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Research article Special Issues

Nonlinear robust control of trajectory-following for autonomous ground electric vehicles with active front steering system

  • Received: 03 January 2023 Revised: 18 February 2023 Accepted: 26 February 2023 Published: 09 March 2023
  • MSC : 93B52, 93C95, 93D05

  • This paper presents a nonlinear robust H-infinity control strategy for improving trajectory following performance of autonomous ground electric vehicles (AGEV) with active front steering system. Since vehicle trajectory dynamics inherently influenced by various driving maneuvers and road conditions, the main objective is to deal with the trajectory following control challenges of parametric uncertainties, system nonlinearities, and external disturbance. The AGEV system dynamics and its uncertain vehicle trajectory following system are first modeled and constructed, in which parameter uncertainties related to the physical limits of tire are considered and handled, then the control-oriented vehicle trajectory following augmented system with dynamic error is developed. The resulting nonlinear robust H-infinity state-feedback controller (NHC) of vehicle trajectory-following system is finally designed by H-infinity performance index and nonlinear compensation under AGEV system requirements, and solved utilizing a set of linear matrix inequalities derived from quadratic H-infinity performance and Lyapunov stability. Simulations for double lane change and serpentine scenes are carried out to verify the effectiveness of the proposed controller with a high-fidelity, CarSim®, full-vehicle model. It is found from the results that the proposed NHC provides improved vehicle trajectory following performance compared with the linear quadratic regulator (LQR) controller and robust H-infinity state-feedback controller (RHC).

    Citation: Xianjian Jin, Qikang Wang, Zeyuan Yan, Hang Yang. Nonlinear robust control of trajectory-following for autonomous ground electric vehicles with active front steering system[J]. AIMS Mathematics, 2023, 8(5): 11151-11179. doi: 10.3934/math.2023565

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  • This paper presents a nonlinear robust H-infinity control strategy for improving trajectory following performance of autonomous ground electric vehicles (AGEV) with active front steering system. Since vehicle trajectory dynamics inherently influenced by various driving maneuvers and road conditions, the main objective is to deal with the trajectory following control challenges of parametric uncertainties, system nonlinearities, and external disturbance. The AGEV system dynamics and its uncertain vehicle trajectory following system are first modeled and constructed, in which parameter uncertainties related to the physical limits of tire are considered and handled, then the control-oriented vehicle trajectory following augmented system with dynamic error is developed. The resulting nonlinear robust H-infinity state-feedback controller (NHC) of vehicle trajectory-following system is finally designed by H-infinity performance index and nonlinear compensation under AGEV system requirements, and solved utilizing a set of linear matrix inequalities derived from quadratic H-infinity performance and Lyapunov stability. Simulations for double lane change and serpentine scenes are carried out to verify the effectiveness of the proposed controller with a high-fidelity, CarSim®, full-vehicle model. It is found from the results that the proposed NHC provides improved vehicle trajectory following performance compared with the linear quadratic regulator (LQR) controller and robust H-infinity state-feedback controller (RHC).



    In this paper, we mainly focus our interest on the existence and concentration of normalized solutions of the following nonlinear elliptic problem involving a Kirchhoff term:

    {(ε2a+εbR3|v|2dx)ΔvK(x)|v|2σv=λvin R3,|v|22=R3v2dx=m0εα,v(x)0as|x|, (1.1)

    where a,b,α are positive real numbers and σ(0,2), λ is unkown and appears as a Lagrange multiplier. Equation (1.1) is related to the stationary solutions of

    utt(a+bR3|u|2)Δu=g(x,t). (1.2)

    Equation (1.2) was first proposed by Kirchhoff in [13] and regarded as an extension of the classical D'Alembert's wave equation, which describes free vibrations of elastic strings. Kirchhoff-type problems also appear in other fields like biological systems. To better understand the physical background, we refer the readers to [1,2,4,14]. From a mathematical point of view, problem (1.1) is not a pointwise identity because of the appearance of the term (R3|u|2)Δu. Due to such a characteristic, Kirchhoff- type equations constitute nonlocal problems. Compared with the semilinear states (i.e., setting b=0 in the above two equations), the nonlocal term creates some additional mathematical difficulties which make the study of such problems particularly interesting.

    In the literature about the following related unconstrained Kirchhoff problems, there have been a lot of results on the existence and concentration of solutions for small values of ε.

    (ε2a+εbR3|u|2dx)Δu+V(x)u=f(u),xR3. (1.3)

    In physics, such solutions are called the semiclassical states for small values of ε. In [10], the existence, multiplicity and concentration behavior of positive solutions to the Kirchhoff problem (1.3) have been studied by He and Zou, where V(x) is a continuous function and f is a subcritical nonlinear term. For the critical case, Wang et al., in [28] obtained some multiplicity and concentration results of positive solutions for the Kirchhoff problem (1.3). And He et al., in [11] obtained the concentration of solutions in the critical case. Recently, multi-peak solutions were established by Luo et al., in [18] for the following problem:

    (ε2a+εbR3|u|2dx)Δu+V(x)u=|u|p2u,xR3. (1.4)

    In [15] Li et al., revisited the singular perturbation problem (1.4), where V(x) satisfies some suitable assumptions. They established the uniqueness and nondegeneracy of positive solutions to the following limiting Kirchhoff problem:

    (a+bR3|u|2dx)Δu+u=|u|p2u,xR3.

    By the Lyapunov-Schmidt reduction method and a local Pohozaev identity, single-peak solutions were obtained for (1.4). In the past decades, other related results have also been widely studied, such as the existence of ground states, positive solutions, multiple solutions and sign-changing solutions to (1.4). We refer the reader to [7,9,10,16,29] and the references therein.

    In recent years, the problems on normalized solutions have attracted much attention from many researchers. In [25,26], Stuart considered the problem given by

    {Δu+λu=f(u),xRN,RN|u|2dx=c (1.5)

    in the mass-subcritical case and obtained the existence of normalized solutions by seeking a global minimizer of the energy functional. In [12], Jeanjean considered the mass supercritical case and studied the existence of normalized solutions to problem (1.5) by using the mountain pass lemma. For the Sobolev critical case, Soave in [24] considered normalized ground state solutions of problem (1.5) with f(u)=μ|u|q2u+|u|22u, where 2=2N/(N2),N3 is the Sobolev critical exponent. For f(u)=g(u)+|u|22u with a mass critical or supercritical state but Sobolev subcritical nonlinearity g, we refer the reader to [19]. Now, we would like to mention some related results on Kirchhoff problems. The authors of [29,30] considered the problem in the mass subcritical and mass critical cases:

    {(a+bRN|v|2dx)Δv=λv+f(v)in RN,|v|22=RNv2dx=c2, (1.6)

    with a,b>0 and p(2,2). The existence and non-existence of normalized solutions are obtained. In [20], the Kirchhoff problem (1.6) was investigated for f(u)=μ|u|q2u+|u|22u and N=3. With the aid of a subcritical approximation approach, the existence of normalized ground states can be obtained for μ>0 large enough. Moreover, the asymptotic behavior of ground state solutions is also considered as c. As for further results on Sobolev critical Kirchhoff equations and high energy normalized solutions, we refer the reader to [21,22,32].

    In what follows, we turn our attention to normalized multi-bump solutions of the Kirchhoff problem (1.1). For the related results on Schrödinger equations, we refer the reader to the references [27,31]. In [31], the following nonlinear Schrödinger equation was studied by Zhang and Zhang:

    {2ΔvK(x)|v|2σv=λvin RN,|v|22=RNv2dx=m0α,v(x)0as|x|. (1.7)

    For the case that the parameter goes to 0, the authors of [31] constructed normalized multi-bump solutions around the local maximum points of K by employing the variational gluing methods of Séré [23] and Zelati and Rabinowitz [5,6], as well as the penalization technique [31]. Soon afterward, Tang et al., in [27] considered normalized solutions to the nonlinear Schrödinger problem

    Δu+λa(x)u+μu=|u|2σu,xRN (1.8)

    with an L2-constraint. By taking the limit as λ+, they derive the existence of normalized multi-bump solutions with each bump concentrated around the local minimum set of a(x).

    Motivated by [27,31], the present paper is devoted to the existence and concentration behavior of the multi-bump solutions for the Kirchhoff problem (1.1). In contrast to the nonlinear Schrödinger problems, the Kirchhoff term brings us some additional difficulties. We intend to obtain the existence of multi-bump solutions for (1.1).

    Before stating our main result, we give the following assumptions:

    (A) α(3,2σ) if σ(0,23) and α(2σ,3) if σ(23,2).

    (K) K(R3,(0,+))L(R3) and there are 2 mutually disjoint bounded domains ΩiR3, i=1,2,, such that

    ki:=maxxΩiK(x)>maxxΩiK(x).

    Denote Ki={xΩi|K(x)=ki}, which is nonempty and compact and set

    β:=2ασ23σ.

    Now, we state our main result as follows.

    Theorem 1.1. Assume that (A) and (K). There is ε0>0 such that for each ε(0,ε0), it follows that (1.1) admits a solution (λε,vε)R×H1(R3) with the following properties:

    (a) vε admits exactly local maximum points Pi,ε, i=1,2,, that satisfy

    limε0dist(Pi,ε,Ki)=0.

    (b) μ=ε2σ(3α)23σλεμ0 and ε3α23σvε(εβ)i=1ui(εβPi,ε)H10 as ε0, where

    μ0=m2σ23σ0a3σ23σ(i=1θ1σi|U|22)2σ23σ,
    ui=θ12σiμ12σU(μa),i=1,2,,,

    and UH1(R3) is a positive solution to

    {ΔU+U=|U|2σUinR3,U(0)=maxxR3U(x),limxU(x)=0. (1.9)

    (c) There are constants C,c>0 that are independent of ε such that

    |vε|Cε3α23σexp{cεβdist(x,i=1Ki)}.

    The proof of Theorem 1.1 is similar to that in [31]. By virtue of the change of variables techinque, we have

    u()=ε3α23σv(εβ).

    Equation(1.1) is transformed into the following problem:

    {(a+ε(3α)(σ2)23σb|u|22)ΔuK(εβx)|u|2σu=λε2σ(3α)23σuin R3,|u|22=m0,u(x)0as|x|.

    Let

    :=εβ,μ=ε2σ(3α)23σλ,d=(3α)(σ2)2ασ.

    Then, under the assumption (A) and given β>0 and d>0, we have the following:

    {(a+db|u|22)ΔuK(x)|u|2σu=μuin R3,|u|22=m0,u(x)0as|x|. (1.10)

    Define the energy functional

    E(u)=a2R3|u|2+db4(R3|u|2)212σ+2R3K(x)|u|2σ+2.

    Then, a solution (μ,u) of (1.10) can be obtained as a critical point of E that is restrained on

    M:={uH1(R3)||u|22=m0}.

    By adopting similar deformation arguments in [5,6,23,31], we show that the Lagrange multiplier μ satisfies

    μ=μ0+o(1),u=i=1ui(qi,)+o(1)in  H1(R3),

    where qi, satisfies the condition that dist(qi,,Ki)0 as 0, i=1,2,,.

    This paper is organized as follows: In Section 2, we study the existence and variational structure of solutions to the limit equation of Eq (1.1). In Section 3, we introduce the penalized function which satisfies the Palais-Smale condition. In Section 4, we prove the existence of a critical point of the penalized function in the subcritical and supercritical cases. In Section 5, we show that the critical point is a solution to the original problem through the application of a decay estimate.

    Notation : In this paper, we make use of the following notations:

    |u|p:=(R3|u|p)1p, where uLp(R3), p[1,);

    u:=(R3|u|2+|u|2)12, where uH1(R3);

    b±=max{0,±b} for bR;

    B(x,ρ) denotes an open ball centered at xR3 with radius ρ>0;

    ● For a domain DR3, we denote 1D:={xR3|xD};

    ● Unless stated otherwise, δ and C are general constants.

    Let m0,θ1,θ2,,θ be a series of positive numbers. We consider the following system:

    {aΔviθi|vi|2σvi=μviin R3,i=1|vi|22=m0,vi(x)>0,lim|x|vi(x)=0,i=1,2,,. (2.1)

    Next, we refer the reader to [31] to show Lemmas 2.1–2.3 as follows.

    Lemma 2.1. For σ(0,23)(23,2), system (2.1) has a unique solution (μ,v1,v2,,v)R×H1(R3) up to translations of each vi,i=1,2,,, where

    μ=m2σ23σ0a3σ23σ(i=1θ1σi|U|22)2σ23σ,vi(x)=θ12σiμ12σU(μax), (2.2)

    and UH1(R3) is the unique positive radial solution to (1.9).

    By using (2.2), we can obtain the mass distribution for each vi,i=1,2,, in the limit system (2.1), as follows:

    |vi|22=m0θ1σii=1θ1σi

    and for each i=1,2,,,vi is the ground state of

    Iθi(u)=a2|u|22θi2σ+2|u|2σ+22σ+2

    on

    Mi:={uH1(R3)||u|22=|vi|22}.

    Lemma 2.2. i=1Iθi(vi) is continuous and strictly decreasing with respect to m0 and θi,i=1,2,,, where vi is determined as in Lemma 2.1.

    We next characterize the energy level of i=1Iθi(vi). Let

    s=(s1,s2,,s)(0,+)

    and for each si>0, the minimizing problem

    bsi=inf{Iθi(v)||v|22=s2i,|v|22=3θiσ(2σ+2)a|v|2σ+22σ+2}

    is achieved for each i=1,2,, given some radial function wsi. In particular, vi=ws0i for s0i=|vi|2. Moreover, if σ(0,23), then

    bsi=inf{Iθi(v)|vH1(R3),|v|22=s2i}

    and if σ(23,2), then

    bsi=inf{supt>0Iθi(t32v(t))|vH1(R3),|v|22=s2i}.

    Set

    S1+:={s=(s1,s2,,s)(0,m0)|i=1s2i=m0,i=1,2,,},

    and define E(s):=i=1Iθi(wsi) for sS1+.

    Lemma 2.3. Denote s0=(s01,s02,,s0)=(|v1|2,|v2|2,,|v|2). For each sS1+{s0}, the following statements hold:

    (a) If σ(0,23), then i=1Iθi(vi)=E(s0)>E(s);

    (b) If σ(23,2), then i=1Iθi(vi)=E(s0)<E(s).

    In this section, we adopt the penalization argument and the deformation approach in [31] to obtain a constrained localized Palais-Smale sequence. Denote (μ0,ui) as the solution of the limit system (2.1) with m0=1 and θi=ki,i=1,2,,, where (ki)i=1 denotes positive numbers given by (K). Next, we set b0:=i=1Ii(ui), where

    Ii(u):=Iki(u)=a2|u|22ki2σ+2|u|2σ+22σ+2.

    Then, we will find a positive solution (μ,u) to the following system:

    {(a+db|u|22)ΔuK(x)|u|2σu=μuin R3,|u|22=1,u(x)0as|x|, (3.1)

    satisfying

    μ=μ0+o(1),u(x)=i=1ui(xqi,)+o(1)in  H1(R3) 

    with qi,qiKi.

    Set M:={uH1(R3)||u|2=1} and for i=1,2,, and τ>0, define

    (Ki)τ:={xR3|dist(x,Ki)τ}Ωi.

    Define the following equation for each ρ(0,110min1iuiL2(B1(0))):

    Z(ρ)={u=i=1ui(xqi,)+vM|qi,(Ki)τ,vρ}.

    For uH1(R3), consider the penalized energy functional I:H1(R3)R is given by

    I(u):=E(u)+G(u),

    where

    G(u)=(1R3χ(x)(|u|2+u2)dx1)2+,

    and

    χ={0xR3i=11Ωi,1xR3i=11Ωi.

    We also denote

    J(u)=12|u|22for uH1(R3).

    Note that if uM with u2H1(R3i=11Ωi)< is a critical point of I|M, then it solves (3.1) for some μ. Denote the tangent space of M at uM by

    TuM={vH1(R3)|R3uv=0}.

    Lemma 3.1. For any LR, there exists L>0 such that for any fixed (0,L), if a sequence {un,}Z(ρ) such that

    I(un,)L,I|M(un,)Tun,M0, (3.2)

    as n, then un, has a strong convergent subsequence in H1(R3).

    Proof. Set un,=i=1ui(xzn,i,)+vn, with zn,i,(Ki)τ and vn,ρ. It follows from un,Z(ρ) that un,ρ+i=1ui, which is bounded. Then, by

    I(un,)+12σ+2R3K(x)|un,|2σ+2=a2|un,|22+db4|un,|42+G(un,),

    we have that G(un,)I(un,)+12σ+2R3K(x)|un,|2σ+2CL for some CL>0 that is independent of and n. From the assumption (3.2), for some μn,R, we deduce that

    I(un,)+μn,J(un,)0in H1,as n. (3.3)

    We have

    |μn,|=I(un,)un,+o(1)aR3|un,|2+db(R3|un,|2)2R3K(x)|un,|2σ+2+G(un,)un,C(un,2+un,4+un,2σ+2+G(un,)+G(un,)12)CL,

    where CL>0 is independent of and n. Then up to a subsequence, μn,μ in R and un,u=i=1ui(xzi,)+v in H1(R3) with zn,i,zi,1(Ki)τ and vn,v.

    Next, for any φH1(R3), note that limnI(un,)φ+μn,J(un,)φ=0, (μ,u) satisfies

    aR3uφ+dbR3|u|2R3uφR3K(x)|u|2σuφ+R3μuφ+QR3χ(uφ+uφ)=0, (3.4)

    where Q=41limnG(un,)120. Then, we claim that L and μL are two positive constants such that μ>μL for each (0,L). Otherwise, we assume that μμ0 as 0 up to a subsequence. Because u is bounded in H1(R3), we can assume that u(+z1,)u. Note that

    lim inf0u(+zi,)L2(B1(0))uiL2(B1(0))ρ>0.

    We can obtain that u0 if ρ>0 is small. Then set φ=ψ(xz1,) in (3.4) for each ψC0(R3) and take the limit 0, that is

    lim0[aR3uψ(xz1,)+dbR3|u|2R3uψ(xz1,)R3K(x)|u|2σuψ(xz1,)+R3μuψ(xz1,)+QR3χ(uψ(xz1,)+uψ(xz1,))]=0.

    Using the boundedness of u and d>0, we have

    dbR3|u|2R3uψ(xz1,)=o(1).

    We see that u is a nontrivial solution to aΔu+μu=k0|u|2σu in H1(R3) for some k0>0, which is impossible by Lemma 2.1.

    Setting φ=un,u in (3.4), we have

    aR3u(un,u)+dbR3|u|2R3u(un,u)R3K(x)|u|2σu(un,u)+R3μu(un,u)+QR3χ(u(un,u)+u(un,u))=0. (3.5)

    Then it follows from (3.3) that

    I(un,)+μn,J(un,),un,u=o(1)un,u.

    That is,

    aR3un,(un,u)+dbR3|un,|2R3un,(un,u)R3K(x)|un,|2σun,(un,u)+R3μn,un,(un,u)+Qn,R3χ(un,(un,u)+un,(un,u))=o(1)un,u. (3.6)

    We can show that for n large enough,

    R3|un,,|2R3un,(un,u)R3|u|2R3u(un,u)=R3|un,|2R3un,(un,u)R3|un,|2R3u(un,u)+R3|un,|2R3u(un,u)R3|u|2R3u(un,u)=R3|un,|2R3|un,u|2+(R3|un,|2R3|u|2)R3u(un,u)on(1), (3.7)

    where using the fact that un,u in H1(R3), it follows R3u(un,u)0. Thus from (3.5)–(3.7), we have

    aR3|(un,u)|2+μR3|un,u|2R3K(x)|un,u|2σ+2+QhR3χh[|(un,u)|2+|un,uh|2]+dbR3|un,|2R3|un,u|2=o(1).

    Noting also that {\int}_{\mathbb{R}^{3}}K(\hbar x)\left| u_{n, \, \hbar}-u_{\hbar}\right|^{2\sigma+2}\le C||u_{n, \, \hbar}-u_{\hbar}||^{2\sigma+2} and

    \begin{align*} ||u_{n,\,\hbar}-u_{\hbar}|| & = ||\sum\limits_{i = 1}^{\ell}u_{i}(\cdot-z_{n,\,i,\,\hbar})+v_{n,\,\hbar}-\sum\limits_{i = 1}^{\ell}u_{i}(\cdot -z_{i,\,\hbar})-v_{\hbar}||\\ &\le\sum\limits_{i = 1}^{\ell}||u_{i}(\cdot-z_{n,\,i,\,\hbar})-u_{i}(\cdot-z_{i,\,\hbar})||+||v_{n,\,\hbar}||+||v_{\hbar}||\\ &\le 2\rho+o_{n}(1), \end{align*}

    the following inequality holds:

    \begin{align*} C^{*}||u_{n,\,\hbar}-u_{\hbar}||^{2}&\le a\displaystyle {\int}_{\mathbb{R}^{3}}^{}\left | \nabla (u_{n,\,\hbar}- u_{\hbar})\right|^{2}+\mu_\hbar\displaystyle {\int}_{\mathbb{R}^{3}}|u_{n,\,\hbar}-u_{\hbar}|^{2}\\ &\le C||u_{n,\,\hbar}-u_{\hbar}||^ {2\sigma+2}+o(1), \end{align*}

    where C^{*} is a positive constant since a > 0 and \mu_{\hbar} > 0 . Making \rho smaller if necessary given C||u_{n, \, \hbar}-u_{\hbar}||^{2\sigma} < C^{*}/2 , it follows that u_{n, \, \hbar}\to u_{\hbar} in H^{1}(\mathbb{R}^{3}) . This completes the proof of Lemma 3.1.

    Proposition 3.2. For some \rho > 0 small and by letting \left \{ \hbar _{n}\right \}\subset\mathbb{R} , \left \{ \mu_{n}\right \}\subset\mathbb{R} and \left \{u _{n}\right \}\subset Z(\rho) satisfy that

    \begin{equation} \begin{array}{ll} \hbar _{n}\to 0^{+},\quad \underset{n\to \infty }{\limsup} I_{\hbar _{n}} (u_{n})\le b_{0}, \end{array} \end{equation} (3.8)
    \begin{equation} \begin{array}{ll} \left \| I'_{\hbar _{n} } (u_{n})+\mu _{n}J'(u_{n}) \right \|_{H^{-1}}\to 0, \end{array} \end{equation} (3.9)

    as n\to\infty . Then, \mu _{n}\to\mu _{0} holds, \lim_{n \to \infty } I_{\hbar _{n}} (u_{n}) = b_{0} and for some z_{n, \, i}\in \mathbb{R}^{3}, i = 1, \, 2, \, \cdots, \, \ell, we have

    \left \| u_{n}-\sum\limits_{i = 1}^{\ell}u_{i}(\cdot-z_{n,\,i}) \right \|\to 0\; and\ \mathrm{dist}(\hbar _{n}z_{n,\,i},\,\mathcal{K}_{i} )\to 0.

    Proof. The proof is similar to that in [31]. For the sake of completeness, we shall give the details.

    Step 1. We claim that \mu _{n}\to\overset{\sim}{\mu} > 0 .

    As \left \{u _{n}\right \}\subset Z(\rho) , we can write that u_{n} = { \sum_{i = 1}^{\ell}}u_{i}(x-z_{n, \, i })+v_{n} with z_{n, \, i}\in\frac{1}{\hbar}(\mathcal{K}_{i})^{\tau } and \left \| v_{n} \right \| \le \rho . It follows from u_{n}\in Z(\rho) and the boundedness of I_{\hbar _{n} } (u_{n}) that \left \| u_{n} \right \| and G_{\hbar_{n} }(u_{n}) are bounded. Besides, by (3.9) and J'(u_{n})u_{n} = 1 , we know that \mu _{n} is bounded. Then up to a subsequence, we can assume that \mu _{n}\to \overset{\sim}{\mu} in \mathbb{R} and u_{n}(\cdot +z_{n, \, i})\rightharpoonup w_{i}\in H^{1}(\mathbb{R}^{3}) . For \rho < \frac{1}{10}\min_{1\le i\le \ell}\left \| u_{i} \right \|_{L^{2}(B_{1}(0)) } , we have

    \liminf _{n\to \infty } \left \| u_{n}(\cdot +z_{n,\,i}) \right \|_{L^{2}(B_{1}(0))}\ge\left \| u_{i} \right \|_{L^{2}(B_{1}(0))}-\rho > 0.

    Notice that for any R > 0 , we can obtain that \left \| u_{i}-w_{i} \right\|_{L^{2}(B_{R}(0))}\le\rho . Hence,

    \begin{equation} \left \| u_{i} \right \|_{2}-\rho \le \left\|w_{i}\right\|_{2}\le\left\|u_{i}\right\|_{2}+\rho. \end{equation} (3.10)

    Then, if \rho is small enough, we know that w_{i} \ne0 . Next, testing (3.9) with \varphi(x-z_{n, \, i}) for each \varphi\in C_{0}^{\infty}(\mathbb{R}^{3}) , we deduce that

    \hbar_{n}^{d}b\displaystyle {\int}_{R^{3}}^{}\left|\nabla u_{n}(x+z_{n,\,i})\right|^{2}\displaystyle {\int}_{R^{3}}^{}\nabla u_{n}(x+z_{n,\,i})\nabla \varphi = o(1).

    Thus, w_{i} is a solution to -a\Delta w_{i}+\overset{\sim}{\mu} w_{i} = \overset{\sim}{k_{i}}\left | w_{i} \right |^{2\sigma}w_{i} in H^{1}(\mathbb{R}^{3}) with \underset{n\to\infty}\lim K(\hbar_{n}z_{n, i}) \to\overset{\sim}{k_{i}} \in[\underline{k}, \, \bar{k}] , where \underline{k} = \min _{x \in U_{i = 1}^{\ell} \bar{\Omega}_{i}} K(x) > 0 and \bar{k} = \max _{1 \leq i \leq \ell} k_{i} . Then, combining the Pohozaev identity with

    a\left|w_{i}\right|_{2}^{2}+\overset{\sim}{\mu}\left|w_{i}\right|_{2}^{2} = \overset{\sim}{k_{i}}\left|w_{i}\right|_{2\sigma+2}^{2\sigma+2},

    it follows that there exists a positive contant \overset{\sim }{\mu} .

    Step 2. u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i}) \to 0\ \mbox{in}\ L^{2\sigma+2}(\mathbb{R}^{3})\ \mbox{and}\ \mbox{dist}(\hbar _{n}z_{n, \, i}, \, \mathcal{K}_{i})\to 0 .

    We show that

    \tilde{v}_{n}: = u_{n}-\sum\limits_{i = 1}^{\ell}w_{i}(\cdot -z_{n,\,i})\to\ 0\ \mbox{in}\ L^{2\sigma +2}(\mathbb{R}^{3}).

    Otherwise, by Lions' lemma [17], there exists a sequence of points \left \{ z_{n} \right \}\subset\mathbb{R}^{3} such that

    \underset{n\to \infty }{\limsup }\left \|u_{n}-\sum\limits_{i = 1}^{\ell}w_{i}(\cdot -z_{n,\,i}) \right \| _{L^{2}({B_{1}(z_{n})})}^{2} > 0.

    Noting that \left | z_{n}- z_{n, \, i} \right |\to \infty i = 1, \, 2, \, \cdots, \, \ell , we have

    \begin{equation} \underset{n\to\infty }{\limsup }\displaystyle {\int}_{B_{1}(0)}^{}\left | u_{n}(\cdot+z_{n}) \right |^{2} > 0. \end{equation} (3.11)

    By (3.8), G_{\hbar_{n}}(u_{n})\le C holds for some C > 0 that is independent of \hbar . Then, we have that \mbox{ dist}\left (\hbar_{n}z_{n}, \, \cup _{i = 1}^{\ell}\Omega_{i} \right)\to0 . Up to a subsequence, we assume that \tilde{v}_{n}(x+z_{n})\rightharpoonup v_{0}\ne0 in H^{1}(\mathbb{R}^{3}) and K(\hbar_{n}z_{n}) \to k_{0}\in[\underline{k}, \, \bar{k}] , where k_{0} = k(y_{0}), \, y_{0}\in\cup _{i = 1}^{\ell}\Omega_{i} . Let D: = \left\{x\in\mathbb{R}^{3}|x_{3}\ge-M\right\} . For some i_{0} , if \underset{n \to \infty}\lim\frac{\mbox{dist}(\hbar_{n} z_{n}, \, \partial\Omega_{{i}_{0}})}{\hbar_{n}} = M < \infty , we get that \hbar_{n}z_{n}\to z_{0} as n\to\infty , where z_{0}\in\partial\Omega_{{i}_{0}} . Next, without loss of generality we can assume that v_{0}\in H_{0}^{1}(D) . Testing (3.9) with \varphi (\cdot-z_{n}) for any \varphi\in C_{0}^{\infty}(D) , we have

    \begin{equation*} \begin{aligned} &\underset{n\to\infty}\lim\Big[ a\displaystyle {\int}_{\mathbb{R}^{3}}\nabla u_{n}\nabla\varphi(x-z_{n})+\hbar_{n}^{d}b \displaystyle {\int}_{\mathbb{R}^{3}}|\nabla u_{n}|^{2}\displaystyle {\int}_{\mathbb{R}^{3}}\nabla u_{n}\nabla\varphi (x-z_{n})\\ &- \displaystyle {\int}_{\mathbb {R}^{3}}K(\hbar_{n} x)|u_{n}|^{2\sigma}u_{n}\varphi(x-z_{n})+\displaystyle {\int}_{\mathbb {R}^{3}}\mu_{n} u_{n}\varphi(x-z_{n})\\ &+Q_{\hbar_{n} }\displaystyle {\int}_{\mathbb{R} ^{3} }\chi _{\hbar_{n} }(\nabla u_{n } \nabla\varphi(x-z_{n})+u_{n } \varphi(x-z_{n}))\Big] = 0. \end{aligned} \end{equation*}

    Then by applying \left\|u_{n}\right\|_{{H}^{1}(\mathbb{R}^{3}\setminus\frac{1}{\hbar_{n}}\cup_{i = 1}^{\ell}\Omega_{i})}\le C\hbar_{n} and

    \hbar_{n}^{d}b \displaystyle {\int}_{\mathbb{R}^{3}}|\nabla u_{n}|^{2}\displaystyle {\int}_{\mathbb{R}^{3}}\nabla u_{n}\nabla\varphi (x-z_{n}) = o(1),

    we can obtain that v_{0} is a solution of -a\Delta u +\overset{\sim}{\mu} u = k_{0}\left | u\right |^{2\sigma}u in H_{0}^{1}(D) , which is impossible since this equation does not have a nontrivial solution on the half space according to [8]. Thus \underset{n \to \infty} \lim\mbox{ dist}(\hbar_{n} z_{n}, \, \partial\Omega_{{i}_{0}}) = +\infty and z_{n}\in\frac{1}{\hbar_{n}}\Omega_{{i}_{0}} . Now we test (3.9) with \varphi (\cdot-z_{n}) for any \varphi\in C_{0}^{\infty}(\mathbb{R}^{3}) to get

    -a\Delta v_{0} +\overset{\sim}{\mu} v_{0} = k_{0}\left | v_{0}\right |^{2\sigma}v_{0},

    where \overset{\sim}{\mu} > 0 , and \left | v_{0} \right |_{2}^{2} > C_{1} for some C_{1} > 0 that is independent of \rho .

    If we have chosen \rho small enough, then by the Brézis-Lieb lemma,

    \begin{align*} 1 = \underset{n\to\infty}{\lim}\left | u_{n} \right |_{2}^{2}& = \underset{n\to\infty}{\lim}\left|u_{n}(\cdot+z_{n,\,1})-v_{0}(\cdot+z_{n,\,1})\right|_{2}^{2}+\left|v_{0}\right|_{2}^{2}+o(1)\\ &\ge\sum\limits_{i = 1}^{\ell}\left | w_{i} \right |_{2}^{2}+\left | v_{0} \right |_{2}^{2}\\ &\ge\sum\limits_{i = 1}^{\ell}\left | u_{i} \right |_{2}^{2}-2\rho\sum\limits_{i = 1}^{\ell}\left | u_{i} \right |_{2}^{2}+\ell\rho^{2}+C_{1}\\ & > 1, \end{align*}

    which is a contradiction.

    Step 3. \left \| u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i}) \right \|\to 0\ \mbox{and}\ \lim_{n \to \infty } I_{\hbar_{n}} (u_{n}) = b_{0} .

    Testing (3.9) with u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i}) , given

    \hbar_{n}^{d}b\displaystyle {\int}_{\mathbb{R}^{3}}|\nabla u_{n}|^{2}\displaystyle {\int}_{\mathbb{R}^{3}}\nabla u_{n}\nabla (u_{n}-\sum\limits_{i = 1}^{\ell}w_{i}(x-z_{n,\,i})) = o(1),

    we can get that

    a(|\nabla u_{n}|_{2}^{2}-\sum\limits_{i = 1}^{\ell}|\nabla w_{i}|_{2}^{2})+\overset{\sim}{\mu}(|u_{n}|_{2}^{2}-\sum\limits_{i = 1}^{\ell}|w_{i}|_{2}^{2})\le o_{n}(1).

    Next, we have

    a|\nabla (u_{n}-\sum\limits_{i = 1}^{\ell}w_{i}(\cdot-z_{n,\,i}))|_{2}^{2}+\overset{\sim}{\mu}|u_{n}-\sum\limits_{i = 1}^{\ell}w_{i}(\cdot-z_{n,\,i})|_{2}^{2} = o_{n}(1),

    i.e., u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i})\to 0\ \mbox{in}\ H^{1}(\mathbb{R}^{3}) .

    On the other hand, by Lemma 2.2, we obviously get that \underset{n \to \infty } \lim I_{\hbar_{n}} (u_{n}) = b_{0} .

    In this section, let \rho be fixed in Proposition 3.2. We present the result as follows.

    Proposition 4.1. There exists \hbar _{0} > 0 such that for \hbar\in(0, \, \hbar _{0}) , I_{\hbar }|_{\mathcal{M} } has a critical point u_{\hbar }\in Z(\rho) . Moreover, \lim_{\hbar \to 0}I(u _{\hbar }) = b_{0} and the Lagrange multiplier \mu _{\hbar }\in\mathbb{R} satisfies

    \begin{equation} \lim\limits_{\hbar \to 0}\mu _{\hbar } = \mu _{0}, \quad I'_{\hbar }(u_{\hbar )}+\mu _{\hbar }J'(u_{\hbar }) = 0. \end{equation} (4.1)

    Remark 4.2. By Proposition 3.2, it is easy to verify that (4.1) holds if u_{\hbar} is a critical point of I_{\hbar}|_\mathcal{M} such that \limsup _{\hbar\to0}I_{u_{\hbar}}\le b_{0} .

    The proof of Proposition 4.1 can be obtained as in [31] by considering the following contradiction: \left \{ \hbar_{n} \right \} with \hbar_{n}\to0 such that for some sequence {b_{\hbar_{n}}}\to b_{0} , I_{\hbar} admits no critical points in \left \{ u\in Z(\rho)|I_{\hbar_{n}}(u)\le b_{\hbar_{n}} \right \} . For brevity, we denote \hbar = \hbar_{n}. Then from Lemma 3.1 and Proposition 3.2, there respectively exist \kappa _{0} > 0 and v > 0 independent of \hbar and v_{\hbar} > 0 such that

    \begin{equation} \begin{aligned} \left \| I_{\hbar}|_{\mathcal{M} }^{'}(u) \right \|_{T^{*}_{u}\mathcal{M}}&\ge v_{\hbar}, \mbox{for}\ u\in Z(\rho)\cap \left [ b_{0}-2\kappa _{0}\le I_{\hbar}\le b_{\hbar} \right ],\\ \left \| I_{\hbar}|_{\mathcal{M} }^{'}(u) \right \|_{T^{*}_{u}\mathcal{M}}&\ge v, \mbox{for}\ u\in (Z(\rho)\setminus Z(\rho/4))\cap \left [ b_{0}-2\kappa _{0}\le I_{\hbar}\le b_{\hbar} \right ], \end{aligned} \end{equation} (4.2)

    where

    [b_{1}\le I_{\hbar}] = \left \{ u\in H^{1}(\mathbb{R}^{3})|b_{1}\le I_{\hbar}(u) \right \},
    [I_{\hbar}\le b_{2}] = \left \{ u\in H^{1}(\mathbb{R}^{3})|I_{\hbar}(u)\le b_{2} \right \},
    \left [b_{1}\le I_{\hbar}\le b_{2}\right ] = \left \{ u\in H^{1}(\mathbb{R}^{3}) |b_{1}\le I_{\hbar}(u)\le b_{2} \right \},

    for any b_{1}, \, b_{2}\in \mathbb{R} .

    Thanks to (4.2), one can get the following deformation lemma.

    Lemma 4.3. Let v_{\hbar} and v be given as in (4.2). For any \kappa\in\left(0, \, \min\left\{\kappa_{0}, \, \frac{\rho v}{16}\right\}\right), there exists \hbar_{\kappa} > 0 such that for \hbar\in \left(0, \, \hbar_{\kappa}\right) there is a deformation \eta:\mathcal{M}\to\mathcal{M} that satisfied the following conditions:

    (a) \eta(u) = u if u\in\mathcal{M}\setminus (Z(\rho)\cap \left [ b_{0}-2\kappa \le I_{\hbar} \right ]).

    (b) \ I_{\hbar}(\eta (u))\le I_{\hbar}(u) if u\in\mathcal{M}.

    (c) \eta(u)\in Z(\rho)\cap \left [ I_{\hbar}\le b_{0}-\kappa \right ] if u\in Z(\rho/4)\cap\left [ I_{\hbar}\le b_{\hbar} \right ].

    To give the proof of Lemma 4.3, we borrow some ideas from [5,6,31] in the L^{2} -subcritical case and L^{2} -supercritical case.

    For every \delta > 0 , we denote

    S_{\delta}: = \left\{s\in S_{+}^{\ell-1}|\left|s-s^{0}\right|\le \delta\right\},

    where s^{0} = (|u_{1}|_{2}, \, \cdots, \, |u_{\ell}|_{2}) . Fix q_{i}\in \mathcal{K}_{i} and q_{i, \, \hbar} = \frac{1}{\hbar}q_{i} for i = 1, \, 2, \, \cdots, \, \ell and define the (\ell-1) -dimensional initial path by

    \xi_{\hbar}(s) = B_{\hbar}\sum\limits_{i = 1 }^{\ell}w_{s_{i}}(\cdot-q_{i,\,\hbar}),

    where B_{\hbar}: = \left | \sum_{i = 1}^{\ell}w_{s_{i}}(\cdot-q_{{i}, \, \hbar}) \right |_{2}^{-1} . Note that we can fix \delta > 0 small enough such that

    \xi_{\hbar}(s)\in Z(\rho/4)\ \mbox{for}\ s\in S_{\delta}

    and

    B_{\hbar}\to1 \ \mbox{as}\ \hbar\to0 \ \mbox{uniformly in} \ S_{\delta}.

    Define

    b_{\hbar}: = \underset{s\in S_{\delta}}{\max} I_{\hbar}(\xi_{\hbar}(s)).

    Lemma 4.4. \lim_{\hbar\to0}b_{\hbar} = b_{0} and fix any \kappa\in(0, \, \min\left\{\kappa_{0}, \, \frac{\rho v}{16}\right\}) such that

    \begin{equation} \underset{s\in\partial S_{\delta}}{\sup }I_{\hbar}(\xi_{\hbar}(s)) < b_{0}-2\kappa, \end{equation} (4.3)

    where \partial S_{\delta}: = \left\{s\in S_{+}^{\ell}|\left|s-s^{0}\right| = \delta\right\} .

    Proof. Since

    \hbar^{d}b\left(\displaystyle {\int}_{\mathbb{R}^{3}}\left|\nabla\xi_{\hbar}\right|^{2}\right)^{2}\to0\quad \mbox{as}\quad \hbar\to0,

    one can deduce that

    I_{\hbar}(\xi_{\hbar}(s))\to \sum\limits_{i = 1}^{\ell}I_{i}(w_{s_{i}})\ \mbox{as}\ \hbar\to0 \ \mbox{uniformly for} \ s\in S_{\delta}.

    By Lemma 2.3(a), we have

    \underset{s\in\partial S_{\delta}}{\sup }I_{\hbar}(\xi_{\hbar}(s)) < b_{0}-2\kappa.

    Proof of Proposition 4.1 in the L^{2} -subcritical case. By Lemma 4.3 and (4.3), we have

    \begin{equation} \eta(\xi_{\hbar}(s)) = \xi_{\hbar}(s)\ \mbox{for}\ s\in \partial S_{\delta}, \end{equation} (4.4)
    \begin{equation} I_{\hbar}(\eta(\xi_{\hbar}(s)))\le b_{0}-\kappa\ and\ \eta(\xi_{\hbar}(s))\in Z(\rho)\ \mbox{for} \ s\in S_{\delta}. \end{equation} (4.5)

    Define

    \Psi _{i,\hbar} = \left ( \displaystyle {\int}_{\frac{1}{\hbar}\Omega_{i} }^{}\left | u \right |^{2} \right )^{\frac{1}{2}} \left (\sum\limits_{i = 1}^{\ell} \displaystyle {\int}_{\frac{1}{\hbar}\Omega_{i} }^{}\left | u \right |^{2}\right )^{-\frac{1}{2}},\ \mbox{for}\ u\in \mathcal{M}.

    Similar to the case in [31], there exists s^{1}\in S_{\delta} such that \Psi_{i, \, \hbar}(\eta(\xi_{\hbar}(s^{1}))) = s_{i}^{0} = \left|u_{i}\right|_{2} . Denote

    \begin{equation} u_{0,\,\hbar}: = \eta(\xi_{\hbar}(s^{1})), \ u_{i,\,\hbar}: = \gamma_{i,\,\hbar}u_{0,\,\hbar}, \end{equation} (4.6)

    where \gamma _{i, \, \hbar}\in C_{0}^{\infty}(\frac{1}{\hbar}(\Omega_{i}^{'}), \, \left[0, \, 1\right]) is a cut-off function such that \gamma _{i, \, \hbar} = 1 on \frac{1}{\hbar}\Omega_{i} and \left|\nabla\gamma_{i, \, \hbar}\right|\le C\hbar for each i = 1, \, 2, \, \cdots, \, \ell and some C > 0 ; \Omega_{i}^{'} is an open neighborhood of \bar{\Omega}_{i} . By (4.5), we have that G_{\hbar}(u_{0, \, \hbar})\le C for some C > 0 that is independent of \hbar , which implies that

    \begin{equation} \left \| u_{0,\,\hbar} \right \|_{H^{1}(\mathbb{R}^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{\hbar}\Omega_{i})}\le C\hbar. \end{equation} (4.7)

    Then

    \begin{equation} \left | u_{i,\,\hbar} \right |_{2} = \left|u_{i}\right|_{2}+o_{\hbar}(1) \end{equation} (4.8)

    and

    \begin{equation} I_{i}(u_{i})\le I_{i}(u_{i,\,\hbar})+o_{\hbar}(1). \end{equation} (4.9)

    Hence from (4.5)–(4.9), we have

    b_{0}-\kappa\ge I_{\hbar}(u_{0,\,\hbar}) \ge\sum\limits_{i = 1}^{\ell}I_{i}(u_{i,\,\hbar})+o_{\hbar}(1) \ge\sum\limits_{i = 1}^{\ell}I_{i}(u_{i})+o_{\hbar}(1) = b_{0}+o_{\hbar}(1),

    which is a contradiction. This completes the proof.

    Fix q_{i}\in \mathcal{K}_{i} and denote q_{i, \, \hbar} = \frac{1}{\hbar}q_{i} ; we set

    \zeta_{\hbar}(s) = \bar{B}_{\hbar} \sum\limits_{i = 1 }^{\ell}{t_{i}}^{3/2 }u_{i}(t_{i} (\cdot-q_{i,\,\hbar}) )\ \mbox{for}\ t = (t_{1},\,t_{2},\,\cdots,\,t_{\ell})\in\left(0,\,+\infty\right)^{\ell},

    where \bar{B}_{\hbar}: = \left |{ \sum_{i = 1}^{\ell}}{t_{i}^{3/2} }u_{i}(t_{i} (\cdot-q_{i, \, \hbar})) \right |_{2}^{-1} .

    Define

    b_{\hbar}: = \underset{t\in [ 1-\bar{\delta },\,1+\bar{\delta } ] ^{\ell}}{\max} I_{\hbar}(\zeta_{\hbar}(t)).

    Note that we can fix \bar{\delta } > 0 small enough such that

    \zeta _{\hbar}(t)\in Z(\rho/4) \ \mbox{for}\ t\in [ 1-\bar{\delta },\,1+\bar{\delta } ] ^{\ell},

    and \bar{B}_{\hbar}\to1 holds. Note also that

    I_{i}(u_{i}) > I_{i}(t_{i}^{3/2}u_{ i}(t_{i}\cdot))\ \mbox{for} \ t_{i}\in [ 1-\bar{\delta } ,\,1+\bar{\delta } ]\setminus\left\{1\right\}.

    Since

    \hbar^{d}b\left(\displaystyle {\int}_{\mathbb{R}^{3}}\left|\nabla\zeta_{\hbar}\right|^{2}\right)^{2}\to0\quad \mbox{as}\quad \hbar\to0,

    and

    I_{\hbar}(\zeta_{\hbar}(t))\to \sum\limits_{i = 1}^{\ell}I_{i}(t_{i}^{3/2}u_{i}(t_{i}\cdot))\ \mbox{as}\ \hbar\to0\ \mbox{uniformly for}\ t\in [ 1-\bar{\delta } ,\,1+\bar{\delta } ] ^{\ell},

    one can get the result as in [31].

    Lemma 4.5. \lim_{\hbar\to0}b_{\hbar} = b_{0} and fix any \kappa\in(0, \, \min\left\{\kappa_{0}, \, \frac{\rho v}{16}\right\}) such that

    \begin{equation} \underset{t\in\partial [1-\bar{\delta},\,1+\bar{\delta}]^{\ell} }{\sup }I_{\hbar}(\zeta_{\hbar}(t)) < b_{0}-2\kappa. \end{equation} (4.10)

    Proof of Proposition 4.1 in the L^{2} -supercritical case. By Lemma 4.3 and (4.10),

    \begin{equation} \eta(\zeta_{\hbar}(t)) = \zeta_{\varepsilon}(t)\ \mbox{if}\ t\in\partial [1-\bar{\delta},\,1+\bar{\delta}]^{\ell}, \end{equation} (4.11)
    \begin{equation} I_{\hbar}(\eta(\zeta_{\hbar}(t)))\le b_{0}-\kappa\ \mbox{and}\ \eta(\zeta_{\hbar}(t))\in Z(\rho)\ \mbox{for}\ t\in [1-\bar{\delta},\,1+\bar{\delta}]^{\ell}. \end{equation} (4.12)

    Define

    \Phi _{i,\hbar} = \left ( \displaystyle {\int}_{\frac{1}{\hbar}\Omega_{i} }^{}\left |\nabla u \right |^{2} \right )^{\frac{1}{2-3\sigma}} \left (\frac{3\sigma k_{i}}{(2+2\sigma)a} \displaystyle {\int}_{\frac{1}{\hbar}\Omega_{i} }^{}\left | u \right |^{2\sigma+2}\right )^{-\frac{1}{2-3\sigma}},\ \mbox{for}\ u\in \mathcal{M}.

    Similar to the case in [31], there exists t^{1}\in[1-\bar{\delta}, \, 1+\bar{\delta}]^{\ell} such that

    \begin{equation} \Phi_{i,\,\hbar}(\eta(\zeta_{\hbar}(t^{1}))) = 1, \ i = 1,\,2,\,\cdots,\,\ell. \end{equation} (4.13)

    We denote

    \bar{u}_{0,\hbar}: = \eta(\zeta_{\hbar}(t^{1})), \quad \bar{u}_{i,\,\hbar}: = \gamma_{i,\,\hbar}\bar{u}_{0,\,\hbar}\left(\sum\limits_{i = 1}^{\ell}\left|\gamma_{i,\,\hbar}\bar{u}_{0,\hbar}\right|_{2}^{2}\right)^{-\frac{1}{2}}.

    Similar to (4.7) and (4.8), we have

    \begin{equation} \left\|\bar{u}_{0,\,\hbar}\right\|_{H^{1}(\mathbb{R}^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{\hbar}\Omega _{i})} = o_{\hbar}(1) \end{equation} (4.14)

    and

    \begin{equation} \sum\limits_{i = 1}^{\ell}\left | \gamma_{i,\,\hbar}\bar{u}_{0,\,\hbar} \right |_{2}^{2} = 1+o_{\hbar}(1). \end{equation} (4.15)

    From (4.13)–(4.15), we have

    t_{i,\,\hbar}: = \left(\left|\nabla\bar{u}_{i,\,\hbar}\right|_{2}^{2}\right)^{\frac{1}{2-3\sigma}}\left(\frac{3\sigma k_{i}}{(2+2\sigma)a}\left|\bar{u}_{i,\,\hbar}\right|_{2\sigma+2}^{2\sigma+2}\right)^{\frac{1}{3\sigma-2}} = \Phi_{i,\,\hbar}(\bar{u}_{0,\hbar})+o_{\hbar}(1) = 1+o_{\hbar}(1).

    A direct calculation shows that

    t^{*}: = \left(\left|t_{1,\,\hbar}^{-\frac{3}{2}}\bar{u}_{1,\,\hbar}(t_{1,\,\hbar}^{-1}\cdot)\right|_{2},\,\left|t_{2,\,\hbar}^{-\frac{3}{2}}\bar{u}_{2,\,\hbar}(t_{2,\,\hbar}^{-1}\cdot)\right|_{2},\,\cdots,\,\left|t_{\ell,\,\hbar}^{-\frac{3}{2}}\bar{u}_{\ell,\,\hbar}(t_{\ell,\,\hbar}^{-1}\cdot)\right|_{2} \right)\in S_{+}^{\ell-1}

    and

    \left|\nabla \left(t_{i,\,\hbar}^{-\frac{3}{2}}\bar{u}_{i,\,\hbar}(t_{i,\,\hbar}^{-1}\cdot)\right)\right|_{2}^{2} = \frac{3\sigma k_{i}}{(2+2\sigma)a}\left|t_{i,\,\hbar}^ {-\frac{3}{2}}\bar{u}_{i,\,\hbar}(t_{i,\,\hbar}^{-1}\cdot)\right|_{2\sigma+2}^{2\sigma+2}.

    Hence by the definition of b_{s_{i}} , we have

    \sum\limits_{i = 1}^{\ell}I_{i}(u_{i}) = b_{0}\le\sum\limits_{i = 1}^{\ell}I_{i}\left(t_{i,\,\hbar}^{-\frac{3}{2}}\bar{u}_{i,\,\hbar}(t_{i,\,\hbar}^{-1}\cdot)\right) = \sum\limits_{i = 1}^{\ell}I_{i}(\bar{u}_{i,\,\hbar})+o_{\hbar}(1).

    Similarly, one can get a contradiction.

    Let u_{\hbar} be the critical point of the modified function I_{\hbar} given in Proposition 4.1.

    Completion of proof of Theorem 1.1.

    Proof. We show that there exists c > 0 independent of \hbar such that

    \begin{equation} \left \| u_{\hbar } \right \|^{2}_{H^{1}(\mathbb{R }^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{ \hbar}(\mathcal{K}_{i} )^{{\tau}} )}\le e^{-\frac{C }{\hbar } }. \end{equation} (5.1)

    We adopt some arguments from [3,31]. Set \left \lfloor 2\hbar ^{-1}\tau \right \rfloor-1: = n_{\hbar } . For n = 1, \, 2, \, \cdots, \, n_{\hbar } , we take \phi _{n}\in C^{1}(\mathbb{R }^{3}, \, \left [ 0, \, 1 \right ]) such that

    \left\{\begin{matrix} \phi _{n}(x) = 0,&\mbox{if}\ x\in\ \mathbb{R}^{3}\setminus E_{n}, \\ \phi _{n}(x) = 1,&\mbox{if}\ x\in E_{n+1},\\ \left | \nabla \phi _{n}(x) \right |\le 2 ,&x\in\mathbb{R}^{3}, \end{matrix}\right.

    where E_{n}: = \left \{ x\in \mathbb{R}^{3}|\mbox{dist}(x, \, \cup _{i = 1}^{\ell}\frac{1}{ \hbar}(\mathcal{K}_{i})^{\frac{\tau}{2} }) > n-1\right \} . Then by Proposition 3.2,

    \begin{equation} \lim\limits_{\hbar \to 0} \left \| u_{\hbar } \right \|_{H^{1}(E_{1}) }\le\lim\limits_{\hbar\to 0}\sum\limits_{i = 1}^{\ell} \left \| u_{i}\right \|_{H^{1}(\mathbb{R}^{3}\setminus B_{\hbar\tau}(0)) } = 0. \end{equation} (5.2)

    Note that for each n = 1, \, 2, \, \cdots, \, n_{\hbar } ,

    \mbox{supp}\chi_{\hbar } = \mathbb{R}^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{\hbar }\Omega _{i}\subset\mathbb{R}^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{\hbar }(\mathcal{K}_{i} )^{\tau}\subset \phi _{n}^{-1}(1).

    Since \left \langle I'_{\hbar }(u_{\hbar })+\mu_{\hbar }J'(u_{\hbar }), \, \phi _{n}u_{\hbar }\right \rangle = 0 , we have

    \begin{equation} \begin{aligned} &a\displaystyle {\int}_{\mathbb{R} ^{3} }^{ }\nabla u_{\hbar }\nabla( \phi _{n} u_{\hbar })+\hbar^{d}b\displaystyle {\int}_{\mathbb{R} ^{3} }^{ }\left | \nabla u_{\hbar } \right |^{2}\displaystyle {\int}_{\mathbb{R} ^{3} }^{ }\nabla u_{\hbar }\nabla( \phi _{n} u_{\hbar })\\ &- \displaystyle {\int}_{\mathbb{R} ^{3} }^{ }K(\hbar x)\left | u_{\hbar } \right |^{2\sigma +2}\phi _{n}+\displaystyle {\int}_{\mathbb{R} ^{3} }^{ }\mu _{\hbar }u_{\hbar}^{ 2}\phi _{n}\\ = &-4\hbar ^{-1}G_{\hbar }(u_{\hbar })^{\frac{1}{2} }\displaystyle {\int}_{\mathbb{R} ^{3} }^{ }\chi _{\hbar } (\nabla u_{\hbar }\nabla( \phi _{n} u_{\hbar })+u_{\hbar}^{ 2}\phi _{n})\\ = &-4\hbar ^{-1}G_{\hbar }(u_{\hbar })^{\frac{1}{2} }\displaystyle {\int}_{\mathbb{R}^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{\hbar }\Omega _{i}}^{ } (\nabla u_{\hbar }\nabla( \phi _{n} u_{\hbar})+u_{\hbar}^{ 2}\phi _{n}) \\ = &-4\hbar ^{-1}G_{\hbar }(u_{\hbar })^{\frac{1}{2} }\displaystyle {\int}_{\mathbb{R}^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{\hbar }\Omega _{i}}^{ }(\left | \nabla u_{\hbar } \right |^{2}+u_{\hbar}^{ 2})\le 0. \end{aligned} \end{equation} (5.3)

    Therefore, by (5.3) and the Sobolev embedding,

    \begin{align*} &\min\left \{ a,\,\frac{\mu _{0}}{2}\right \}\left \| u_{\hbar }\right \|_{H^{1} (E_{n+1} )}^{2}\\ &\le \displaystyle {\int}_{\mathbb{R}^{3}}\phi _{n} (a\left | \nabla u_{\hbar } \right |^{2}+{\mu}_{\hbar } u_{\hbar}^{ 2}) \\ & \le \displaystyle {\int}_{\mathbb{R} ^{3} }K(\hbar x)\left | u_{\hbar } \right |^{2\sigma +2}\phi _{n}-a\displaystyle {\int}_{\mathbb{R}^{3}} u_{\hbar }\nabla u_{\hbar }\nabla\phi _{n}-\hbar^{d}b\displaystyle {\int}_{\mathbb{R} ^{3} }\left | \nabla u_{\hbar } \right |^{2}\displaystyle {\int}_{\mathbb{R} ^{3} }\nabla u_{\hbar }\nabla( \phi _{n} u_{\hbar }) \\ &\le C\left \| u_{\hbar } \right \|_{H^{1}(E_{n}) }^{2\sigma +2} + a\left \| u_{\hbar } \right \|_{H^{1}(E_{n}) }^{2}- a\left \| u_{\hbar } \right \|_{H^{1}(E_{n+1}) }^{2}-\hbar^{d}b\displaystyle {\int}_{\mathbb{R} ^{3} }\left | \nabla u_{\hbar } \right |^{2}\displaystyle {\int}_{\mathbb{R} ^{3} }\nabla u_{\hbar }\nabla( \phi _{n} u_{\hbar }) \\ &\le (a+C\left \| u_{\hbar } \right \|_{H^{1}(E_{1}) }^{2\sigma}+o_{\hbar}(1))\left \| u_{\hbar } \right \|_{H^{1}(E_{n}) }^{2}-(a+o_{\hbar}(1))\left \| u_{\hbar } \right \|_{H^{1}(E_{n+1}) }^{2}, \end{align*}

    where -\hbar^{d}b {\int}_{\mathbb{R} ^{3} }^{ }\left | \nabla u_{\hbar } \right |^{2} {\int}_{\mathbb{R} ^{3} }\nabla u_{\hbar }\nabla(\phi _{n} u_{\hbar })\le o_{\hbar}(1)(2||u_{\hbar}||^{2}_{H^{1}(E_{n})}-||u_{\hbar}||^{2}_{H^{1}(E_{n+1})}) as \hbar\rightarrow0 . By (5.2), we have

    \left\|u_{\hbar }\right \|_{H^{1}(E_{n+1})}^{2} \le \theta _{\hbar}^{-1} \left \| u_{\hbar } \right \|_{H^{1}(E_{n}) }^{2},

    where

    \theta _{\hbar} : = \frac{a+ \min\left \{ a,\,\frac{\mu _{0} }{2} \right \} +o_{\hbar}(1)}{a+o_{\hbar}(1) }\rightarrow1+\min\left \{ 1,\,\frac{\mu _{0} }{2a} \right \} \,\,\mbox{as}\,\,\hbar\rightarrow0.

    Nothing that n_{\hbar }\ge\frac{ \tau }{\hbar} for small values of \hbar , one can take some \theta_0 > 1 and obtain

    \left \| u_{\hbar } \right \|_{H^{1}(\mathbb{R }^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{ \hbar}(\mathcal{K}_{i} )^{{\tau}} )}^{2}\le \left \| u_{\hbar } \right \|_{H^{1}(E_{n_{\hbar }+1 }) }^{2}\le \theta _{0}^{-n_{\hbar } }\left \| u_{\hbar } \right \|_{H^{1}(E_{1}) }^{2}\le e^{-\frac{\tau \ln_{}{\theta _{0} } }{\hbar } }.

    It follows that for small values of \hbar , G_{\hbar}(u_{\hbar}) = 0 . So u_{\hbar} is a solution to the original problem (3.1) for small values of \hbar .

    Zhidan Shu: Writing-original draft and Writing-review & editing; Jianjun Zhang: Methodology and Supervision. All authors equally contributed to this manuscript and approved the final version.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to express their sincere gratitude to the anonymous referee for his/her valuable suggestions and comments.

    The authors declare no conflicts of interest.



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