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+εb∫R3|∇v|2dx)Δv−K(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+b∫R3|∇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+εb∫R3|∇u|2dx)Δu+V(x)u=f(u),x∈R3. | (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+εb∫R3|∇u|2dx)Δu+V(x)u=|u|p−2u,x∈R3. | (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+b∫R3|∇u|2dx)Δu+u=|u|p−2u,x∈R3. |
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),x∈RN,∫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|q−2u+|u|2∗−2u, where 2∗=2N/(N−2),N≥3 is the Sobolev critical exponent. For f(u)=g(u)+|u|2∗−2u 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+b∫RN|∇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|q−2u+|u|2∗−2u 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:
{−ℏ2Δv−K(x)|v|2σv=−λvin RN,|v|22=∫RNv2dx=m0ℏα,v(x)→0as|x|→∞. | (1.7) |
For the case that the parameter ℏ 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,x∈RN | (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 ℓ≥2 mutually disjoint bounded domains Ωi⊂R3, i=1,2,⋯,ℓ such that
ki:=maxx∈ΩiK(x)>maxx∈∂ΩiK(x). |
Denote Ki={x∈Ωi|K(x)=ki}, which is nonempty and compact and set
β:=2−ασ2−3σ. |
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 ℓ local maximum points Pi,ε, i=1,2,⋯,ℓ that satisfy
limε→0dist(Pi,ε,Ki)=0. |
(b) μ=ε2σ(3−α)2−3σλε→μ0 and ‖ε3−α2−3σvε(εβ⋅)−∑ℓi=1ui(⋅−ε−βPi,ε)‖H1→0 as ε→0, where
μ0=m2σ2−3σ0a−3σ2−3σ(ℓ∑i=1θ−1σi|U|22)−2σ2−3σ, |
ui=θ−12σiμ12σU(√μa⋅),i=1,2,⋯,ℓ, |
and U∈H1(R3) is a positive solution to
{−ΔU+U=|U|2σUinR3,U(0)=maxx∈R3U(x),limx→∞U(x)=0. | (1.9) |
(c) There are constants C,c>0 that are independent of ε such that
|vε|≤Cε−3−α2−3σexp{−cε−βdist(x,∪ℓi=1Ki)}. |
The proof of Theorem 1.1 is similar to that in [31]. By virtue of the change of variables techinque, we have
u(⋅)=ε3−α2−3σv(εβ⋅). |
Equation(1.1) is transformed into the following problem:
{−(a+ε(3−α)(σ−2)2−3σb|∇u|22)Δu−K(εβx)|u|2σu=−λε2σ(3−α)2−3σuin R3,|u|22=m0,u(x)→0as|x|→∞. |
Let
ℏ:=εβ,μ=ε2σ(3−α)2−3σλ,d=(3−α)(σ−2)2−ασ. |
Then, under the assumption (A) and given β>0 and d>0, we have the following:
{−(a+ℏdb|∇u|22)Δu−K(ℏx)|u|2σu=−μuin R3,|u|22=m0,u(x)→0as|x|→∞. | (1.10) |
Define the energy functional
Eℏ(u)=a2∫R3|∇u|2+ℏdb4(∫R3|∇u|2)2−12σ+2∫R3K(ℏx)|u|2σ+2. |
Then, a solution (μℏ,uℏ) of (1.10) can be obtained as a critical point of Eℏ that is restrained on
M:={u∈H1(R3)||u|22=m0}. |
By adopting similar deformation arguments in [5,6,23,31], we show that the Lagrange multiplier μℏ satisfies
μℏ=μ0+oℏ(1),uℏ=ℓ∑i=1ui(⋅−qi,ℏ)+oℏ(1)in H1(R3), |
where qi,ℏ satisfies the condition that dist(ℏqi,ℏ,Ki)→0 as ℏ→0, i=1,2,⋯,ℓ.
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 u∈Lp(R3), p∈[1,∞);
● ‖u‖:=(∫R3|∇u|2+|u|2)12, where u∈H1(R3);
● b±=max{0,±b} for b∈R;
● B(x,ρ) denotes an open ball centered at x∈R3 with radius ρ>0;
● For a domain D⊂R3, we denote 1ℏD:={x∈R3|ℏx∈D};
● Unless stated otherwise, δ and C are general constants.
Let m0,θ1,θ2,⋯,θℓ be a series of positive numbers. We consider the following system:
{−aΔvi−θi|vi|2σvi=−μviin R3,∑ℓi=1|vi|22=m0,vi(x)>0,lim|x|→∞vi(x)=0,i=1,2,⋯,ℓ. | (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,⋯,vℓ)∈R×H1(R3)ℓ up to translations of each vi,i=1,2,⋯,ℓ, where
μ=m2σ2−3σ0a−3σ2−3σ(ℓ∑i=1θ−1σi|U|22)−2σ2−3σ,vi(x)=θ−12σiμ12σU(√μax), | (2.2) |
and U∈H1(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,⋯,ℓ in the limit system (2.1), as follows:
|vi|22=m0θ−1σi∑ℓi=1θ−1σi |
and for each i=1,2,⋯,ℓ,vi is the ground state of
Iθi(u)=a2|∇u|22−θi2σ+2|u|2σ+22σ+2 |
on
Mi:={u∈H1(R3)||u|22=|vi|22}. |
Lemma 2.2. ∑ℓi=1Iθi(vi) is continuous and strictly decreasing with respect to m0 and θi,i=1,2,⋯,ℓ, where vi is determined as in Lemma 2.1.
We next characterize the energy level of ∑ℓi=1Iθi(vi). Let
s=(s1,s2,⋯,sℓ)∈(0,+∞)ℓ |
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,⋯,ℓ given some radial function wsi. In particular, vi=ws0i for s0i=|vi|2. Moreover, if σ∈(0,23), then
bsi=inf{Iθi(v)|v∈H1(R3),|v|22=s2i} |
and if σ∈(23,2), then
bsi=inf{supt>0Iθi(t32v(t⋅))|v∈H1(R3),|v|22=s2i}. |
Set
Sℓ−1+:={s=(s1,s2,⋯,sℓ)∈(0,√m0)ℓ|ℓ∑i=1s2i=m0,i=1,2,⋯,ℓ}, |
and define E(s):=∑ℓi=1Iθi(wsi) for s∈Sℓ−1+.
Lemma 2.3. Denote s0=(s01,s02,⋯,s0ℓ)=(|v1|2,|v2|2,⋯,|vℓ|2). For each s∈Sℓ−1+∖{s0}, the following statements hold:
(a) If σ∈(0,23), then ∑ℓi=1Iθi(vi)=E(s0)>E(s);
(b) If σ∈(23,2), then ∑ℓi=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,⋯,ℓ, where (ki)ℓi=1 denotes positive numbers given by (K). Next, we set b0:=∑ℓi=1Ii(ui), where
Ii(u):=Iki(u)=a2|∇u|22−ki2σ+2|u|2σ+22σ+2. |
Then, we will find a positive solution (μℏ,uℏ) to the following system:
{−(a+ℏdb|∇u|22)Δu−K(ℏx)|u|2σu=−μuin R3,|u|22=1,u(x)→0as|x|→∞, | (3.1) |
satisfying
μℏ=μ0+oℏ(1),uℏ(x)=ℓ∑i=1ui(x−qi,ℏ)+oℏ(1)in H1(R3) |
with ℏqi,ℏ→qi∈Ki.
Set M:={u∈H1(R3)||u|2=1} and for i=1,2,⋯,ℓ and τ>0, define
(Ki)τ:={x∈R3|dist(x,Ki)≤τ}⊂Ωi. |
Define the following equation for each ρ∈(0,110min1≤i≤ℓ‖ui‖L2(B1(0))):
Z(ρ)={u=ℓ∑i=1ui(x−qi,ℏ)+v∈M|ℏqi,ℏ∈(Ki)τ,‖v‖≤ρ}. |
For u∈H1(R3), consider the penalized energy functional Iℏ:H1(R3)→R is given by
Iℏ(u):=Eℏ(u)+Gℏ(u), |
where
Gℏ(u)=(ℏ−1∫R3χℏ(x)(|∇u|2+u2)dx−1)2+, |
and
χℏ={0x∉R3∖∪ℓi=11ℏΩi,1x∈R3∖∪ℓi=11ℏΩi. |
We also denote
J(u)=12|u|22for u∈H1(R3). |
Note that if uℏ∈M with ‖uℏ‖2H1(R3∖∪ℓi=11ℏΩi)<ℏ is a critical point of Iℏ|M, then it solves (3.1) for some μℏ. Denote the tangent space of M at u∈M by
TuM={v∈H1(R3)|∫R3uv=0}. |
Lemma 3.1. For any L∈R, there exists ℏL>0 such that for any fixed ℏ∈(0,ℏL), if a sequence {un,ℏ}⊂Z(ρ) such that
Iℏ(un,ℏ)≤L,‖Iℏ|′M(un,ℏ)‖T∗un,ℏM→0, | (3.2) |
as n→∞, then un,ℏ has a strong convergent subsequence in H1(R3).
Proof. Set un,ℏ=∑ℓi=1ui(x−zn,i,ℏ)+vn,ℏ with ℏzn,i,ℏ∈(Ki)τ and ‖vn,ℏ‖≤ρ. It follows from un,ℏ∈Z(ρ) that ‖un,ℏ‖≤ρ+∑ℓi=1‖ui‖, which is bounded. Then, by
Iℏ(un,ℏ)+12σ+2∫R3K(ℏx)|un,ℏ|2σ+2=a2|∇un,ℏ|22+ℏdb4|∇un,ℏ|42+Gℏ(un,ℏ), |
we have that Gℏ(un,ℏ)≤Iℏ(un,ℏ)+12σ+2∫R3K(ℏx)|un,ℏ|2σ+2≤CL for some CL>0 that is independent of ℏ and n. From the assumption (3.2), for some μn,ℏ∈R, we deduce that
I′ℏ(un,ℏ)+μn,ℏJ′(un,ℏ)→0in H−1,as n→∞. | (3.3) |
We have
|μn,ℏ|=I′ℏ(un,ℏ)un,ℏ+o(1)≤a∫R3|∇un,ℏ|2+ℏdb(∫R3|∇un,ℏ|2)2−∫R3K(ℏx)|un,ℏ|2σ+2+G′ℏ(un,ℏ)un,ℏ≤C(‖un,ℏ‖2+‖un,ℏ‖4+‖un,ℏ‖2σ+2+Gℏ(un,ℏ)+Gℏ(un,ℏ)12)≤C∗L, |
where C∗L>0 is independent of ℏ and n. Then up to a subsequence, μn,ℏ→μℏ in R and un,ℏ⇀uℏ=∑ℓi=1ui(x−zi,ℏ)+vℏ in H1(R3) with zn,i,ℏ→zi,ℏ∈1ℏ(Ki)τ and vn,ℏ⇀vℏ.
Next, for any φ∈H1(R3), note that limn→∞I′ℏ(un,ℏ)φ+μn,ℏJ′(un,ℏ)φ=0, (μℏ,uℏ) satisfies
a∫R3∇uℏ∇φ+ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇φ−∫R3K(ℏx)|uℏ|2σuℏφ+∫R3μℏuℏφ+Qℏ∫R3χℏ(∇uℏ∇φ+uℏφ)=0, | (3.4) |
where Qℏ=4ℏ−1limn→∞Gℏ(un,ℏ)12≥0. Then, we claim that ℏL and μL are two positive constants such that μℏ>μL for each ℏ∈(0,ℏL). Otherwise, we assume that μℏ→μ≤0 as ℏ→0 up to a subsequence. Because uℏ is bounded in H1(R3), we can assume that uℏ(⋅+z1,ℏ)⇀u. Note that
lim infℏ→0‖uℏ(⋅+zi,ℏ)‖L2(B1(0))≥‖ui‖L2(B1(0))−ρ>0. |
We can obtain that u≠0 if ρ>0 is small. Then set φ=ψ(x−z1,ℏ) in (3.4) for each ψ∈C∞0(R3) and take the limit ℏ→0, that is
limℏ→0[a∫R3∇uℏ∇ψ(x−z1,ℏ)+ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇ψ(x−z1,ℏ)−∫R3K(ℏx)|uℏ|2σuℏψ(x−z1,ℏ)+∫R3μℏuℏψ(x−z1,ℏ)+Qℏ∫R3χℏ(∇uℏ∇ψ(x−z1,ℏ)+uℏψ(x−z1,ℏ))]=0. |
Using the boundedness of uℏ and d>0, we have
ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇ψ(x−z1,ℏ)=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,ℏ−uℏ in (3.4), we have
a∫R3∇uℏ∇(un,ℏ−uℏ)+ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇(un,ℏ−uℏ)−∫R3K(ℏx)|uℏ|2σuℏ(un,ℏ−uℏ)+∫R3μℏuℏ(un,ℏ−uℏ)+Qℏ∫R3χℏ(∇uℏ∇(un,ℏ−uℏ)+uℏ(un,ℏ−uℏ))=0. | (3.5) |
Then it follows from (3.3) that
⟨I′ℏ(un,ℏ)+μn,ℏJ′(un,ℏ),un,ℏ−uℏ⟩=o(1)‖un,ℏ−uℏ‖. |
That is,
a∫R3∇un,ℏ∇(un,ℏ−uℏ)+ℏdb∫R3|∇un,ℏ|2∫R3∇un,ℏ∇(un,ℏ−uℏ)−∫R3K(ℏx)|un,ℏ|2σun,ℏ(un,ℏ−uℏ)+∫R3μn,ℏun,ℏ(un,ℏ−uℏ)+Qn,ℏ∫R3χℏ(∇un,ℏ∇(un,ℏ−uℏ)+un,ℏ(un,ℏ−uℏ))=o(1)‖un,ℏ−uℏ‖. | (3.6) |
We can show that for n large enough,
∫R3|∇un,,ℏ|2∫R3∇un,ℏ∇(un,ℏ−uℏ)−∫R3|∇uℏ|2∫R3∇uℏ∇(un,ℏ−uℏ)=∫R3|∇un,ℏ|2∫R3∇un,ℏ∇(un,ℏ−uℏ)−∫R3|∇un,ℏ|2∫R3∇uℏ∇(un,ℏ−uℏ)+∫R3|∇un,ℏ|2∫R3∇uℏ∇(un,ℏ−uℏ)−∫R3|∇uℏ|2∫R3∇uℏ∇(un,ℏ−uℏ)=∫R3|∇un,ℏ|2∫R3|∇un,ℏ−∇uℏ|2+(∫R3|∇un,ℏ|2−∫R3|∇uℏ|2)∫R3∇uℏ∇(un,ℏ−uℏ)≥on(1), | (3.7) |
where using the fact that un,ℏ⇀uℏ in H1(R3), it follows ∫R3∇uℏ∇(un,ℏ−uℏ)→0. Thus from (3.5)–(3.7), we have
a∫R3|∇(un,ℏ−uℏ)|2+μℏ∫R3|un,ℏ−uℏ|2−∫R3K(ℏx)|un,ℏ−uℏ|2σ+2+Qh∫R3χh[|∇(un,ℏ−uℏ)|2+|un,ℏ−uh|2]+ℏdb∫R3|∇un,ℏ|2∫R3|∇un,ℏ−∇uℏ|2=o(1). |
Noting also that ∫R3K(ℏx)|un,ℏ−uℏ|2σ+2≤C||un,ℏ−uℏ||2σ+2 and
||un,ℏ−uℏ||=||ℓ∑i=1ui(⋅−zn,i,ℏ)+vn,ℏ−ℓ∑i=1ui(⋅−zi,ℏ)−vℏ||≤ℓ∑i=1||ui(⋅−zn,i,ℏ)−ui(⋅−zi,ℏ)||+||vn,ℏ||+||vℏ||≤2ρ+on(1), |
the following inequality holds:
C∗||un,ℏ−uℏ||2≤a∫R3|∇(un,ℏ−uℏ)|2+μℏ∫R3|un,ℏ−uℏ|2≤C||un,ℏ−uℏ||2σ+2+o(1), |
where C∗ is a positive constant since a>0 and μℏ>0. Making ρ smaller if necessary given C||un,ℏ−uℏ||2σ<C∗/2, it follows that un,ℏ→uℏ in H1(R3). This completes the proof of Lemma 3.1.
Proposition 3.2. For some ρ>0 small and by letting {ℏn}⊂R, {μn}⊂R and {un}⊂Z(ρ) satisfy that
ℏn→0+,lim supn→∞Iℏn(un)≤b0, | (3.8) |
‖I′ℏn(un)+μnJ′(un)‖H−1→0, | (3.9) |
as n→∞. Then, μn→μ0 holds, limn→∞Iℏn(un)=b0 and for some zn,i∈R3, i=1,2,⋯,ℓ, we have
‖un−ℓ∑i=1ui(⋅−zn,i)‖→0and dist(ℏnzn,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 μn→∼μ>0.
As {un}⊂Z(ρ), we can write that un=∑ℓi=1ui(x−zn,i)+vn with zn,i∈1ℏ(Ki)τ and ‖vn‖≤ρ. It follows from un∈Z(ρ) and the boundedness of Iℏn(un) that ‖un‖ and Gℏn(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 μn→∼μ in R and un(⋅+zn,i)⇀wi∈H1(R3). For ρ<110min1≤i≤ℓ‖ui‖L2(B1(0)), we have
lim infn→∞‖un(⋅+zn,i)‖L2(B1(0))≥‖ui‖L2(B1(0))−ρ>0. |
Notice that for any R>0, we can obtain that ‖ui−wi‖L2(BR(0))≤ρ. Hence,
‖ui‖2−ρ≤‖wi‖2≤‖ui‖2+ρ. | (3.10) |
Then, if ρ is small enough, we know that wi≠0. Next, testing (3.9) with φ(x−zn,i) for each φ∈C∞0(R3), we deduce that
ℏdnb∫R3|∇un(x+zn,i)|2∫R3∇un(x+zn,i)∇φ=o(1). |
Thus, wi is a solution to −aΔwi+∼μwi=∼ki|wi|2σwi in H1(R3) with limn→∞K(ℏnzn,i)→∼ki∈[k_,ˉk], where k_=minx∈Uℓi=1ˉΩiK(x)>0 and ˉk=max1≤i≤ℓki. 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. un−∑ℓi=1wi(⋅−zn,i)→0 in L2σ+2(R3) and dist(ℏnzn,i,Ki)→0.
We show that
˜vn:=un−ℓ∑i=1wi(⋅−zn,i)→ 0 in L2σ+2(R3). |
Otherwise, by Lions' lemma [17], there exists a sequence of points {zn}⊂R3 such that
lim supn→∞‖un−ℓ∑i=1wi(⋅−zn,i)‖2L2(B1(zn))>0. |
Noting that |zn−zn,i|→∞ i=1,2,⋯,ℓ, we have
lim supn→∞∫B1(0)|un(⋅+zn)|2>0. | (3.11) |
By (3.8), Gℏn(un)≤C holds for some C>0 that is independent of ℏ. Then, we have that dist(ℏnzn,∪ℓi=1Ωi)→0. Up to a subsequence, we assume that ˜vn(x+zn)⇀v0≠0 in H1(R3) and K(ℏnzn)→k0∈[k_,ˉk], where k0=k(y0),y0∈∪ℓi=1Ωi. Let D:={x∈R3|x3≥−M}. For some i0, if limn→∞dist(ℏnzn,∂Ωi0)ℏn=M<∞, we get that ℏnzn→z0 as n→∞, where z0∈∂Ωi0. Next, without loss of generality we can assume that v0∈H10(D). Testing (3.9) with φ(⋅−zn) for any φ∈C∞0(D), we have
limn→∞[a∫R3∇un∇φ(x−zn)+ℏdnb∫R3|∇un|2∫R3∇un∇φ(x−zn)−∫R3K(ℏnx)|un|2σunφ(x−zn)+∫R3μnunφ(x−zn)+Qℏn∫R3χℏn(∇un∇φ(x−zn)+unφ(x−zn))]=0. |
Then by applying ‖un‖H1(R3∖1ℏn∪ℓi=1Ωi)≤Cℏn and
ℏdnb∫R3|∇un|2∫R3∇un∇φ(x−zn)=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(ℏnzn,∂Ωi0)=+∞ and zn∈1ℏnΩi0. Now we test (3.9) with φ(⋅−zn) for any φ∈C∞0(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)≥ℓ∑i=1|wi|22+|v0|22≥ℓ∑i=1|ui|22−2ρℓ∑i=1|ui|22+ℓρ2+C1>1, |
which is a contradiction.
Step 3. ‖un−∑ℓi=1wi(⋅−zn,i)‖→0 and limn→∞Iℏn(un)=b0.
Testing (3.9) with un−∑ℓi=1wi(⋅−zn,i), given
ℏdnb∫R3|∇un|2∫R3∇un∇(un−ℓ∑i=1wi(x−zn,i))=o(1), |
we can get that
a(|∇un|22−ℓ∑i=1|∇wi|22)+∼μ(|un|22−ℓ∑i=1|wi|22)≤on(1). |
Next, we have
a|∇(un−ℓ∑i=1wi(⋅−zn,i))|22+∼μ|un−ℓ∑i=1wi(⋅−zn,i)|22=on(1), |
i.e., un−∑ℓi=1wi(⋅−zn,i)→0 in H1(R3).
On the other hand, by Lemma 2.2, we obviously get that limn→∞Iℏn(un)=b0.
In this section, let ρ be fixed in Proposition 3.2. We present the result as follows.
Proposition 4.1. There exists ℏ0>0 such that for ℏ∈(0,ℏ0), Iℏ|M has a critical point uℏ∈Z(ρ). Moreover, limℏ→0I(uℏ)=b0 and the Lagrange multiplier μℏ∈R satisfies
limℏ→0μℏ=μ0,I′ℏ(uℏ)+μℏJ′(uℏ)=0. | (4.1) |
Remark 4.2. By Proposition 3.2, it is easy to verify that (4.1) holds if uℏ is a critical point of Iℏ|M such that lim supℏ→0Iuℏ≤b0.
The proof of Proposition 4.1 can be obtained as in [31] by considering the following contradiction: {ℏn} with ℏn→0 such that for some sequence bℏn→b0, Iℏ admits no critical points in {u∈Z(ρ)|Iℏn(u)≤bℏn}. For brevity, we denote ℏ=ℏn. Then from Lemma 3.1 and Proposition 3.2, there respectively exist κ0>0 and v>0 independent of ℏ and vℏ>0 such that
‖Iℏ|′M(u)‖T∗uM≥vℏ,for u∈Z(ρ)∩[b0−2κ0≤Iℏ≤bℏ],‖Iℏ|′M(u)‖T∗uM≥v,for u∈(Z(ρ)∖Z(ρ/4))∩[b0−2κ0≤Iℏ≤bℏ], | (4.2) |
where
[b1≤Iℏ]={u∈H1(R3)|b1≤Iℏ(u)}, |
[Iℏ≤b2]={u∈H1(R3)|Iℏ(u)≤b2}, |
[b1≤Iℏ≤b2]={u∈H1(R3)|b1≤Iℏ(u)≤b2}, |
for any b1,b2∈R.
Thanks to (4.2), one can get the following deformation lemma.
Lemma 4.3. Let vℏ and v be given as in (4.2). For any κ∈(0,min{κ0,ρv16}), there exists ℏκ>0 such that for ℏ∈(0,ℏκ) there is a deformation η:M→M that satisfied the following conditions:
(a) η(u)=u if u∈M∖(Z(ρ)∩[b0−2κ≤Iℏ]).
(b) Iℏ(η(u))≤Iℏ(u) if u∈M.
(c) η(u)∈Z(ρ)∩[Iℏ≤b0−κ] if u∈Z(ρ/4)∩[Iℏ≤bℏ].
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:={s∈Sℓ−1+||s−s0|≤δ}, |
where s0=(|u1|2,⋯,|uℓ|2). Fix qi∈Ki and qi,ℏ=1ℏqi for i=1,2,⋯,ℓ and define the (ℓ−1)-dimensional initial path by
ξℏ(s)=Bℏℓ∑i=1wsi(⋅−qi,ℏ), |
where Bℏ:=|∑ℓi=1wsi(⋅−qi,ℏ)|−12. Note that we can fix δ>0 small enough such that
ξℏ(s)∈Z(ρ/4) for s∈Sδ |
and
Bℏ→1 as ℏ→0 uniformly in Sδ. |
Define
bℏ:=maxs∈SδIℏ(ξℏ(s)). |
Lemma 4.4. limℏ→0bℏ=b0 and fix any κ∈(0,min{κ0,ρv16}) such that
sups∈∂SδIℏ(ξℏ(s))<b0−2κ, | (4.3) |
where ∂Sδ:={s∈Sℓ+||s−s0|=δ}.
Proof. Since
ℏdb(∫R3|∇ξℏ|2)2→0asℏ→0, |
one can deduce that
Iℏ(ξℏ(s))→ℓ∑i=1Ii(wsi) as ℏ→0 uniformly for s∈Sδ. |
By Lemma 2.3(a), we have
sups∈∂SδIℏ(ξℏ(s))<b0−2κ. |
Proof of Proposition 4.1 in the L2-subcritical case. By Lemma 4.3 and (4.3), we have
η(ξℏ(s))=ξℏ(s) for s∈∂Sδ, | (4.4) |
Iℏ(η(ξℏ(s)))≤b0−κ and η(ξℏ(s))∈Z(ρ) for s∈Sδ. | (4.5) |
Define
Ψi,ℏ=(∫1ℏΩi|u|2)12(ℓ∑i=1∫1ℏΩi|u|2)−12, for u∈M. |
Similar to the case in [31], there exists s1∈Sδ such that Ψi,ℏ(η(ξℏ(s1)))=s0i=|ui|2. Denote
u0,ℏ:=η(ξℏ(s1)), ui,ℏ:=γi,ℏu0,ℏ, | (4.6) |
where γi,ℏ∈C∞0(1ℏ(Ω′i),[0,1]) is a cut-off function such that γi,ℏ=1 on 1ℏΩi and |∇γi,ℏ|≤Cℏ for each i=1,2,⋯,ℓ and some C>0; Ω′i is an open neighborhood of ˉΩi. By (4.5), we have that Gℏ(u0,ℏ)≤C for some C>0 that is independent of ℏ, which implies that
‖u0,ℏ‖H1(R3∖∪ℓi=11ℏΩi)≤Cℏ. | (4.7) |
Then
|ui,ℏ|2=|ui|2+oℏ(1) | (4.8) |
and
Ii(ui)≤Ii(ui,ℏ)+oℏ(1). | (4.9) |
Hence from (4.5)–(4.9), we have
b0−κ≥Iℏ(u0,ℏ)≥ℓ∑i=1Ii(ui,ℏ)+oℏ(1)≥ℓ∑i=1Ii(ui)+oℏ(1)=b0+oℏ(1), |
which is a contradiction. This completes the proof.
Fix qi∈Ki and denote qi,ℏ=1ℏqi; we set
ζℏ(s)=ˉBℏℓ∑i=1ti3/2ui(ti(⋅−qi,ℏ)) for t=(t1,t2,⋯,tℓ)∈(0,+∞)ℓ, |
where ˉBℏ:=|∑ℓi=1t3/2iui(ti(⋅−qi,ℏ))|−12.
Define
bℏ:=maxt∈[1−ˉδ,1+ˉδ]ℓIℏ(ζℏ(t)). |
Note that we can fix ˉδ>0 small enough such that
ζℏ(t)∈Z(ρ/4) for t∈[1−ˉδ,1+ˉδ]ℓ, |
and ˉBℏ→1 holds. Note also that
Ii(ui)>Ii(t3/2iui(ti⋅)) for ti∈[1−ˉδ,1+ˉδ]∖{1}. |
Since
ℏdb(∫R3|∇ζℏ|2)2→0asℏ→0, |
and
Iℏ(ζℏ(t))→ℓ∑i=1Ii(t3/2iui(ti⋅)) as ℏ→0 uniformly for t∈[1−ˉδ,1+ˉδ]ℓ, |
one can get the result as in [31].
Lemma 4.5. limℏ→0bℏ=b0 and fix any κ∈(0,min{κ0,ρv16}) such that
supt∈∂[1−ˉδ,1+ˉδ]ℓIℏ(ζℏ(t))<b0−2κ. | (4.10) |
Proof of Proposition 4.1 in the L2-supercritical case. By Lemma 4.3 and (4.10),
η(ζℏ(t))=ζε(t) if t∈∂[1−ˉδ,1+ˉδ]ℓ, | (4.11) |
Iℏ(η(ζℏ(t)))≤b0−κ and η(ζℏ(t))∈Z(ρ) for t∈[1−ˉδ,1+ˉδ]ℓ. | (4.12) |
Define
Φi,ℏ=(∫1ℏΩi|∇u|2)12−3σ(3σki(2+2σ)a∫1ℏΩi|u|2σ+2)−12−3σ, for u∈M. |
Similar to the case in [31], there exists t1∈[1−ˉδ,1+ˉδ]ℓ such that
Φi,ℏ(η(ζℏ(t1)))=1, i=1,2,⋯,ℓ. | (4.13) |
We denote
ˉu0,ℏ:=η(ζℏ(t1)),ˉui,ℏ:=γi,ℏˉu0,ℏ(ℓ∑i=1|γi,ℏˉu0,ℏ|22)−12. |
Similar to (4.7) and (4.8), we have
‖ˉu0,ℏ‖H1(R3∖∪ℓi=11ℏΩi)=oℏ(1) | (4.14) |
and
ℓ∑i=1|γi,ℏˉu0,ℏ|22=1+oℏ(1). | (4.15) |
From (4.13)–(4.15), we have
ti,ℏ:=(|∇ˉui,ℏ|22)12−3σ(3σki(2+2σ)a|ˉui,ℏ|2σ+22σ+2)13σ−2=Φi,ℏ(ˉu0,ℏ)+oℏ(1)=1+oℏ(1). |
A direct calculation shows that
t∗:=(|t−321,ℏˉu1,ℏ(t−11,ℏ⋅)|2,|t−322,ℏˉu2,ℏ(t−12,ℏ⋅)|2,⋯,|t−32ℓ,ℏˉuℓ,ℏ(t−1ℓ,ℏ⋅)|2)∈Sℓ−1+ |
and
|∇(t−32i,ℏˉui,ℏ(t−1i,ℏ⋅))|22=3σki(2+2σ)a|t−32i,ℏˉui,ℏ(t−1i,ℏ⋅)|2σ+22σ+2. |
Hence by the definition of bsi, we have
ℓ∑i=1Ii(ui)=b0≤ℓ∑i=1Ii(t−32i,ℏˉui,ℏ(t−1i,ℏ⋅))=ℓ∑i=1Ii(ˉui,ℏ)+oℏ(1). |
Similarly, one can get a contradiction.
Let uℏ be the critical point of the modified function Iℏ given in Proposition 4.1.
Completion of proof of Theorem 1.1.
Proof. We show that there exists c>0 independent of ℏ such that
‖uℏ‖2H1(R3∖∪ℓi=11ℏ(Ki)τ)≤e−Cℏ. | (5.1) |
We adopt some arguments from [3,31]. Set ⌊2ℏ−1τ⌋−1:=nℏ. For n=1,2,⋯,nℏ, we take ϕn∈C1(R3,[0,1]) such that
{ϕn(x)=0,if x∈ R3∖En,ϕn(x)=1,if x∈En+1,|∇ϕn(x)|≤2,x∈R3, |
where En:={x∈R3|dist(x,∪ℓi=11ℏ(Ki)τ2)>n−1}. Then by Proposition 3.2,
limℏ→0‖uℏ‖H1(E1)≤limℏ→0ℓ∑i=1‖ui‖H1(R3∖Bℏτ(0))=0. | (5.2) |
Note that for each n=1,2,⋯,nℏ,
suppχℏ=R3∖∪ℓi=11ℏΩi⊂R3∖∪ℓi=11ℏ(Ki)τ⊂ϕ−1n(1). |
Since ⟨I′ℏ(uℏ)+μℏJ′(uℏ),ϕnuℏ⟩=0, we have
a∫R3∇uℏ∇(ϕnuℏ)+ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇(ϕnuℏ)−∫R3K(ℏx)|uℏ|2σ+2ϕn+∫R3μℏu2ℏϕn=−4ℏ−1Gℏ(uℏ)12∫R3χℏ(∇uℏ∇(ϕnuℏ)+u2ℏϕn)=−4ℏ−1Gℏ(uℏ)12∫R3∖∪ℓi=11ℏΩi(∇uℏ∇(ϕnuℏ)+u2ℏϕn)=−4ℏ−1Gℏ(uℏ)12∫R3∖∪ℓi=11ℏΩi(|∇uℏ|2+u2ℏ)≤0. | (5.3) |
Therefore, by (5.3) and the Sobolev embedding,
min{a,μ02}‖uℏ‖2H1(En+1)≤∫R3ϕn(a|∇uℏ|2+μℏu2ℏ)≤∫R3K(ℏx)|uℏ|2σ+2ϕn−a∫R3uℏ∇uℏ∇ϕn−ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇(ϕnuℏ)≤C‖uℏ‖2σ+2H1(En)+a‖uℏ‖2H1(En)−a‖uℏ‖2H1(En+1)−ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇(ϕnuℏ)≤(a+C‖uℏ‖2σH1(E1)+oℏ(1))‖uℏ‖2H1(En)−(a+oℏ(1))‖uℏ‖2H1(En+1), |
where −ℏdb∫R3|∇uℏ|2∫R3∇uℏ∇(ϕnuℏ)≤oℏ(1)(2||uℏ||2H1(En)−||uℏ||2H1(En+1)) as ℏ→0. By (5.2), we have
‖uℏ‖2H1(En+1)≤θ−1ℏ‖uℏ‖2H1(En), |
where
θℏ:=a+min{a,μ02}+oℏ(1)a+oℏ(1)→1+min{1,μ02a}asℏ→0. |
Nothing that nℏ≥τℏ for small values of ℏ, one can take some θ0>1 and obtain
‖uℏ‖2H1(R3∖∪ℓi=11ℏ(Ki)τ)≤‖uℏ‖2H1(Enℏ+1)≤θ−nℏ0‖uℏ‖2H1(E1)≤e−τlnθ0ℏ. |
It follows that for small values of ℏ, Gℏ(uℏ)=0. So uℏ is a solution to the original problem (3.1) for small values of ℏ.
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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