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Service scheduling optimization for multiple tower cranes considering the interval time of the cross-tasks


  • Received: 02 November 2022 Revised: 18 December 2022 Accepted: 28 December 2022 Published: 18 January 2023
  • The key issues that have always affected the production yield of the construction industry are delays and cost overruns, especially when dealing with large-scale projects and super-high buildings in which multiple tower cranes with overlapping areas are often deployed because of urgent due date and limited space. The service scheduling of tower cranes, which act as the crucial site equipment for lifting and transporting materials, is one of the main problems not only related to the construction progress and project cost but also affecting equipment health, and it may bring security risks. The current work presents a multi-objective optimization model for a multiple tower cranes service scheduling problem (MCSSP) with overlapping areas while achieving maximum interval time of cross-tasks and minimum makespan. For the solving procedure, NSGA-Ⅱ is employed with double-layer chromosome coding and simultaneous coevolutionary strategy design, which can obtain a satisfactory solution through effectively allocating tasks within overlapping areas to each crane and then prioritizing all the assigned tasks. The makespan was minimized, and stable operation of tower cranes without collision was achieved by maximizing the cross-tasks interval time. A case study of the megaproject Daxing International Airport in China has been conducted to evaluate the proposed model and algorithm. The computational results illustrated the Pareto front and its non-dominant relationship. The Pareto optimal solution outperforms the results of the single objective classical genetic algorithm in terms of overall performance of makespan and interval time of cross-tasks. It also can be seen that significant improvement in the time interval of cross-tasks can be achieved at the cost of a tiny increase in makespan, which means effective avoidance of the tower cranes entering the overlapping area at the same time. This can help eliminate collision, interference and frequent start-up and braking of tower cranes, leading to safe, stable and efficient operation on the construction site.

    Citation: Jing Yin, Jiahao Li, Yifan Fang, Ahui Yang. Service scheduling optimization for multiple tower cranes considering the interval time of the cross-tasks[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 5993-6015. doi: 10.3934/mbe.2023259

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  • The key issues that have always affected the production yield of the construction industry are delays and cost overruns, especially when dealing with large-scale projects and super-high buildings in which multiple tower cranes with overlapping areas are often deployed because of urgent due date and limited space. The service scheduling of tower cranes, which act as the crucial site equipment for lifting and transporting materials, is one of the main problems not only related to the construction progress and project cost but also affecting equipment health, and it may bring security risks. The current work presents a multi-objective optimization model for a multiple tower cranes service scheduling problem (MCSSP) with overlapping areas while achieving maximum interval time of cross-tasks and minimum makespan. For the solving procedure, NSGA-Ⅱ is employed with double-layer chromosome coding and simultaneous coevolutionary strategy design, which can obtain a satisfactory solution through effectively allocating tasks within overlapping areas to each crane and then prioritizing all the assigned tasks. The makespan was minimized, and stable operation of tower cranes without collision was achieved by maximizing the cross-tasks interval time. A case study of the megaproject Daxing International Airport in China has been conducted to evaluate the proposed model and algorithm. The computational results illustrated the Pareto front and its non-dominant relationship. The Pareto optimal solution outperforms the results of the single objective classical genetic algorithm in terms of overall performance of makespan and interval time of cross-tasks. It also can be seen that significant improvement in the time interval of cross-tasks can be achieved at the cost of a tiny increase in makespan, which means effective avoidance of the tower cranes entering the overlapping area at the same time. This can help eliminate collision, interference and frequent start-up and braking of tower cranes, leading to safe, stable and efficient operation on the construction site.



    In this paper, we consider the following Cauchy problem for the nonlinear Schrödinger equation with an attractive inverse-power potential

    {itψ+Δψ+γ|x|σψ+|ψ|αψ=0,(t,x)R×RN,ψ(0,x)=ψ0(x),xRN, (1.1)

    where N3, ψ:[0,T)×RNC is an unknown complex valued function with 0<T, ψ0H1(RN), γ(0,+), σ(0,2) and 4N<α<4N2.

    In the case σ=1, i.e., the operator Δ+γ|x| with Coulomb potential, (1.1) describes the situation where the wave function of an electron, satisfying the Schrödinger evolution equation, is influenced by m nuclei, see [19] for a broader introduction. In the case 0<σ<2, i.e., the operator Δ+γ|x|σ with slowly decaying potentials, we refer the readers to [15] and references therein. It also attracted a great deal of attention from mathematicians in recent years, see, e.g. [4,13,14,23].

    Recently, this type of equations has been studied widely in [1,4,5,6,7,8,9,10,11,13,17,20,22,23,24,25,26]. In particular, Eq (1.1) enjoys a class of special solutions, which are called standing waves, namely solutions of the form ψ(t,x)=eiωtu(x), where ωR is a frequency and uH1(RN) satisfies the elliptic equation

    Δu+ωuγ|x|σu|u|αu=0. (1.2)

    The Eq (1.2) is variational and its action functional is defined by

    Sω(u):=Eγ(u)+ω2u2L2, (1.3)

    where the corresponding energy functional Eγ(u) is defined by

    Eγ(u):=12u2L2γ2RN|u(x)|2|x|σdx1α+2uα+2Lα+2. (1.4)

    For the evolutional type Eq (1.1), one of the most interesting problems is to consider the stability of standing waves, which is defined as follows:

    Definition 1.1. Assume u is a solution of (1.2). The standing wave eiωtu(x) is called orbitally stable in H1(RN), if for every ϵ>0, there exists δ>0 such that if ψ0H1(RN) satisfies

    ψ0uH1δ,

    then the solution ψ(t) to (1.1) with ψ|t=0=ψ0 satisfies

    suptRinfθR,yRNψ(t,)eiθu(y)H1ϵ.

    There are two main methods to study the stability of standing waves. One is the stability or instability criterion in [16] proposed by Grillakis, Shatah and Strauss. It says that the standing wave eiωtuω is stable if ωuω2L2>0 and unstable if not, see [12] for more details. However, it is hard to estimate the sign of ωuω2L2 for nonlinear Schrödinger equations without scaling invariance, e.g. (1.1). The other is the constrained minimization approach introduced by Cazenave and Lions in [3]. In this paper, we thus take into account the orbital stability of the set of minimizers by using the method from Cazenave and Lions in [3].

    Now we recall the following definition of the orbital stability of the set M.

    Definition 1.2. The set MH1(RN) is called orbitally stable if for every ϵ>0, there exists δ>0 such that for any initial data ψ0H1(RN) satisfying

    infuMψ0uH1<δ,

    the corresponding solution u to (1.1) satisfies

    infuMψ(t)uH1<ϵ,t>0.

    Based on the above definition, in order to study the stability, we require that the solution of (1.1) exists globally, at least for initial data ψ0 enough close to M. In the L2-subcritical case, all solutions for (1.1) exist globally. Therefore, the stability of standing waves has been studied widely in this case, see, e.g. [3,4,14,23]. However, in the L2-supercritical case, we know that the solution of (1.1) with small initial data exists globally, but for some large initial data, the solution of (1.1) may blow up in finite time by the local well-posedness theory of NLS, see [2] for further inference.

    For Eq (1.1), in the mass subcritical and critical cases, i.e., 0<α4N, Dinh in [4] and Li-Zhao in [23] studied the stability of set of minimizers by using the concentration compactness principle. In the mass supercritical case, i.e., 4N<α<4N2, Fukaya-Ohta in [14] proved the strong instability of standing wave eiωtu under the assumption 2λEγ(uλ)|λ=10 with uλ(x):=λN2u(λx). Therefore, whether there are stable standing waves is an interesting problem in the mass supercritical case. In this paper, we will solve this problem by considering the following minimization problem

    mγ(a):=infuS(a)Eγ(u), (1.5)

    where

    S(a)={uH1(RN),u2L2=a}. (1.6)

    In the L2-supercritical case, the energy Eγ(u) is unbounded from below on S(a). Actually, when 4N<α<4N2, taking uS(a) and setting uλ(x):=λN2u(λx), then u2L2=uλ2L2=a, and we will find that

    Eγ(uλ)=λ22u2L2λσγ2RN|u(x)|2|x|σdxλNα2α+2uα+2Lα+2,

    as λ, where 2<Nα2<2NN2. Thus, we can not discuss the global minimization problem (1.5) to study the existence and stability of standing waves for (1.1). However, inspired by the thought in [18], we consider the further constrained minimization problem:

    mγ(a,r0):=infuV(a)Eγ(u). (1.7)

    The sets V(a) and V(a) are given by

    V(a):={uS(a):u2L2<r0},V(a):={uS(a):u2L2=r0},

    for an appropriate r0>0, depending only on a0 but not on a(0,a0). However, compared with the case (i.e., γ=0) considered in [18], the energy functional of (1.2) is not invariant under the scaling transform due to the inhomogeneous nonlinearity γ|x|σu. Furthermore, we cannot prove the preconceived limiting problem of the energy functional under translation sequences. In fact, we can solve the minimization problem (1.7) by proving the boundedness of the translation sequences. We now denote all the energy minimizers of (1.7) by

    M(a):={uV(a):Eγ(u)=mγ(a,r0)}.

    Our main results are as follows:

    Theorem 1.3. Let N3, 0<σ<2 and 4N<α<4N2. Then there exists a γ0>0 sufficiently small such that 0<γ<γ0 and for any a(0,a0), the following properties hold:

    (i) M(a)V(a).

    (ii) M(a) is orbitally stable.

    This paper is organized as follows: In Section 2, we firstly give some variational problems and then prove that the solution ψ(t) of (1.1) with the initial data ψ0 exists globally. In Section 3, we will show that M(a) is orbitally stable.

    In this section, we first establish the following classical inequalities: If N2 and α[2,2NN2), then the following Gagliardo-Nirenberg inequality holds that

    uα+2Lα+2C(α)uα+2Nα2L2uNα2L2,uH1(RN), (2.1)

    where C(α) is the sharp constant. Let 1p<, if σ<N is such that 0σp, then |u()|p||σL1(RN) for any uH1(RN). Moreover,

    RN|u(x)|p|x|σdxC1upσLpuσLp,uH1(RN), (2.2)

    where C1=(pNσ)σ.

    Secondly, by (2.1) and (2.2), we have

    Eγ(u)u2L2(12γC12u2σL2uσ2L2C(α)α+2uα+2Nα2L2uNα22L2)=u2L2f(u2L2,u2L2). (2.3)

    Next, setting

    η0=2σ,η1=σ2,η2=α+2Nα2,η3=Nα22,

    we now consider the function f(a,r) defined on (0,)×(0,) by

    f(a,r)=12γC12aη02rη12C(α)α+2aη22rη32, (2.4)

    and, for each a(0,), its restriction ga(r) defined on (0,) by rga(r)=f(a,r). For further reference, note that for any N3, η0(0,2), η1(2,0), η2(0,4N) and η3(0,4N2).

    Lemma 2.1. [2,Theorem 9.2.6] Let N3, 0<σ<2 and 4N<α<4N2. Then for any initial data ψ0H1(RN), there exists T=T(ψ0H1) such that (1.1) admits a unique solution with ψ(0)=ψ0. Let [0,T) be the maximal time interval on which the solution ψ is well-defined, if T<, then ψ(t)H1 as tT.Moreover, for all 0<t<T, the solution ψ(t) satisfies the following conservations of mass and energy

    ψ(t)2L2=ψ02L2,Eγ(ψ(t))=Eγ(ψ0),

    where the energy Eγ(u) is defined by (1.4).

    Lemma 2.2. For every a>0, the function ga(r)=f(a,r) has a unique global maximum and the maximum value satisfies

    {maxr>0ga(r)>0,ifa<a0,maxr>0ga(r)=0,ifa=a0,maxr>0ga(r)<0,ifa>a0,

    where

    a0:=[12K]2(η3η1)η0η3η1η2>0, (2.5)

    with

    K:=γC12[C1η1(α+2)2C(α)η3]η1η3η1+C(α)α+2[C1η1(α+2)2C(α)η3]η3η3η1. (2.6)

    Proof. According to the definition of ga(r), we have

    ga(r)=γC1η14aη02rη121C(α)η32(α+2)aη22rη321.

    Hence, the equation ga(r)=0 has a unique solution given by

    ra=[γC1η1(α+2)2C(α)η3]2η3η1aη0η2η3η1. (2.7)

    Noticing that ga(r) as r0 and ga(r) as r, we obtain that ra is the unique global maximum point of ga(r). Actually, the maximum value is

    maxr>0ga(r)=12γC12aη02rη12aC(α)α+2aη22rη32a=12γC12aη02[C1η1(α+2)2C(α)η3]η1η3η1aη1(η0η2)2(η3η1)C(α)α+2aη22[C1η1(α+2)2C(α)η3]η3η3η1aη3(η0η2)2(η3η1)=12Kaη0η3η1η22(η3η1).

    According to the definition of a0, we have maxr>0ga0(r)=0. Hence, the lemma follows.

    Now let a0>0 be given by (2.5) and r0:=ra0>0 being determined by (2.7). Note that by the proof of Lemma 2.2, we have f(a0,r0)=0 and f(a,r0)0 for all a(0,a0). We denote

    B(r0):={uH1(RN):u2L2<r0}andV(a):=S(a)B(r0).

    We now consider the following local minimization problem:

    mγ(a,r0):=infuV(a)Eγ(u),a(0,a0). (2.8)

    And if (a1,r1)(0,)×(0,) be such that f(a1,r1)0, then for any a2(0,a1], by Lemma 2.2 and direct calculations, we have

    f(a2,r2)0ifr2(a2a1r1,r1). (2.9)

    Theorem 2.3. Let N3, 0<σ<2, 4N<α<4N2 and a(0,a0), there exists a γ0>0 sufficiently small such that 0<γ<γ0. Then, there exists uH1(RN) such that Eγ(u)=mγ(a,r0).

    Lemma 2.4. For any a(0,a0), the following property holds,

    infuV(a)Eγ(u)<0infuV(a)Eγ(u). (2.10)

    Proof. For any uV(a), we have u2L2=r0. Thus, by (2.3) and Lemma 2.2, we have

    Eγ(u)u2L2f(a,u2L2)=r0f(a,r0)r0f(a0,r0)=0. (2.11)

    Now let uS(a) be arbitrary but fixed. We denote uλ(x):=λN2u(λx) for λ(0,). It is obvious that uλS(a) for any λ(0,). We set the map on (0,) by

    Eγ(uλ)=λ22u2L2γλσ2RN|u|2|x|σdxλNα2α+2uα+2Lα+2. (2.12)

    Noticing that 0<σ<2,2<Nα2<2NN2, thus Eγ(uλ)<0 and uλ2L2=λ2u2L2<r0 for sufficiently small λ>0. The proof follows.

    Lemma 2.5. Let 4N<α<4N2, r0>0 be determined as Lemma 2.2 and 0<a<a0 be as in Lemma 2.4. Then, there exists δ>0 such that for any initial data ψ0H1(RN) and infuM(a)ψ0uH1<δ, the solution ψ(t) of (1.1) with the initial data ψ0 exists globally.

    Proof. Firstly, we denote the right hand of (2.10) by A. According to the continuity of energy functional Eγ(u) with respect to uM(a), we deduce Eγ(u)=mγ(a,r0)<A and u2L2<r0. Moreover, there exists δ>0 such that for any ψ0H1(RN) and ψ0uH1<δ, we have

    Eγ(ψ0)<Aandψ02L2<r0. (2.13)

    Secondly, let us prove this result by contradiction. If not, there exists ψ0H1(RN) such that ψ0uH1<δ and the solution ψ(t) with an initial value of ψ0 blows up in finite time. By continuity, there exists T1>0 such that ψ(T1)2L2>r0. We now assume the initial value ~ψ0=aψ0L2ψ0. When δ>0 is sufficiently small, we have

    Eγ(~ψ0)<Aand~ψ0S(a).

    When a<ψ0L2, ~ψ02L2<ψ02L2<r0. When a>ψ0L2, considering 0<a<a0, we have ~ψ02L2<r0. This implies that ˜ψ0V(a). Since the solution of (1.1) is continuously dependent on the initial data and ψ(T1)2L2>r0, there exists T2>0 such that ˜ψ(T2)2L2>r0, where ˜ψ(t) is the solution of (1.1) that satisfies ˜ψ(t)|t=0=˜ψ0. We consequently deduce from the continuity that there exists T3>0 such that ˜ψ(T3)2L2=r0. This indicates that ˜ψ(T3)V(a). Then we infer from Lemma 2.4 that A>Eγ(˜ψ0)>Eγ(˜ψ(T3))infuV(a)Eγ(u), which is a contradiction. The proof follows.

    Lemma 2.6. It holds that

    (i) a(0,a0), amγ(a,r0) is a continuous mapping.

    (ii) Leta(0,a0). For all μ(0,a), we have

    mγ(a,r0)<mγ(μ,r0)+mγ(aμ,r0).

    Proof. (ⅰ) Let a(0,a0) be arbitrary and {an}n1(0,a0) be such that ana. According to the definition of mγ(an,r0), we know that there exists unV(a) such that

    Eγ(un)mγ(an,r0)+εandEγ(un)<0foranyε>0sufficientlysmall. (2.14)

    We set vn:=aanun and hence vnS(a). We find that vnV(a). Indeed, if ana, then

    vn2L2=aanun2L2un2L2<r0.

    If an<a, by Lemma 2.2, (2.4) and (2.9) we have f(an,r)f(an,r0)f(a0,r0)=0 for any r[anar0,r0]. Indeed, since f(a,r) is a non-increasing function, then we have

    f(an,r0)f(a0,r0)=0. (2.15)

    And then by direct calculations we have

    f(an,r)f(an,r0). (2.16)

    Hence, we deduce from (2.3) and (2.14) that f(an,un2L2)<0, thus un2L2<anar0 and

    vn2L2=aanun2L2<aananar0=r0.

    Since vnV(a) we can write

    mγ(a,r0)Eγ(vn)=Eγ(un)+[Eγ(vn)Eγ(un)],

    where

    Eγ(vn)Eγ(un)=12(aan1)un2L2γ2(aan1)RN|un|2|x|σdx1α+2[(aan)α+221]unα+2Lα+2.

    Since un2L2<r0, also unα+2Lα+2 and RN|un|2|x|σdx are uniformly bounded. Thus, we have

    mγ(a,r0)Eγ(vn)=Eγ(un)+on(1)asn. (2.17)

    Combining (2.14) and (2.17), we get

    mγ(a,r0)mγ(an,r0)+ε+on(1).

    Now, let uV(a) be such that

    Eγ(u)mγ(a,r0)+εandEγ(u)<0.

    Set un:=anau and hence unS(an). Clearly, u2L2<r0 and ana imply un2L2<r0 for n large enough, so that unV(a). Also, Eγ(un)Eγ(u). We thus have

    mγ(an,r0)Eγ(un)=Eγ(u)+[Eγ(un)Eγ(u)]mγ(a,r0)+ε+on(1). (2.18)

    Therefore, since ε>0 is arbitrary, we deduce that mγ(an,r0)mγ(a,r0). The point (ⅰ) follows.

    (ⅱ) Note that, fixed μ(0,a), it is sufficient to prove that the following holds

    mγ(θμ,r0)<θmγ(μ,r0),θ(1,aμ]. (2.19)

    Indeed, if (2.19) holds, then we have

    mγ(a,r0)<aμaaaμmγ(aμ,r0)+μaaμmγ(μ,r0)=mγ(aμ,r0)+mγ(μ,r0).

    Next, we prove that (2.19) holds. According to the definition of mγ(μ,r0), we have that there exists uV(μ) such that

    Eγ(u)mγ(μ,r0)+εandEγ(u)<0foranyε>0sufficientlysmall. (2.20)

    By (2.9), f(a,r)0 for any r[μar0,r0]. Hence, we can deduce from (2.3) and (2.20) that

    u2L2<μar0. (2.21)

    Now we set v(x):=u(θ1Nx). On the one hand, we note that v2L2=θu2L2=θμ and also, because of (2.21), v2L2=θu2L2<r0. Thus vV(θμ). On the other hand, we obtain from (2.2) that

    γRN|u(x)|2|x|σdxγC1μ2σ2uσL2<γ0C1a2σ2uσL2,

    and it follows easily that

    u2L2γRN|u(x)|2|x|σdx>u2L2γ0C1a2σ2uσL2>0

    for 0<γ<γ0 and γ0 small sufficiently. We can obtain that

    mγ(θμ,r0)lim infnEγ(v)=lim infn(θEγ(u)+u2L2(θ1σN2θ2)γ(θ1σN2θ2)RN|u(x)|2|x|σdx)=θmγ(μ,r0)+(θ1σN2θ2)lim infn(u2L2γRN|u(x)|2|x|σdx)<θmγ(μ,r0).

    Lemma 2.7. Let {vn}n1B(r0) be such that RN|vn|2|x|σdx0. Then, there exist a β0>0 such that

    Eγ(vn)β0vn2L2+on(1).

    Proof. Indeed, using the Sobolev inequality, we obtain that

    Eγ(vn)=12vn2L21α+2vnα+2Lα+2+on(1)vn2L2(12C(α)α+2aη22raη32)+on(1).

    Now, since f(a0,r0)=0, we have

    β0:=12C(α)α+2aη220r0η32=γC12aη020rη120>0.

    Lemma 2.8. For any a(0,a0), let {un}n1B(r0) be such that un2L2a and Eγ(un)mγ(a,r0). Then, there exists a β1>0 and a sequence {yn}n1RN such that

    B(yn,R)|un|2dxβ1>0,forsomeR>0. (2.22)

    Proof. We assume by contradiction that (2.22) does not hold. Since {un}n1B(r0) and un2L2a, the sequence {un}n1 is bounded in H1(RN). From Lemma I.1 in [21] and since 4N+2<α+2<2NN2, we deduce that unLα+20 as n. At this point, Lemma 2.7 implies Eγ(un)on(1). This contradicts the fact that mγ(a,r0)<0 and the lemma follows.

    Theorem 2.9. For any a(0,a0), if {un}n1B(r0) is such that un2L2a and Eγ(un)mγ(a,r0) then, up to translation, unu in H1(RN).

    Proof. Let {un}n1B(r0) be a minimizing sequence of the energy functional Eγ(u), that is,

    un2L2=aandlimnEγ(un)=mγ(a,r0).

    By similar proof as [23], we known that {un}n1 is bounded in H1(RN). By Lemma 2.6, Lemma 2.8 and Rellich compactness theorem, there exist sequences {unk}k1H1(RN) and {ynk}k1RN such that for any ϵ>0, there exists R(ϵ)>0 such that for all k1,

    B(ynk,R(ϵ))|unk|2dxβ2ϵ>0. (2.23)

    Denote ˜unk()=unk(+ynk), then there exists ˜u such that ˜unk˜u weakly in H1(RN), ˜unk˜u strongly in Lrloc(RN) with r[2,2NN2), combining with (2.23) we have

    B(0,R(ϵ))|˜u|2dxβ2ϵ>0.

    Thus RN|˜u|2dx=β2. Indeed, we assume by contradiction that RN|˜u|2dx=¯β2<β2. According to Brézis-Lieb Lemma and ˜unk=(˜unk˜u)+˜u, we have

    ˜unk2H1=˜unk˜u2H1+˜u2H1+on(1),
    ˜unkqLq=˜unk˜uqLq+˜uqLq+on(1),1q<,
    ˜unkp+2Lp+2=˜unk˜up+2Lp+2+˜up+2Lp+2+on(1),
    RN|˜unk|2|x|σdx=RN|˜unk˜u|2|x|σdx+RN|˜u|2|x|σdx+on(1).

    Then,

    E(˜unk)=E(˜unk˜u)+E(˜u)+on(1).

    Hence,

    mγ(β2,r0)mγ(β2¯β2,r0)+mγ(¯β2,r0),

    which contradicts Lemma 2.6. Thus RN|˜u|2dx=β2, i.e., ˜unk˜u strongly in L2(RN). By the Gagliardo-Nirenberg inequality, ˜unk˜u strongly in Ls(RN), where s[2,2NN2). Next, our aim is to show that {ynk}k1RN is bounded. If it was not the case, we deduce that

    RN|unk|2|x|σdx=RN|˜unk|2|x+ynk|σdx0ask.

    Hence limnEγ(unk)=limnEγ(unk), where Eγ(unk) is the corresponding energy functional of the sequence {unk}k1H1(RN) via translation transformation. On the other hand, we know that limnEγ(un) is attained by a nontrivial function ur0B(r0). Hence,

    limkEγ(unk)=limnEγ(˜unk)=limk12RN|˜unk|2dxlimk1α+2RN|˜unk|α+2dx12RN|˜u|2dx1α+2RN|˜u|α+2dx.

    According to the definition of Eγ(unk), we see that ˜u is a minimizer of limkEγ(unk). Consequently,

    limkEγ(unk)<limkEγ(unk)RN|ur0|2|x|σdx<limkEγ(unk),

    which contradicts limkEγ(unk)limkEγ(unk). Thus, {ynk}k1RN is bounded. We can deduce, up to subsequences, limkynk=y0 for some y0RN. Consequently,

    unk(x)˜u(xy0)Lsunk(x)˜u(xynk)Ls+˜u(xynk)˜u(xy0)Ls=unk(x+ynk)˜u(x)Ls+˜u(x+y0ynk)˜u(x)Ls0 (2.24)

    as k for any s[2,2NN2), that is, unk converges strongly to ˜u(xy0) in Ls(RN) for s[2,2NN2). We denote u(x)=˜u(xy0). Hence,

    limkEγ(unk)12RN|u|2dxγ2RN|u|2|x|σdx1α+2RN|u|α+2dx.

    We see that Eγ(u)=mγ(a,r0) and hence unku in H1(RN).

    Proof of Theorem 2.3. The existence of γ0 and a minimizer for Eγ(u) on V(a) were proved in Lemma 2.6, Lemma 2.7 and Theorem 2.9.

    Proof of Theorem 1.3. (ⅰ) The property that M(a)V(a) is non-empty follows from Theorem 2.3.

    (ⅱ) We will show that M(a) is orbitally stable. Firstly, we note that the solution ψ of (1.1) exists globally by Lemma 2.5. Now we suppose by contradiction that there exist ϵ0>0, a sequence of initial data {ψ0,n}n1H1(RN) and a sequence {tn}n1R such that for all n1,

    infuM(a)ψ0,nuH1<1n,infuM(a)ψn(tn)uH1ϵ0, (3.1)

    where ψn(t) is the solution to (1.1) with initial data ψ0,n. Next, we claim that there exists vM(a) such that

    limnψ0,nvH1=0.

    Indeed, by (3.1), we see that for each n1, there exists vnV(a) such that

    ψ0,nvnH1<2n. (3.2)

    Since {vn}n1M(a) is a minimizing sequence of (1.7) and by the same argument as in Lemma 2.3, there exists vM(a) such that

    limnvnvH1=0. (3.3)

    By (3.2) and (3.3), the claim follows. And then, due to vV(a), we have ~ψn=aψn(tn)ψn(tn)L2V(a) and

    limnEγ(~ψn)=limnEγ(ψn(tn))=limnEγ(ψ0,n)=Eγ(v)=mγ(a,r0),

    which implies that {~ψn}n1 is a minimizing sequence for (1.7). Thanks to the compactness of all minimizing sequence of (1.7), there is a ˜uV(a) satisfies

    ~ψn˜uinH1(RN).

    Moreover, by the definition of ~ψn, it follows easily that

    ~ψnψn(tn)0inH1(RN).

    Consequently, we have

    ψn(tn)˜uinH1(RN),

    which contradicts (3.1). The proof is now complete.

    In this work, we study the stability of the set of energy minimizers in the mass supercritical case. In the mass supercritical case, the energy functional is unbounded from below on S(a). Thus, we consider the further constrained minimization problem to study the existence and stability of standing waves for (1.1). And, the energy functional is not invariant under the scaling transform due to inverse-power potentials γ|x|σu. It is intrinsically difficult for us to prove the compactness of minimizing sequences. By a rather delicate analysis, we can overcome this difficulty by proving the boundedness of any translation sequence.

    This work is supported by the Outstanding Youth Science Fund of Gansu Province (No. 20JR10RA111).

    The author declares no conflicts of interest.



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