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A review of different deep learning techniques for sperm fertility prediction

  • Received: 02 January 2023 Revised: 15 April 2023 Accepted: 17 April 2023 Published: 08 May 2023
  • MSC : 68T07

  • Sperm morphology analysis (SMA) is a significant factor in diagnosing male infertility. Therefore, healthy sperm detection is of great significance in this process. However, the traditional manual microscopic sperm detection methods have the disadvantages of a long detection cycle, low detection accuracy in large orders, and very complex fertility prediction. Therefore, it is meaningful to apply computer image analysis technology to the field of fertility prediction. Computer image analysis can give high precision and high efficiency in detecting sperm cells. In this article, first, we analyze the existing sperm detection techniques in chronological order, from traditional image processing and machine learning to deep learning methods in segmentation and classification. Then, we analyze and summarize these existing methods and introduce some potential methods, including visual transformers. Finally, the future development direction and challenges of sperm cell detection are discussed. We have summarized 44 related technical papers from 2012 to the present. This review will help researchers have a more comprehensive understanding of the development process, research status, and future trends in the field of fertility prediction and provide a reference for researchers in other fields.

    Citation: Muhammad Suleman, Muhammad Ilyas, M. Ikram Ullah Lali, Hafiz Tayyab Rauf, Seifedine Kadry. A review of different deep learning techniques for sperm fertility prediction[J]. AIMS Mathematics, 2023, 8(7): 16360-16416. doi: 10.3934/math.2023838

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  • Sperm morphology analysis (SMA) is a significant factor in diagnosing male infertility. Therefore, healthy sperm detection is of great significance in this process. However, the traditional manual microscopic sperm detection methods have the disadvantages of a long detection cycle, low detection accuracy in large orders, and very complex fertility prediction. Therefore, it is meaningful to apply computer image analysis technology to the field of fertility prediction. Computer image analysis can give high precision and high efficiency in detecting sperm cells. In this article, first, we analyze the existing sperm detection techniques in chronological order, from traditional image processing and machine learning to deep learning methods in segmentation and classification. Then, we analyze and summarize these existing methods and introduce some potential methods, including visual transformers. Finally, the future development direction and challenges of sperm cell detection are discussed. We have summarized 44 related technical papers from 2012 to the present. This review will help researchers have a more comprehensive understanding of the development process, research status, and future trends in the field of fertility prediction and provide a reference for researchers in other fields.



    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 R3K(x)|un,u|2σ+2C||un,u||2σ+2 and

    ||un,u||=||i=1ui(zn,i,)+vn,i=1ui(zi,)v||i=1||ui(zn,i,)ui(zi,)||+||vn,||+||v||2ρ+on(1),

    the following inequality holds:

    C||un,u||2aR3|(un,u)|2+μR3|un,u|2C||un,u||2σ+2+o(1),

    where C is a positive constant since a>0 and μ>0. Making ρ smaller if necessary given C||un,u||2σ<C/2, it follows that un,u in H1(R3). This completes the proof of Lemma 3.1.

    Proposition 3.2. For some ρ>0 small and by letting {n}R, {μn}R and {un}Z(ρ) satisfy that

    n0+,lim supnIn(un)b0, (3.8)
    In(un)+μnJ(un)H10, (3.9)

    as n. Then, μnμ0 holds, limnIn(un)=b0 and for some zn,iR3, i=1,2,,, we have

    uni=1ui(zn,i)0and dist(nzn,i,Ki)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 μnμ>0.

    As {un}Z(ρ), we can write that un=i=1ui(xzn,i)+vn with zn,i1(Ki)τ and vnρ. It follows from unZ(ρ) and the boundedness of In(un) that un and Gn(un) are bounded. Besides, by (3.9) and J(un)un=1, we know that μn is bounded. Then up to a subsequence, we can assume that μnμ in R and un(+zn,i)wiH1(R3). For ρ<110min1iuiL2(B1(0)), we have

    lim infnun(+zn,i)L2(B1(0))uiL2(B1(0))ρ>0.

    Notice that for any R>0, we can obtain that uiwiL2(BR(0))ρ. Hence,

    ui2ρwi2ui2+ρ. (3.10)

    Then, if ρ is small enough, we know that wi0. Next, testing (3.9) with φ(xzn,i) for each φC0(R3), we deduce that

    dnbR3|un(x+zn,i)|2R3un(x+zn,i)φ=o(1).

    Thus, wi is a solution to aΔwi+μwi=ki|wi|2σwi in H1(R3) with limnK(nzn,i)ki[k_,ˉk], where k_=minxUi=1ˉΩiK(x)>0 and ˉk=max1iki. Then, combining the Pohozaev identity with

    a|wi|22+μ|wi|22=ki|wi|2σ+22σ+2,

    it follows that there exists a positive contant μ.

    Step 2. uni=1wi(zn,i)0 in L2σ+2(R3) and dist(nzn,i,Ki)0.

    We show that

    ˜vn:=uni=1wi(zn,i) 0 in L2σ+2(R3).

    Otherwise, by Lions' lemma [17], there exists a sequence of points {zn}R3 such that

    lim supnuni=1wi(zn,i)2L2(B1(zn))>0.

    Noting that |znzn,i| i=1,2,,, we have

    lim supnB1(0)|un(+zn)|2>0. (3.11)

    By (3.8), Gn(un)C holds for some C>0 that is independent of . Then, we have that  dist(nzn,i=1Ωi)0. Up to a subsequence, we assume that ˜vn(x+zn)v00 in H1(R3) and K(nzn)k0[k_,ˉk], where k0=k(y0),y0i=1Ωi. Let D:={xR3|x3M}. For some i0, if limndist(nzn,Ωi0)n=M<, we get that nznz0 as n, where z0Ωi0. Next, without loss of generality we can assume that v0H10(D). Testing (3.9) with φ(zn) for any φC0(D), we have

    limn[aR3unφ(xzn)+dnbR3|un|2R3unφ(xzn)R3K(nx)|un|2σunφ(xzn)+R3μnunφ(xzn)+QnR3χn(unφ(xzn)+unφ(xzn))]=0.

    Then by applying unH1(R31ni=1Ωi)Cn and

    dnbR3|un|2R3unφ(xzn)=o(1),

    we can obtain that v0 is a solution of aΔu+μu=k0|u|2σu in H10(D), which is impossible since this equation does not have a nontrivial solution on the half space according to [8]. Thus limn dist(nzn,Ωi0)=+ and zn1nΩi0. Now we test (3.9) with φ(zn) for any φC0(R3) to get

    aΔv0+μv0=k0|v0|2σv0,

    where μ>0, and |v0|22>C1 for some C1>0 that is independent of ρ.

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

    1=limn|un|22=limn|un(+zn,1)v0(+zn,1)|22+|v0|22+o(1)i=1|wi|22+|v0|22i=1|ui|222ρi=1|ui|22+ρ2+C1>1,

    which is a contradiction.

    Step 3. uni=1wi(zn,i)0 and limnIn(un)=b0.

    Testing (3.9) with uni=1wi(zn,i), given

    dnbR3|un|2R3un(uni=1wi(xzn,i))=o(1),

    we can get that

    a(|un|22i=1|wi|22)+μ(|un|22i=1|wi|22)on(1).

    Next, we have

    a|(uni=1wi(zn,i))|22+μ|uni=1wi(zn,i)|22=on(1),

    i.e., uni=1wi(zn,i)0 in H1(R3).

    On the other hand, by Lemma 2.2, we obviously get that limnIn(un)=b0.

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

    Proposition 4.1. There exists 0>0 such that for (0,0), I|M has a critical point uZ(ρ). Moreover, lim0I(u)=b0 and the Lagrange multiplier μR satisfies

    lim0μ=μ0,I(u)+μJ(u)=0. (4.1)

    Remark 4.2. By Proposition 3.2, it is easy to verify that (4.1) holds if u is a critical point of I|M such that lim sup0Iub0.

    The proof of Proposition 4.1 can be obtained as in [31] by considering the following contradiction: {n} with n0 such that for some sequence bnb0, I admits no critical points in {uZ(ρ)|In(u)bn}. For brevity, we denote =n. Then from Lemma 3.1 and Proposition 3.2, there respectively exist κ0>0 and v>0 independent of and v>0 such that

    I|M(u)TuMv,for uZ(ρ)[b02κ0Ib],I|M(u)TuMv,for u(Z(ρ)Z(ρ/4))[b02κ0Ib], (4.2)

    where

    [b1I]={uH1(R3)|b1I(u)},
    [Ib2]={uH1(R3)|I(u)b2},
    [b1Ib2]={uH1(R3)|b1I(u)b2},

    for any b1,b2R.

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

    Lemma 4.3. Let v and v be given as in (4.2). For any κ(0,min{κ0,ρv16}), there exists κ>0 such that for (0,κ) there is a deformation η:MM that satisfied the following conditions:

    (a) η(u)=u if uM(Z(ρ)[b02κI]).

    (b)  I(η(u))I(u) if uM.

    (c) η(u)Z(ρ)[Ib0κ] if uZ(ρ/4)[Ib].

    To give the proof of Lemma 4.3, we borrow some ideas from [5,6,31] in the L2-subcritical case and L2-supercritical case.

    For every δ>0, we denote

    Sδ:={sS1+||ss0|δ},

    where s0=(|u1|2,,|u|2). Fix qiKi and qi,=1qi for i=1,2,, and define the (1)-dimensional initial path by

    ξ(s)=Bi=1wsi(qi,),

    where B:=|i=1wsi(qi,)|12. Note that we can fix δ>0 small enough such that

    ξ(s)Z(ρ/4) for sSδ

    and

    B1 as 0 uniformly in Sδ.

    Define

    b:=maxsSδI(ξ(s)).

    Lemma 4.4. lim0b=b0 and fix any κ(0,min{κ0,ρv16}) such that

    supsSδI(ξ(s))<b02κ, (4.3)

    where Sδ:={sS+||ss0|=δ}.

    Proof. Since

    db(R3|ξ|2)20as0,

    one can deduce that

    I(ξ(s))i=1Ii(wsi) as 0 uniformly for sSδ.

    By Lemma 2.3(a), we have

    supsSδI(ξ(s))<b02κ.

    Proof of Proposition 4.1 in the L2-subcritical case. By Lemma 4.3 and (4.3), we have

    η(ξ(s))=ξ(s) for sSδ, (4.4)
    I(η(ξ(s)))b0κ and η(ξ(s))Z(ρ) for sSδ. (4.5)

    Define

    Ψi,=(1Ωi|u|2)12(i=11Ωi|u|2)12, for uM.

    Similar to the case in [31], there exists s1Sδ such that Ψi,(η(ξ(s1)))=s0i=|ui|2. Denote

    u0,:=η(ξ(s1)), ui,:=γi,u0,, (4.6)

    where γi,C0(1(Ωi),[0,1]) is a cut-off function such that γi,=1 on 1Ωi and |γi,|C for each i=1,2,, and some C>0; Ωi is an open neighborhood of ˉΩi. By (4.5), we have that G(u0,)C for some C>0 that is independent of , which implies that

    u0,H1(R3i=11Ωi)C. (4.7)

    Then

    |ui,|2=|ui|2+o(1) (4.8)

    and

    Ii(ui)Ii(ui,)+o(1). (4.9)

    Hence from (4.5)–(4.9), we have

    b0κI(u0,)i=1Ii(ui,)+o(1)i=1Ii(ui)+o(1)=b0+o(1),

    which is a contradiction. This completes the proof.

    Fix qiKi and denote qi,=1qi; we set

    ζ(s)=ˉBi=1ti3/2ui(ti(qi,)) for t=(t1,t2,,t)(0,+),

    where ˉB:=|i=1t3/2iui(ti(qi,))|12.

    Define

    b:=maxt[1ˉδ,1+ˉδ]I(ζ(t)).

    Note that we can fix ˉδ>0 small enough such that

    ζ(t)Z(ρ/4) for t[1ˉδ,1+ˉδ],

    and ˉB1 holds. Note also that

    Ii(ui)>Ii(t3/2iui(ti)) for ti[1ˉδ,1+ˉδ]{1}.

    Since

    db(R3|ζ|2)20as0,

    and

    I(ζ(t))i=1Ii(t3/2iui(ti)) as 0 uniformly for t[1ˉδ,1+ˉδ],

    one can get the result as in [31].

    Lemma 4.5. lim0b=b0 and fix any κ(0,min{κ0,ρv16}) such that

    supt[1ˉδ,1+ˉδ]I(ζ(t))<b02κ. (4.10)

    Proof of Proposition 4.1 in the L2-supercritical case. By Lemma 4.3 and (4.10),

    η(ζ(t))=ζε(t) if t[1ˉδ,1+ˉδ], (4.11)
    I(η(ζ(t)))b0κ and η(ζ(t))Z(ρ) for t[1ˉδ,1+ˉδ]. (4.12)

    Define

    Φi,=(1Ωi|u|2)123σ(3σki(2+2σ)a1Ωi|u|2σ+2)123σ, for uM.

    Similar to the case in [31], there exists t1[1ˉδ,1+ˉδ] such that

    Φi,(η(ζ(t1)))=1, i=1,2,,. (4.13)

    We denote

    ˉu0,:=η(ζ(t1)),ˉui,:=γi,ˉu0,(i=1|γi,ˉu0,|22)12.

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

    ˉu0,H1(R3i=11Ωi)=o(1) (4.14)

    and

    i=1|γi,ˉu0,|22=1+o(1). (4.15)

    From (4.13)–(4.15), we have

    ti,:=(|ˉui,|22)123σ(3σki(2+2σ)a|ˉui,|2σ+22σ+2)13σ2=Φi,(ˉu0,)+o(1)=1+o(1).

    A direct calculation shows that

    t:=(|t321,ˉu1,(t11,)|2,|t322,ˉu2,(t12,)|2,,|t32,ˉu,(t1,)|2)S1+

    and

    |(t32i,ˉui,(t1i,))|22=3σki(2+2σ)a|t32i,ˉui,(t1i,)|2σ+22σ+2.

    Hence by the definition of bsi, we have

    i=1Ii(ui)=b0i=1Ii(t32i,ˉui,(t1i,))=i=1Ii(ˉui,)+o(1).

    Similarly, one can get a contradiction.

    Let u be the critical point of the modified function I given in Proposition 4.1.

    Completion of proof of Theorem 1.1.

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

    u2H1(R3i=11(Ki)τ)eC. (5.1)

    We adopt some arguments from [3,31]. Set 21τ1:=n. For n=1,2,,n, we take ϕnC1(R3,[0,1]) such that

    {ϕn(x)=0,if x R3En,ϕn(x)=1,if xEn+1,|ϕn(x)|2,xR3,

    where En:={xR3|dist(x,i=11(Ki)τ2)>n1}. Then by Proposition 3.2,

    lim0uH1(E1)lim0i=1uiH1(R3Bτ(0))=0. (5.2)

    Note that for each n=1,2,,n,

    suppχ=R3i=11ΩiR3i=11(Ki)τϕ1n(1).

    Since I(u)+μJ(u),ϕnu=0, we have

    aR3u(ϕnu)+dbR3|u|2R3u(ϕnu)R3K(x)|u|2σ+2ϕn+R3μu2ϕn=41G(u)12R3χ(u(ϕnu)+u2ϕn)=41G(u)12R3i=11Ωi(u(ϕnu)+u2ϕn)=41G(u)12R3i=11Ωi(|u|2+u2)0. (5.3)

    Therefore, by (5.3) and the Sobolev embedding,

    min{a,μ02}u2H1(En+1)R3ϕn(a|u|2+μu2)R3K(x)|u|2σ+2ϕnaR3uuϕndbR3|u|2R3u(ϕnu)Cu2σ+2H1(En)+au2H1(En)au2H1(En+1)dbR3|u|2R3u(ϕnu)(a+Cu2σH1(E1)+o(1))u2H1(En)(a+o(1))u2H1(En+1),

    where dbR3|u|2R3u(ϕnu)o(1)(2||u||2H1(En)||u||2H1(En+1)) as 0. By (5.2), we have

    u2H1(En+1)θ1u2H1(En),

    where

    θ:=a+min{a,μ02}+o(1)a+o(1)1+min{1,μ02a}as0.

    Nothing that nτ for small values of , one can take some θ0>1 and obtain

    u2H1(R3i=11(Ki)τ)u2H1(En+1)θn0u2H1(E1)eτlnθ0.

    It follows that for small values of , G(u)=0. So u is a solution to the original problem (3.1) for small values of .

    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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