int_{\Omega}\left(U_h^n\right)^2\phi_j^n\phi_i^n+o(1) = \int_{\widetilde{\Omega}_{h,n}}(U_{1,0})^2\widetilde{\phi}_j^n\widetilde{\phi}_i^n+o(1)\to0 \end{split}

    as n\to+\infty and for every i, j\in I_h , since \widetilde{\phi}_i^n, \widetilde{\phi}_j^n\rightharpoonup 0 in \mathcal{D}^{1, 2}(\mathbb{R}^n) and U_{1, 0}^2\in L^2(\mathbb{R}^n) , so that

    \begin{equation} |I|\to0\quad{\rm{and}}\quad|II|\to0\quad{\rm{as }}\;n\to+\infty\,. \end{equation} (A.15)

    As for term IV , for every i\in I_h , j\in I_k , h\neq k , by Hölder and Sobolev inequalities and by [17, Lemma A.2–A.4]

    \begin{split} \left|\int_{\Omega}PU_k^n PU_h^n\phi_j^n\phi_i^n\right|\leq&\left(\int_{\Omega}(PU_k^n)^2(PU_h^n)^2\right)^\frac{1}{2}\left(\int_{\Omega}(\phi_j^n)^2(\phi_i^n)^2\right)^\frac{1}{2}\leq C\left(\int_{\Omega}(U_k^n)^2(U_h^n)^2\right)^\frac{1}{2}\\ \leq&C'\left(O(\delta_{h,n}^2)\int_\Omega(U_k^n)^2+O(\delta_{k,n}^2)\int_\Omega(U_k^n)^2+O(\delta_{h,n}^2\delta_{k,n}^2)\right)^\frac{1}{2}\\ \leq&C''\left(O(\delta_{h,n})\delta_{k,n}\sqrt{|\ln\delta_{k,n}|}+O(\delta_{k,n})\delta_{h,n}\sqrt{|\ln\delta_{h,n}|}+O(\delta_{h,n}\delta_{k,n})\right)\,, \end{split}

    thus ensuring

    \begin{equation} |IV|\to0\qquad{\rm{as }}\;n\to+\infty\,. \end{equation} (A.16)

    We are left to discuss term Ⅲ. On the one hand, if for every h = 1, \dots, q it holds

    \max\limits_{(i,j)\in I_h\times I_k\atop h\not = k}\beta_{ij}\leq0\,,

    then we simply have

    III\le0.

    On the other hand, if there exist i\in I_h, j\in I_k with \beta_{ij} > 0 , then

    \beta_{ij}\int_{\Omega}(PU_k^n)^2(\phi_{i}^n)^2\leq\beta_{ij}\left|\phi_{i}^n\right|_4^2\left|U_k^n\right|_4^2\leq C\beta_{ij}\|\phi_{i}^n\|^2\,.

    Let then \overline{\beta} > 0 be a positive constant so that, whenever

    \max\limits_{(i,j)\in I_h\times I_k\atop h\not = k}\beta_{ij}\leq \overline{\beta}\,,

    we have

    \begin{equation} |III|\leq C\sum\limits_{k\neq h}\sum\limits_{j \in I_k }\beta_{ij} c _j^2\|\phi_{i}^n\|^2\leq\frac{1}{2}\|\phi_i^n\|^2\,. \end{equation} (A.17)

    Summing up, coupling (A.14), (A.15), (A.16) and (A.17) with (A.13), we conclude that \|\phi_{i}^n\|\to0 as n\to+\infty , for every i = 1, \dots, m .

    Step 4: invertibility. Note first that (-\Delta)^{-1}:L^{\frac{4}{3}}(\Omega)\to H_0^1(\Omega) is a compact operator, so that \boldsymbol{{\mathcal L}} restricted to \bf K ^\perp is a compact perturbation of the identity. Furthermore, (3.2) implies that \boldsymbol{{\mathcal L}} is injective, and thus surjective by Fredholm alternative. Henceforth, it is invertible, and the continuity of the inverse operator is guaranteed by (3.2).



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