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Review

A meta-analysis of the effect of modelling activities on learning outcomes in mathematics


  • Mathematical modelling is employed to address complex challenges. This study aims to bridge the gap in existing research by providing a more detailed understanding of the role of modelling activities in improving mathematics learning. The present systematic literature review (SLR) can contribute to the existing literature on the influence of modelling activities on learning outcomes in mathematics. We conducted a literature review adhering to the PRISMA protocol and searched databases such as ProQuest, ScienceDirect, Scopus, and SpringerLink. We found 20 studies with a total of 3047 participants (1543 in the experimental group and 1504 in the control group). Using R software, we calculated the effect size by standardised mean difference (SMD) and 95% confidence interval (CI). Our results show significant moderate effects of modelling activities on cognitive skills (effect size = 0.78), significant effects on problem-solving skills (effect size = 0.97), and moderately significant effects on achievement (effect size = 0.70). These findings deepen our understanding by demonstrating how modelling activities can improve cognitive growth, problem-solving skills, and academic achievement, providing strong support for their inclusion in mathematics education. Therefore, educators should incorporate modelling activities as a structured practise to improve students' understanding of mathematical concepts.

    Citation: Riyan Hidayat, Ahmad Fauzi Mohd Ayub, Mohd Afifi bin Bahurudin Setambah, Nurul Hijja Mazlan. A meta-analysis of the effect of modelling activities on learning outcomes in mathematics[J]. STEM Education, 2025, 5(3): 401-424. doi: 10.3934/steme.2025020

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  • Mathematical modelling is employed to address complex challenges. This study aims to bridge the gap in existing research by providing a more detailed understanding of the role of modelling activities in improving mathematics learning. The present systematic literature review (SLR) can contribute to the existing literature on the influence of modelling activities on learning outcomes in mathematics. We conducted a literature review adhering to the PRISMA protocol and searched databases such as ProQuest, ScienceDirect, Scopus, and SpringerLink. We found 20 studies with a total of 3047 participants (1543 in the experimental group and 1504 in the control group). Using R software, we calculated the effect size by standardised mean difference (SMD) and 95% confidence interval (CI). Our results show significant moderate effects of modelling activities on cognitive skills (effect size = 0.78), significant effects on problem-solving skills (effect size = 0.97), and moderately significant effects on achievement (effect size = 0.70). These findings deepen our understanding by demonstrating how modelling activities can improve cognitive growth, problem-solving skills, and academic achievement, providing strong support for their inclusion in mathematics education. Therefore, educators should incorporate modelling activities as a structured practise to improve students' understanding of mathematical concepts.



    In this paper, we consider the following Kirchhoff-type elliptic problem:

    {(a+bΩ|u|2dx)Δu=λ|u|q2uln|u|2+μ|u|2u,xΩ,u(x)=0,xΩ, (1.1)

    where ΩR4 is a bounded domain with smooth boundary Ω, 2<q<4, a, b, λ, and μ are positive parameters.

    As a natural generalization of (1.1), we obtain the following Kirchhoff-type elliptic problem:

    {(a+bΩ|u|2dx)Δu=f(x,u),xΩ,u(x)=0,xΩ. (1.2)

    When the space dimension N=1, the equation in (1.2) is closely related to the stationary version of the following wave equation:

    ρ2ut2(P0h+E2LL0|ux|2dx)2ux2=f(x,u), (1.3)

    which was proposed by Kirchhoff [1] to describe the transversal oscillations of a stretched string. Here ρ>0 is the mass per unit length, P0 is the base tension, E is the Young modulus, h is the area of the cross section, and L is the initial length of the string. Such problems are widely applied in engineering, physics, and other applied sciences (see [2,3,4]). A remarkable feature of problem (1.2) is the presence of the nonlocal term, which brings essential difficulties when looking for weak solutions to it in the framework of variational methods since, in general, one cannot deduce from unu weakly in H10(Ω) the convergence Ω|un|2dxΩ|u|2dx.

    In recent years, various techniques, such as the mountain pass lemma, the Nehari manifold approach, genus theory, Morse theory, etc., have been used to study the existence and multiplicity of weak solutions to Kirchhoff problems with different kinds of nonlinearities in the general dimension. We refer the interested reader to [5,6,7,8,9,10] and the references therein. In particular, when dealing with (1.2) by using variational methods, f is usually required to satisfy the following Aimbrosetti–Rabinowitz condition: i.e., for some ν>4 and R>0, there holds

    0<νF(x,t)tf(x,t), |t|>R,xΩ,

    which implies that f is 4-superlinear about t at infinity, that is,

    limt+F(x,t)t4=+, (1.4)

    where F(x,t)=t0f(x,τ)dτ. This guarantees the boundedness of any (PS) sequence of the corresponding energy functional in H10(Ω). In addition, assume that f satisfies the subcritical growth condition

    |f(x,t)|C(|t|q1+1), tR, xΩ, (1.5)

    where C>0, 2<q<2:=2NN2. Then it follows from [11, Lemma 1] that the functional satisfies the compactness condition. Combining (1.4) with (1.5), one has q>4, which, together with q<2, implies N<4. Hence, problem (1.2) is usually studied in dimension three or less. For example, Chen et al. [12] studied problem (1.2) with N3 and f(x,u)=λh(x)|u|q2u+g(x)|u|p2u, where a, b, λ>0, 1<q<2<p<2, and h,gC(¯Ω) are sign-changing functions. They obtained the existence of multiple positive solutions with the help of Nehari manifolds and fibering maps. Silva [13] considered the existence and multiplicity of weak solutions to problem (1.2) in a bounded smooth domain ΩR3 with f(x,u)=|u|γ2u and γ(2.4) by using fibering maps and the mountain pass lemma. Later, the main results in [13] were extended to the parallel p-Kirchhoff problem in a previous work of ours [14].

    For the critical problem (1.2) with N4, it was shown in [15] that when a and b satisfy appropriate constraints, the interaction between the Kirchhoff operator and the critical term makes some useful variational properties of the energy functional valid, such as the weak lower semi-continuity and the Palais–Smale properties. Naimen [16] considered problem (1.2) with f(x,u)=λuq+μu21(1q<21) when N=4. By applying the variational method and the concentration compactness argument, he proved the existence of solutions to problem (1.2). Later, Naimen and Shibata [17] considered the same problem with N=5, and the existence of two solutions was obtained by using the variational method. Faraci and Silva [18] considered problem (1.2) with f(x,u)=λg(x,u)+|u|22u in ΩRN(N>4). By using variational properties and the fiber maps of the energy functional associated with the problem, the existence, nonexistence, and multiplicity of weak solutions were obtained under some assumptions on a, b, λ, and g(x,u). Li et al. [19] considered a Kirchhoff-type problem without a compactness condition in the whole space RN(N3). By introducing an appropriate truncation on the nonlocal coefficient, they showed that the problem admits at least one positive solution.

    On the other hand, equations with logarithmic nonlinearity have also been receiving increasing attention recently, due to their wide application in describing many phenomena in physics and other applied sciences such as viscoelastic mechanics and quantum mechanics theory ([20,21,22,23]). The logarithmic nonlinearity is sign-changing and satisfies neither the monotonicity condition nor the Ambrosetti–Rabinowitz condition, which makes the study of problems with logarithmic nonlinearity more interesting and challenging. Therefore, much effort has also been made in this direction during the past few years. For example, Shuai [24] considered problem (1.2) with a>0, b=0, and f(x,u)=a(x)uln|u|, where a(x)C(¯Ω) can be positive, negative, or sign-changing in ¯Ω. Among many interesting results, he proved, by the use of the Nehari manifold, the symmetric mountain pass lemma and Clark's Theorem, that the problem possesses two sequences of solutions when a(x) is sign-changing. Later, the first two authors of this paper and their co-author [25] investigated the following critical biharmonic elliptic problem:

    {Δ2u=λu+μulnu2+|u|22u,xΩ,u=uν=0,xΩ, (1.6)

    where 2:=2NN4 is the critical Sobolev exponent for the embedding H20(Ω)L2(Ω). Both the cases μ>0 and μ<0 are considered in [25], and the existence of nontrivial solutions was derived by combing the variational methods with careful estimates on the logarithmic term. It is worth mentioning that the result with μ>0 implies that the logarithmic term plays a positive role in problem (1.6) to admit a nontrivial solution. Later, Zhang et al. [26] not only weakened the existence condition in [25], but also specified the types and the energy levels of solutions by using Brézis–Lieb's lemma and Ekeland's variational principle.

    Inspired mainly by the above-mentioned literature, we consider a critical Kirchhoff problem with a logarithmic perturbation and investigate the combined effect of the nonlocal term, the critical term, and the logarithmic term on the existence of weak solutions to problem (1.1). We think that at least the following three essential difficulties make the study of such a problem far from trivial. The first one is that since the power corresponding to the critical term is equal to that corresponding to the nonlocal term in the energy functional I(u) (see (2.2)), it is very difficult to obtain the boundedness of the (PS) sequences for I(u). The second one is the lack of compactness of the Sobolev embedding H10(Ω)L4(Ω), which prevents us from directly establishing the (PS) condition for I(u). The third one is caused by the logarithmic nonlinearity, which we have mentioned above.

    To overcome these difficulties and to investigate the existence of weak solutions to problem (1.1), we first consider a sequence of approximation problems (see problem (2.7) in Section 2) and obtain a bounded (PS) sequence for each approximation problem based on a result by Jeanjean [27]. Then, with the help of some delicate estimates on the truncated Talenti functions and Brézis–Lieb's lemma, we prove that the (PS) sequence has a strongly convergent subsequence. Then we obtain a solution un to problem (2.7) for almost every νn(σ,1]. Finally, we show that the original problem admits a mountain pass type solution if the sequence of the approximation solution {un} is bounded by the mountain pass lemma.

    The organization of this paper is as follows: In Section 2, some notations, definitions, and necessary lemmas are introduced. The main results of this paper are also stated in this section. In Section 3, we give detailed proof of the main results.

    We start by introducing some notations and definitions that will be used throughout the paper. In what follows, we denote by p the Lp(Ω) norm for 1p. The Sobolev space H10(Ω) will be equipped with the norm u:=uH10(Ω)=u2, which is equivalent to the full one due to Poincaré's inequality. The dual space of H10(Ω) is denoted by H1(Ω). We use and to denote the strong and weak convergence in each Banach space, respectively, and use C, C1, C2, ... to denote (possibly different) positive constants. BR(x0) is a ball of radius R centered at x0. We use ω4 to denote the area of the unit sphere in R4. For all t>0, O(t) denotes the quantity satisfying |O(t)t|C, O1(t) means there exist two positive constants C1 and C2 such that C1tO1(t)C2t, o(t) means |o(t)t|0 as t0, and on(1) is an infinitesimal as n. We use S>0 to denote the best embedding constant from H10(Ω) to L4(Ω), i.e.,

    u4S12u,  uH10(Ω). (2.1)

    In this paper, we consider weak solutions to problem (1.1) in the following sense:

    Definition 2.1. (weak solution) A function uH10(Ω) is called a weak solution to problem (1.1) if for every φH10(Ω), there holds

    aΩuφdx+bu2ΩuφdxλΩ|u|q2uφlnu2dxμΩ|u|2uφdx=0.

    Define the energy functional associated with problem (1.1) by

    I(u)=a2u2+b4u4+2λq2uqqλqΩ|u|qlnu2dxμ4u44,  uH10(Ω). (2.2)

    From 2<q<4, one can see that I(u) is well defined and is a C1 functional in H10(Ω) (see [18]). Moreover, each critical point of I is also a weak solution to problem (1.1).

    We introduce a definition of a local compactness condition, usually called the (PS)c condition.

    Definition 2.2. ((PS)c condition [28]) Assume that X is a real Banach space, I:XR is a C1 functional, and cR. Let {un}X be a (PS)c sequence of I(u), i.e.,

    I(un)c and I(un)0 in X1(Ω) as n,

    where X1 is the dual space of X. We say that I satisfies the (PS)c condition if any (PS)c sequence has a strongly convergent subsequence.

    The following three lemmas will play crucial roles in proving our main results. The first one is the mountain pass lemma, the second one is Brézis–Lieb's lemma, and the third one will be used to deal with the logarithmic term.

    Lemma 2.1. (mountain pass lemma [28]) Assume that (X,X) is a real Banach space, I:XR is a C1 functional, and there exist β>0 and r>0 such that I satisfies the following mountain pass geometry:

    (i) I(u)β>0 if uX=r;

    (ii) There exists a ¯uX such that ¯uX>r and I(¯u)<0.

    Then there exists a (PS)c sequence such that

    c:=infγΓmaxt[0,1]I(γ(t))β,

    where

    Γ={γC([0,1],X):γ(0)=0,γ(1)=¯u}.

    c is called the mountain level. Furthermore, c is a critical value of I if I satisfies the (PS)c condition.

    Lemma 2.2. (Brézis–Lieb's lemma [29]) Let p(0,). Suppose that {un} is a bounded sequence in Lp(Ω) and unu a.e. on Ω. Then

    limn(unppunupp)=upp.

    Lemma 2.3. (1) For all t(0,1], there holds

    |tlnt|1e. (2.3)

    (2) For any α,δ>0, there exists a positive constant Cα,δ such that

    |lnt|Cα,δ(tα+tδ), t>0. (2.4)

    (3) For any δ>0, there holds

    lnttδ1δe, t>0. (2.5)

    (4) For any qR{0}, there holds

    2tqqtqlnt22, t>0. (2.6)

    Proof. (1) Let k1(t):=tlnt, t(0,1]. Then simple analysis shows that k1(t) is decreasing in (0,1e), increasing in (1e,1), and attaining its minimum at tk1=1e with k1(tk1)=1e. Moreover, k1(t)0 for all t(0,1]. Consequently, |k1(t)|1e, t(0,1].

    (2) For any α, δ>0, from

    limt0+lnttδ=0  and  limt+lnttα=0,

    one sees that there exist constants Cδ,Cα>0 and 0<M1<M2<+ such that |lnt|Cδtδ when t(0,M1) and |lnt|Cαtα when t(M2,). Moreover, it is obvious that there exists Cα>0 such that |lnt|Cαtα when t[M1,M2]. Therefore,

    |lnt|(Cα+Cα)tα+CδtδCα,δ(tα+tδ), t>0,

    where Cα,δ=max{Cδ,Cα+Cα}.

    (3) For any δ>0, set k2(t):=lnttδ, t>0. Then direct computation shows that k2(t)>0 in (0,e1δ), k2(t)<0 in (e1δ,+), and k2(t) attain their maximum at tk2=e1δ. Therefore, k2(t)k2(e1δ)=1δe, t>0.

    (4) Let k3(t):=2tqqtqlnt2, t>0, where qR{0}. From

    k3(t)=q2tq1lnt2,  t>0,

    we know that k3(t) has a unique critical point, tk3=1 in (0,+). Moreover, k3(t)>0 in (0,1), k3(t)<0 in (1,+), and k3(t) attain their maximum at tk3. Consequently, k3(t)k3(1)=2. The proof is complete.

    Following the ideas of [16], we consider the following approximation problem:

    {(a+bΩ|u|2dx)Δu=λ|u|q2uln|u|2+νμ|u|2u,xΩ,u(x)=0,xΩ, (2.7)

    where ν(σ,1] for some σ(12,1). Associated functionals are defined by

    Iν(u)=a2u2+b4u4+2λq2uqqλqΩ|u|qlnu2dx14νμu44,  uH10(Ω).

    Noticing that Iν(u)=Iν(|u|) and I(u)=I(|u|), we may assume that u0 in the sequel.

    To obtain the boundedness of the (PS) sequences for Iν, we need the following result by Jeanjean: [27].

    Lemma 2.4. Assume that (X,X) is a real Banach space, and let JR+ be an interval. We consider a family (Iν)νJ of C1-functionals on X of the form

    Iν(u)=A(u)νB(u), νJ,

    where B(u)0 for all uX and either A(u)+ or B(u)+ as uX. Assume, in addition, that there are two points e1,e2 in X such that for all νJ, there holds

    cν:=infγΓmaxt[0,1]Iν(γ(t))max{Iν(e1),Iν(e2)},

    where

     Γ={γC([0,1],X):γ(0)=e1,γ(1)=e2}.

    Then, for almost every νJ, there is a sequence {un}X such that

    (i) {un} is bounded,  (ii) Iν(un)cv, as n,  (iii) Iν(un)0 in X1(Ω) as n,

    where X1 is the dual space of X.

    At the end of this section, we state the main results of this paper, which can be summarized into the following theorem:

    Theorem 2.1. Let b,μ>0 satisfy bS2<μ<2bS2, and take 12<σ<1 such that bS2σ<μ. Assume one of the following (C1), (C2), or (C3) holds.

    (C1) a>0, and λ>0 is small enough.

    (C2) λ>0, and a>0 is large enough.

    (C3) a>0, λ>0, and b<μS2 is sufficiently close to μS2.

    Then problem (2.7) has a solution for almost every ν(σ,1]. In addition, we can find an increasing sequence νn(σ,1] such that νn1 as n and denote by un the corresponding solution to problem (2.7). Then either (i) or (ii) below holds.

    (i) limnunH10(Ω)=;

    (ii) {un} is bounded in H10(Ω) and consequently, problem (1.1) has a nontrivial weak solution.

    In particular, if ΩR4 is strictly star-shaped, then problem (1.1) has a nontrivial weak solution.

    We are going to show that there exists a bounded (PS) sequence of the energy functional Iν for almost every ν(σ,1]. For this, let us introduce the Talenti functions (see [16]). For any ε>0, define

    Uε(x)=812εε2+|x|2, xR4.

    Then Uε(x) is a solution to the critical problem

    Δu=u3,xR4,   (3.1)

    and it satisfies Uε2=Uε44=S2, where S=infuH10(Ω){0}u2u24=Uε2Uε24 (an equivalent characterization of S defined in (2.1)).

    Lemma 3.1. Let τC0(Ω) be a cut-off function such that 0τ(x)1 in Ω with τ(x)=1 if |x|<R and τ(x)=0 if |x|>2R, where R>0 is a constant such that B2R(0)Ω (Here we assume, without loss of generality, that 0Ω). Set uε(x)=τ(x)Uε(x). Then we have

    uε2=S2+O(ε2),uε44=S2+O(ε4),uεqq=O1(ε4q)+O(εq), (3.2)

    and

    Ωuqεlnu2εdx=O1(ε4qln(1ε))+O(εqlnε)+O(ε4q), (3.3)

    where q(2,4).

    Set vε(x)=uε(x)uε4. Then, as ε0,

    vε2=S+O(ε2),vε44=1,vεqq=O1(ε4q)+O(εq),

    and

    Ωvqεlnv2εdx=O1(ε4qln(1ε))+O(εqlnε)+O(ε4q).

    Proof. We only prove (3.2) and (3.3). The proof of other results is similar and can be referred to [30]. Using the properties of the cut-off function τ, one has

       Ωuqεdx=CB2R(0)τqεq(ε2+|x|2)qdx=CBR(0)εq(ε2+|x|2)qdx+CB2R(0)BR(0)τqεq(ε2+|x|2)qdx:=J1+J2.

    By changing the variable and applying the polar coordinate transformation, we can estimate J1 as follows:

    J1=CBRε(0)εqε4(1+|y|2)qε2qdy=Cε4qBRε(0)1(1+|y|2)qdy=Cω4ε4qRε0r3(1+r2)qdr=C1ε4q(+0r3(1+r2)qdr+Rεr3(1+r2)qdr)=C2ε4q+O(εq), (3.4)

    where we have used the fact that

    |+Rεr3(1+r2)qdr|+Rεr32qdr=O(ε2q4),

    and

    +0r3(1+r2)qdrC.

    On the other hand,

    |J2|CεqB2R(0)BR(0)1|x|2qdx=O(εq). (3.5)

    Hence, (3.2) follows from (3.4) and (3.5).

    Next, we shall prove (3.3). According to the definition of uε, we obtain

    Ωuqεlnu2εdx=ΩτqUqεln(U2ετ2)dx=ΩτqUqεlnτ2dx+ΩτqUqεlnU2εdx:=J3+J4.

    By direct computation, we have

    |J3|=|B2R(0)BR(0)τqUqεlnτ2dx|CB2R(0)BR(0)Uqεdx=O(εq).

    Rewrite J4 as follows:

    J4=BR(0)UqεlnU2εdx+B2R(0)BR(0)τqUqεlnU2εdx:=J41+J42.

    By using inequality (2.4) with α1=δ1<4q, one has

    |J42|CB2R(0)BR(0)(Uqδ1ε+Uq+δ1ε)dx=O(εqδ1)+O(εq+δ1)=O(εqδ1),

    where

    B2R(0)BR(0)Uqδ1εdx=CB2R(0)BR(0)εqδ1(ε2+|x|2)qδ1dxCB2R(0)BR(0)εqδ1|x|2q2δ1dx=O(εqδ1),

    and

    B2R(0)BR(0)Uq+δ1εdx=CB2R(0)BR(0)εq+δ1(ε2+|x|2)q+δ1dxCB2R(0)BR(0)εq+δ1|x|2q2δ1dx=O(εq+δ1).

    In addition,

    J41=CBR(0)εq(ε2+|x|2)qln(Cεε2+|x|2)dx=Cε4BRε(0)εqε2q(1+|y|2)qln(Cεε2(1+|y|2))dy=Cε4qBRε(0)1(1+|y|2)qln(C1ε(1+|y|2))dy=Cε4qln(1ε)BRε(0)1(1+|y|2)qdy+Cε4qBRε(0)1(1+|y|2)qln(C1+|y|2)dy=Cε4qln(1ε)(R41(1+|y|2)qdyBcRε(0)1(1+|y|2)qdy)   +Cε4qBRε(0)1(1+|y|2)qln(C1+|y|2)dy=C1ε4qln(1ε)Cε4qln(1ε)BcRε(0)1(1+|y|2)qdy   +Cε4qBRε(0)1(1+|y|2)qln(C1+|y|2)dy. (3.6)

    By direct computation, one obtains

    |BcRε(0)1(1+|y|2)qdy|=|ω4+Rεr3(1+r2)qdr|C+Rεr32qdr=O(ε2q4), (3.7)

    and

    BRε(0)1(1+|y|2)qln(C1+|y|2)dy=R41(1+|y|2)qln(C1+|y|2)dyBcRε(0)1(1+|y|2)qln(C1+|y|2)dyC+|BcRε(0)1(1+|y|2)qln(C1+|y|2)dy|C+O(ε2q42δ2), (3.8)

    where we have used the fact that

    |BcRε(0)1(1+|y|2)qln(C1+|y|2)dy|CBcRε(0)(1(1+|y|2)qδ2+1(1+|y|2)q+δ2)dy=Cω4+Rεr3(1(1+r2)qδ2+1(1+r2)q+δ2)drC1+Rε(r32q+2δ2+r32q2δ2)dr=O(ε2q42δ2),

    by recalling (2.4) with α2=δ2<q2. Substituting (3.7) and (3.8) into (3.6), one arrives at

    J41=C1ε4qln(1ε)Cε4qln(1ε)O(ε2q4)+Cε4q(C+O(ε2q42δ2))=C1ε4qln(1ε)+O(εqlnε)+O(ε4q)+O(εq2δ2).

    Putting the estimates on J3, J41, and J42 together, one obtains

    Ωuqεlnu2εdx=O(εq)+O(εqδ1)+C1ε4qln(1ε)+O(εqlnε)+O(ε4q)+O(εq2δ2).

    Therefore, (3.3) follows by taking δ1 and δ2 suitably small. The proof is complete.

    With the help of Lemma 3.1, the existence of bounded (PS) sequences of Iν can be derived.

    Lemma 3.2. Let b>0, μ>0 satisfy bS2<μ and take σ<1 such that bS2σ<μ. Then there exists a bounded (PS) sequence of the energy functional Iν for almost every ν(σ,1].

    Proof. Applying (2.5) with δ<4q, using the Sobolev embedding inequality, and noticing that ν1, one has

    Iν(u)=a2u2+b4u4+2q2λuqqλqΩuqlnu2dx14νμu44a2u2λqΩuqlnu2dx14μu44a2u2Cuq+δ14μS2u4=u2(a2Cuq+δ214μS2u4).

    Hence, there exist positive constants β and ρ such that

    Iν(u)β for all u=ρ.

    On the other hand, from (2.6), one has tqlnt22q(tq1) for t>0. Let vε be given in Lemma 3.1. Then, as ε0, we have, for all t>0,

    Iν(tvε)=a2t2vε2+b4t4vε4+2q2λtqvεqqλqtqΩvqεln(tvε)2dx14νμt4vε44a2t2vε2+b4t4vε4+2q2λtqvεqq2q2λΩ((tvε)q1)dx14νμt4vε44=a2t2vε2+b4t4vε4+2q2λ|Ω|14νμt4vε44=a2t2(S+O(ε2))+b4t4(S+O(ε2))2+2q2λ|Ω|14νμt4=a2St214(νμbS2)t4+2q2λ|Ω|+O(ε2),

    which ensures that

    Iν(tvε)a2St214(σμbS2)t4+2q2λ|Ω|+1,

    for all sufficiently small ε>0. Fix such an ε and denote it by ε0. Then it follows from bS2σ<μ and the above inequality that

    limtIν(tvε0)= (3.9)

    uniformly for ν(σ,1], which implies that there exists a t>0 such that tvε0>ρ and Iν(tvε0)<0 for all ν(σ,1]. Thus, Iν satisfies the mountain pass geometry around 0, which is determined independently of ν(σ,1]. Now define

    cν:=infγΓmaxt[0,1]Iν(γ(t)) and Γ={γC([0,1],H10(Ω)):γ(0)=0,γ(1)=tvε0}.

    Then, following the mountain pass lemma, we have

    cνβ>max{Iν(γ(0)),Iν(γ(1))}, (3.10)

    for all ν(σ,1]. Set

    Iν(u)=A(u)νB(u),  uH10(Ω),

    where B(u):=14μu44 and A(u):=a2u2+b4u4+2q2λuqqλqΩuqlnu2dx. By a simple analysis, one has

    B(u)0 for all uH10(Ω), and A(u)+, as u. (3.11)

    In view of (3.10), (3.11), and according to Lemma 2.4, there exists a bounded (PS)cν sequence of Iν for almost every ν(σ,1]. The proof is complete.

    Next, we prove the local compactness for Iν(u), which will play a fundamental role in proving the main results.

    Lemma 3.3. Let b>0, μ>0 satisfy bS2<μ<2bS2 and take 12<σ<1 such that bS2σ<μ. Suppose that one of the following (C1), (C2), or (C3) holds.

    (C1) a>0, and λ>0 is small enough.

    (C2) λ>0, and a>0 is large enough.

    (C3) a>0, λ>0, and b<μS2 is close enough to μS2.

    Let {un}H10(Ω) be a bounded (PS) sequence for Iν(u) with ν(σ,1] at the level c with c<c(K), where c(K):=a2S24(νμbS2)=12aK+14bK2νμK24S2>0 and K:=aS2μνbS2, that is, Iν(un)c and Iν(un)0 in H1(Ω) as n. Then there exists a subsequence of {un} (still denoted by {un} itself) and a uH10(Ω) such that unu in H10(Ω) as n.

    Proof. By the boundedness of {un} in H10(Ω) and the Sobolev embedding, one sees that there is a subsequence of {un} (which we still denote by {un}) such that, as n,

    unu in H10(Ω),unu in Ls(Ω), 1s<4,unu a.e. in Ω. (3.12)

    It follows from unu a.e. in Ω as n that

    uqnlnu2nuqlnu2 a.e. in Ω as n. (3.13)

    Moreover, by virtue of (2.4) with α=δ<4q, we get

    |uqnlnu2n|Cδ,δ(uqδn+uq+δn)Cδ,δ(uqδ+uq+δ) in L1(Ω) as n. (3.14)

    With the help of (3.13), (3.14), and Lebesgue's dominating convergence theorem, we obtain

    limnΩuqnlnu2ndx=Ωuqlnu2dx. (3.15)

    Similarly, we have

    limnΩuq1nulnu2ndx=Ωuqlnu2dx. (3.16)

    To prove unu in H10(Ω) as n, set wn=unu. Then {wn} is also a bounded sequence in H10(Ω). So there exists a subsequence of {wn} (which we still denote by {wn}) such that limnwn2=l0. We claim that l=0. Otherwise, according to (3.12) and Brézis–Lieb's lemma, we see that, as n,

    un2=wn2+u2+on(1),un44=wn44+u44+on(1). (3.17)

    It follows from (3.12), (3.16), and (3.17) that

    on(1)=Iν(un),u=au2+bun2u2λΩuqlnu2dxνμu44+on(1)=au2+bwn2u2+bu4λΩuqlnu2dxνμu44+on(1), as n. (3.18)

    By (3.12), (3.15), (3.17), and (3.18), the boundedness of {un} in H10(Ω) and the Sobolev embedding, we obtain

    on(1)=Iν(un),un=au2+awn2+2bwn2u2+bwn4+bu4 λΩuqlnu2dxνμu44νμwn44+on(1)=Iν(un),u+awn2+bwn4+bwn2u2νμwn44+on(1)=awn2+bwn4+bwn2u2νμwn44+on(1)awn2+bwn4+bwn2u2νμS2wn4+on(1)=[(a+bu2)(νμbS2)S2wn2]wn2+on(1), as n. (3.19)

    Then, we have

    limnwn2=l(a+bu2)S2νμbS2. (3.20)

    It follows from Iν(un)c, (3.12), (3.15), and (3.17) that, as n,

    c+on(1)=Iν(un)=a2un2+b4un4+2q2λunqqλqΩuqnlnu2ndx14νμun44=a2wn2+b4wn4+b4wn2u214νμwn44+on(1)    +a2u2+b4u4+b4wn2u2+2q2λuqqλqΩuqlnu2dx14νμu44:=I1+I2,

    where

    I1=a2wn2+b4wn4+b4wn2u214νμwn44+on(1),I2=a2u2+b4u4+b4wn2u2+2q2λuqqλqΩuqlnu2dx14νμu44+on(1).

    By (3.19) and (3.20), we have, as n,

    I1+on(1)=I114Iν(un),un=a4wn2a4(a+bu2)S2νμbS2=a2S24(νμbS2)+abS24(νμbS2)u2. (3.21)

    Applying (2.5) with δ<4q, from (3.20) and the Sobolev embedding, one has, as n,

    I2a2u2+b4u4+b4wn2u2λqCuq+δ14νμS2u4+on(1)a2u2+b4u4+b4(a+bu2)S2νμbS2u2λqCuq+δ14νμS2u4+on(1)=(12+bS24(νμbS2))au2+14(b+b2S2νμbS2νμS2)u4λqCuq+δ+on(1)=2νμbS24(νμbS2)au2+νμ(2bS2νμ)4S2(νμbS2)u4λqCuq+δ+on(1). (3.22)

    In view of (3.21), (3.22), and the assumption of ν(σ,1], we obtain, as n,

         c+on(1)=Iν(un)=I1+I2a2S24(νμbS2)+abS24(νμbS2)u2+2νμbS24(νμbS2)au2   +νμ(2bS2νμ)4S2(νμbS2)u4λqCuq+δ=a2S24(νμbS2)+νμa2(νμbS2)u2+νμ(2bS2νμ)4S2(νμbS2)u4λqCuq+δ=a2S24(νμbS2)+μa2(μbS2ν1)u2+μ(2bS2νμ)4S2(μbS2ν1)u4λqCuq+δa2S24(νμbS2)+μa2(μbS2)u2+μ(2bS2μ)4S2(μbS2)u4λqCuq+δ:=a2S24(νμbS2)+h(u), (3.23)

    where

    h(t)=μa2(μbS2)t2+μ(2bS2μ)4S2(μbS2)t41qλCtq+δ, for t>0.

    By a simple analysis, (C1), (C2), or (C3) imply that

    h(t)>0, for t>0. (3.24)

    Indeed, if the parameters satisfy (C1), then, for any a>0, set

    g1(t):=μaq2C(μbS2)t2qδ+μ(2bS2μ)q4CS2(μbS2)t4qδ, t>0.

    From

    g1(t)=(μq(2bS2μ)(4qδ)4CS2(μbS2)t2μaq(q+δ2)2C(μbS2))t1qδ, t>0,

    we know that g1(t) has a unique critical point

    t1=(2aS2(q+δ2)(4qδ)(2bS2μ))12

    in (0,+). Moreover, g1(t)<0 in (0,t1), g1(t)>0 in (t1,+) and g1(t) attain their minimum at t1. Consequently,

    g1(t)g1(t1)=μaq2C(μbS2)(2aS2(q+δ2)(4qδ)(2bS2μ))q+δ22+μq(2bS2μ)4CS2(μbS2)(2aS2(q+δ2)(4qδ)(2bS2μ))4qδ2>0.

    Consequently, for λ>0 small enough, we have g1(t)>λ for t(0,+). This implies (3.24). By applying a similar argument, one can show that (3.24) is also true if (C2) or (C3) hold. It then follows from (3.23) and (3.24) that cc(K):=a2S24(νμbS2), a contradiction. Thus l=0, i.e., un converges strongly to u in H10(Ω). The proof is complete.

    With the help of the Talenti functions given in Lemma 3.1, we show that the mountain pass level of Iν(u) around 0 is smaller than c(K).

    Lemma 3.4. Let b,μ>0 satisfy bS2<μ and take σ<1 such that bS2σ<μ. Suppose that ν(σ,1]. Then there exists a u>0 such that

    supt0Iν(tu)<c(K), (3.25)

    where c(K) is the positive constant given in Lemma 3.3.

    Proof. Let vε be given in Lemma 3.1. According to the definition of Iν, one sees that limt0Iν(tvε)=0 and limt+Iν(tvε)= uniformly for ε(0,ε1), where ε1 is a sufficiently small but fixed number. Therefore, there exists 0<t1<t2<+, independent of ε, such that

    Iν(tvε)<c(K),  t(0,t1][t2,+). (3.26)

    For t[t1,t2], it follows from Lemma 3.1 that, as n,

    Iν(tvε)=a2t2vε2+b4t4vε4+2q2λtqvεqqλqtqΩvqεln(tvε)2dx14νμt4vε44=a2t2(S+O(ε2))+b4t4(S+O(ε2))2+O1(ε4q)+O(εq)   [O1(ε4qln(1ε))+O(εqlnε)+O(ε4q)]14νμt4=a2St214(νμbS2)t4+O(ε2)+O1(ε4q)+O(εq)   O1(ε4qln(1ε))+O(εqlnε)+O(ε4q):=g(t)O1(ε4qln(1ε))+O(ε4q),

    where g(t):=a2St214(νμbS2)t4. According to the positivity of g(t) for t>0, suitably small, and the fact that limtg(t)=, there exists a t0>0 such that maxt>0g(t)=g(t0). So, one has g(t0)=t0[aS(νμbS2)t20]=0, that is,

    t20=aSνμbS2.

    It follows from the definition of g(t) that

    maxt>0g(t)=g(t0)=a2S24(νμbS2). (3.27)

    Then, for t[t1,t2], one sees

    Iν(tvε)maxt[t1,t2]g(t)O1(ε4qln(1ε))+O(ε4q)maxt>0g(t)O1(ε4qln(1ε))+O(ε4q), as ε0. (3.28)

    From

    limε0ε4qε4qln1ε=0,

    we have

    O1(ε4qln(1ε))+O(ε4q)<0, (3.29)

    for suitably small ε. Fix such an ε>0. It then follows from (3.27), (3.28), and (3.29) that

    Iν(tvε)<c(K), t[t1,t2]. (3.30)

    Take uvε, and we obtain (3.25) by combining (3.26) with (3.30). The proof is complete.

    On the basis of the above lemmas, we can now prove Theorem 2.1.

    Proof of Theorem 2.1. From (3.10) and Lemma 3.4, we obtain

    0<βcνmaxt[0,1]Iν(ttvε)supt0Iν(tvε)<c(K), (3.31)

    where cν is defined in Lemma 3.2. This, together with Lemmas 3.2 and 3.3, shows that there exists a bounded (PS)cν sequence of the energy functional Iν for almost every ν(σ,1], which strongly converges to some nontrivial function in H10(Ω) up to subsequences. Thus, the approximation problem (2.7) has a nontrivial weak solution for almost every ν(σ,1]. Take an increasing sequence νn(σ,1] such that νn1 as n, and denote the corresponding solution by un, which fulfills Iν(un)=cνnβ. It is obvious that either un as n or {un}H10(Ω) is bounded.

    To show that problem (1.1) admits a mountain pass-type solution for the latter case, we first prove

    cνnc1 as n. (3.32)

    Assume by contradiction that

    c1<limncνn,

    where we have used the fact that cν is nonincreasing in ν since B(u) is nonnegative for all uH10(Ω). Let

    θ:=limncνnc1>0. (3.33)

    Following from the definition of c1, there exists a γ1Γ such that

    maxt[0,1]I(γ1(t))<c1+14θ. (3.34)

    Then, in view of the fact that Iνn(γ1(t))=I(γ1(t))+14(1νn)μγ1(t)44 and (3.34), we have

    maxt[0,1]Iνn(γ1(t))<c1+14θ+14(1νn)μmaxt[0,1]γ1(t)44. (3.35)

    Since μγ1(t)44 is continuous in t, we deduce μmaxt[0,1]γ1(t)44C. From this and (3.35), one has

    limnmaxt[0,1]Iνn(γ1(t))c1+12θ.

    On the other hand, by virtue of the definition of cνn, we have

    limnmaxt[0,1]Iνn(γ1(t))limncνn.

    By using the above two inequalities, we obtain

    limncνnc1+12θ,

    a contradiction with (3.33).

    Next, we claim that {un} is a (PS)c1 sequence of I(u). Indeed, by (3.32), one has

        I(un)=a2un2+b4un4+2q2λunqqλqΩuqnlnu2ndx14μun44=a2un2+b4un4+2q2λunqqλqΩuqnlnu2ndx14νnμun44+14(νn1)μun44=Iνn(un)+14(νn1)μun44=c1+on(1), as n.

    Similarly, for any φH10(Ω),

        I(un),φ=aΩunφdx+bun2ΩunφdxλΩuq1nφlnu2ndxμΩu3nφdx=aΩunφdx+bun2ΩunφdxλΩuq1nφlnu2ndx   νnμΩu3nφdx+(νn1)μΩu3nφdx=Iνn(un),φ+(νn1)μΩu3nφdx=on(1), as n.

    Hence, {un} is a bounded (PS)c1 sequence for I. It then follows from (3.31) and Lemma 3.3 that there exists a uH10(Ω) such that unu in H10(Ω) as n, and u is a mountain pass-type solution to problem (1.1).

    Finally, we prove the last part of Theorem 2.1. Take a sequence {νn}(σ,1] such that νn1 as n and denote the corresponding solution to problem (2.7) by {un}. We first show that {un} is bounded in H10(Ω). Assume by contradiction that un as n. Set ˜wn:=unun0. Then ˜wn=1, and there is a subsequence of {˜wn} (which we still denote by {˜wn}) such that ˜wn˜w0 in H10(Ω) as n. Notice that, for all φH10(Ω), we have

    0=Iνn(un),φ=aΩunφdx+bun2ΩunφdxλΩuq1nφlnu2ndxνnμΩu3nφdx=un3[(aun2+b)Ω˜wnφdxλunq4Ω˜wq1nφln˜w2ndx   λunq4lnun2Ω˜wq1nφdxνnμΩ˜w3nφdx], as n.

    Letting n in the above equality and recalling the assumptions that νn1 and un as n, one has

    bΩ˜w0φdx=μΩ˜w30φdx, φH10(Ω), (3.36)

    where we have used the fact that limxxq4lnx=0 since q<4. Since Ω is strictly star-shaped, we know from Pohozaev's identity that ˜w0=0 (see [31]). Then, applying a similar argument to that of the proof of Theorem 1.6 in [16], one obtains μ=bS2, a contradiction with bS2<μ<2bS2. Therefore, {un} is bounded, and problem (1.1) has a nontrivial weak solution. The proof is complete.

    Qi Li, Yuzhu Han and Bin Guo: Methodology; Qi Li: Writing-original draft; Yuzhu Han and Bin Guo: Writing-review & editing.

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

    The authors wish to express their gratitude to the anonymous referee for giving a number of valuable comments and helpful suggestions, which improve the presentation of the manuscript significantly.

    This work is supported by the National Key Research and Development Program of China(grant no.2020YFA0714101).

    The authors declare there is no conflict of interest.



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  • Author's biography Riyan Hidayat is a senior lecturer at the Faculty of Education, Universiti Putra Malaysia (UPM). He earned his bachelor's degree in education (Mathematics) from Universitas Islam Riau (UIR) in Indonesia and his master's degree in education (Mathematics) from Universiti Kebangsaan Malaysia (UKM). He then pursued a doctoral degree in Mathematics Education at Universiti Malaya (UM) in Kuala Lumpur. Dr. Hidayat specializes in statistics, educational psychology, and mathematics teaching; Ahmad Fauzi Mohd Ayub is a professor at the Faculty of Educational Studies Universiti Putra Malaysia. He is also a research associate at the Institute for Mathematical Research. He earned his diploma in computer Science and degree in Mathematics from Univerisit Putra Malaysia. His master's degree in information technology from Universiit Teknologi MARA and he pursued a doctoral degree in Information Science at Universiti Kebangsaan Malaysia. His specialization is in Information Technology, educational technology and Mathematics Education; Mohd Afifi Bahurudin Setambah is a senior lecturer at the Fakulti Pembangunan Manusia, Universiti Pendidikan Sultan Idris. He obtained his bachelor's degree in education (Mathematics) from Universiti Kebangsaan Malaysia (UKM), Malaysia. He later completed his master's degree in mathematics education and PHD in Mathematics Education from Universiti Pendidikan Sultan Idris (UPSI), Malaysia. Dr. Mohd Afifi Bahurudin's areas of expertise include module development, experimental research, critical thinking, and gamification in mathematics education; Nurul Hijja Mazlan is a senior lecturer in Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Shah Alam. She received her Doctor of Philosophy (Ph.D) from Universiti Malaya, Malaysia. She has been a passion academician in Multimedia Technology for 16 years. Her current research interests are in Spatial Computing and Educational Technology; specifically designing and developing interactive spatial computing products in educational settings. She can be contacted at: nurulhijja@uitm.edu.my
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