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Research article

Microplastics in urban New Jersey freshwaters: distribution, chemical identification, and biological affects

  • This proof of concept study was undertaken to test methodologies to characterize potential environmental risk associated with the presence of microplastics in surface waters. The goals of the study were to determine whether urban New Jersey freshwaters contained microplastic pollutants, and if so, to test analytic techniques that could potentially identify chemical compounds associated with this pollution. A third objective was to test whether identified associated compounds might have physiological effects on an aquatic organism. Using field collected microplastic samples obtained from the heavily urbanized Raritan and Passaic Rivers in New Jersey, microplastic densities, types, and sizes at 15 sampling locations were determined. Three types of plastic polymers were identified using pyrolysis coupled with gas chromatography (Pyr-GC/MS). Samples were further characterized using solid phase micro extraction coupled with headspace gas chromatography/ion trap mass spectrometry (HS-SPME-GC/ITMS) to identify organic compounds associated with the: (i) solid microplastic fraction, and (ii) site water fraction. Identical retention times for GC peaks found in both fractions indicated compounds can move between the two phases, potentially available for uptake by aquatic biota in the dissolved phase. Patterns of tentatively identified compounds were similar to patterns obtained in Pyr-GC/MS. Embryonic zebrafish exposed to PyCG/MS- identified pure polymers in the 1–10 ppm range exhibited altered growth and heart defects. Using two analytic methods (SPME GC/MS and Pyr-GC/MS) allows unambiguous identification of compounds associated with microplastic debris and characterization of the major plastic type(s). Specific “fingerprint” patterns can categorize the class of plastics present in a waterbody and identify compounds associated with the particles. This technique can also be used to identify compounds detected in biota that may be the result of ingesting plastics or plastic-associated compounds.

    Citation: B. Ravit, K. Cooper, G. Moreno, B. Buckley, I. Yang, A. Deshpande, S. Meola, D. Jones, A. Hsieh. Microplastics in urban New Jersey freshwaters: distribution, chemical identification, and biological affects[J]. AIMS Environmental Science, 2017, 4(6): 809-826. doi: 10.3934/environsci.2017.6.809

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  • This proof of concept study was undertaken to test methodologies to characterize potential environmental risk associated with the presence of microplastics in surface waters. The goals of the study were to determine whether urban New Jersey freshwaters contained microplastic pollutants, and if so, to test analytic techniques that could potentially identify chemical compounds associated with this pollution. A third objective was to test whether identified associated compounds might have physiological effects on an aquatic organism. Using field collected microplastic samples obtained from the heavily urbanized Raritan and Passaic Rivers in New Jersey, microplastic densities, types, and sizes at 15 sampling locations were determined. Three types of plastic polymers were identified using pyrolysis coupled with gas chromatography (Pyr-GC/MS). Samples were further characterized using solid phase micro extraction coupled with headspace gas chromatography/ion trap mass spectrometry (HS-SPME-GC/ITMS) to identify organic compounds associated with the: (i) solid microplastic fraction, and (ii) site water fraction. Identical retention times for GC peaks found in both fractions indicated compounds can move between the two phases, potentially available for uptake by aquatic biota in the dissolved phase. Patterns of tentatively identified compounds were similar to patterns obtained in Pyr-GC/MS. Embryonic zebrafish exposed to PyCG/MS- identified pure polymers in the 1–10 ppm range exhibited altered growth and heart defects. Using two analytic methods (SPME GC/MS and Pyr-GC/MS) allows unambiguous identification of compounds associated with microplastic debris and characterization of the major plastic type(s). Specific “fingerprint” patterns can categorize the class of plastics present in a waterbody and identify compounds associated with the particles. This technique can also be used to identify compounds detected in biota that may be the result of ingesting plastics or plastic-associated compounds.


    In the present paper, we study the following Kirchhoff system with a coupled critical nonlinearity

    {(1+RN|u|2dx)Δu=λ1u+μ1|u|p2u+βr1|u|r12u|v|r2,(1+RN|v|2dx)Δv=λ2v+μ2|v|q2v+βr2|u|r1|v|r22v, (1.1)

    having prescribed mass

    RN|u|2dx=m1andRN|v|2dx=m2, (1.2)

    where m1,m2>0, λ1,λ2,β>0 and N3, λ1,λ2 are unknown parameters that will appear as Lagrange multipliers.

    Problem (1.1) originates from the steady-state analogy of the equation:

    ρ2ut2(P0h+E2LL0|ux|2dx)2ux2=0, (1.3)

    which was proposed by Kirchhoff in 1883 in [1] as the existence of the classical D'Alembert wave equation for the free vibration of elastic strings. The Kirchhoff model takes into consideration the changes in the length of the string that are caused by transverse vibrations.

    In recent years, lots of interesting results on the normalized solutions for the Kirchhoff type problem that has been obtained. From a physical perspective, the mass RN|u|2dx=m may represent the number of the power supply in the framework of nonlinear optics or Bose-Einstein condensates. Alternatively, finding normalized solutions seems to be particularly meaningful because the L2-norm of such solutions is a preserved quantity of the evolution, and their variational characterization can help to analyze the orbital stability or instability, e.g., see [2,3,4]. In Bose-Einstein condensates, the parameters μi and β both describe the interactions between particles. When β>0, the two components attract each other, while β<0, the two components repel each other.

    Based on the above important background, the problem like (1.1) has been studied in numerous papers. For example, Yang [5] has obtained a couple of positive solutions to the following equation:

    {(a1+b1RN|u|2dx)Δu=λ1u+μ1|u|p2u+βr1|u|r12u|v|r2,(a2+b2RN|v|2dx)Δv=λ2v+μ2|v|q2v+βr2|u|r1|v|r22v, (1.4)

    where ai,bi>0(i=1,2) and 2N4. By proving that (1.4) satisfies the mountain pass structure, they obtained a couple of positive solutions. In particular, as β>0, Cao et al. [6] considered the L2-subcritical case and L2-critical case of the problem by the bifurcation method and showed the existence of normalized solutions when N3. Eq (1.1) can also be formally transformed into the following fractional Kirchhoff equation

    {(a1+b1R3|()s2u|2dx)()su+λu=f(u)+γv,inR3,(a2+b2R3|()s2v|2dx)()sv+μv=g(v)+γu,inR3,u,vHs(R3), (1.5)

    where ai,bi(i=1,2),λ,μ>0. When s[34,1) and γ>0, by assuming that the nonlinear terms f and g satisfy Berestycki-Lions conditions, and combining with Pohožaev identity, Che and Chen in [7] proved problem (1.5) has positive ground state solutions, and the asymptotic behavior of the solution was also studied when γ0+. When s=1, Lü and Peng [8] proved that (1.5) has vector solutions. We refer readers to [9,10] for multiplicity solutions. However, to our knowledge, there are few articles discussing the results regarding N5 for the Kirchhoff-type system. This motivates us to consider the solution of the Kirchhoff system (1.1) for N3 and with a coupled critical nonlinearity, where 2p,q<2+8N and r1+r2=2=2NN2.

    Other forms of (1.1), such as the Schrödinger equation, have also been extensively studied. For example, Li and Zou [11] considered the case with 2<p,r1+r2<2,q2 of the following equation:

    {Δu+λ1u=μ1|u|p2u+βr1|u|r12|v|r2uinRN,Δv+λ2v=μ2|v|q2v+βr2|u|r1|v|r22vinRN,RNu2dx=a21andRNv2dx=a22. (1.6)

    When 2<r1+r2<2=p=q, Bartsch et al. in [12] have proved (1.6) has a normalized ground state solution and have also investigated the asymptotic behavior by the symmetric decreasing rearrangement and the Ekeland variational principle. When 2+4N<p,q<r1+r2<2 and N3, Liu and Fang [13] obtained the existence of positive normalized solutions of (1.6) by revealing the basic behavior of mountain-pass energy. Compared with Schrödinger equations, it is more challenging and interesting to study problem (1.1) due to the nonlocal term RN|u|2dxΔu and RN|v|2dxΔv.

    In order to study the solution of Eq (1.1) satisfying the normalized condition (1.2), it suffices to consider the critical points of the functional

    I(u,v)=12RN(|u|2+|v|2)dx+14[RN(|u|2+|v|2)dx]2μ1pRN|u|pdx  μ2qRN|v|qdxβRN|u|r1|v|r2dx, (1.7)

    on the constraint S(m1,m2)=S(m1)×S(m2), where S(m)={uH1(RN):u22=m} for m>0. In this paper, we employ the Pohožaev manifold, which is defined by (1.8) and plays a crucial role, encompassing all solutions that satisfy the condition (u,v)S(m1,m2)

    P(m1,m2)={(u,v)S(m1,m2):ϑ(u,v)=0}, (1.8)

    where

    ϑ(u,v)=RN(|u|2+|v|2)dx+[RN(|u|2+|v|2)dx]2μ1δpRN|u|pdx  μ2δqRN|v|qdxβ2RN|u|r1|v|r2dx,

    where δt=N(t2)2t. To accommodate the constraint S(m), it becomes crucial to define dilation

    (tu)(x)=eNt2u(etx),fora.e. xRN.

    Consider the following functionals I(u,v) and Lu,v(t)

    Lu,v(t)=I(tu,tv)=12e2tRN(|u|2+|v|2)dx+14e4t[RN(|u|2+|v|2)dx]2  μ1pepδptRN|u|pdxμ2qeqδqtRN|v|qdxβe2tRN|u|r1|v|r2dx,

    for any (u,v)S(m1,m2).

    Remark 1.1. As in [5], if (u, v) is a solution of (1.1), then (u,v)P(m1,m2). We can also see that if (u,v)S(m1,m2), then (eNt2u(etx),eNt2v(etx))S(m1,m2). Furthermore, for fixed (u,v)S(m1,m2), by performing a simple calculation, we can obtain (Lu,v)(0)=ϑ(u,v). Then we have that (tu,tv)P(m1,m2) if and only if t is a critical point of Lu,v(t). In addition, (u,v)P(m1,m2) if t=0 is a critical point of Lu,v(t).

    To prove the existence of a normalized solution to (1.1), we use the following assumptions:

    (H1) N{3,4}, 2<p,q<2+8N, r1+r2=2.

    (H2) N5, 2<p,q<2+2N2, r1+r2=2.

    Here comes our main result:

    Theorem 1.2. Assume that (H1) or (H2) is established. Then, there exist βτ=βτ(m1,m2)>0 and ρτ=ρτ(m1,m2)>0 such that for arbitrary 0<β<βτ, (1.1) has a positive ground state solution (u, v) for λ1,λ2<0, which satisfies

    I(u,v)=inf(u,v)P(m1,m2)I(u,v)=inf(u,v)S(m1,m2)V(ρτ)I(u,v)<0,

    where

    V(r)={(u,v)H1(RN)×H1(RN):u22+v22<r2}.

    Remark 1.3. Due to the additional difficulties caused by the combined effect of the nonlocal term RN|u|2dxΔu, RN|v|2dxΔv and multiple powers, the study is much more challenging; for example, the functional I(u,v) is composed of several distinct terms that exhibit varying scaling behavior with respect to the dilation eNt2u(etx). The intricate interplay among these terms makes it more difficult to ascertain the types of critical points for I(u,v) on S(m1,m2). Furthermore, when proving (˜un,˜vn)(u,v) in D1,2(RN;R2), the inequalities that need to be estimated will also be more difficult.

    Remark 1.4. From a variational point of view, besides the Sobolev critical exponent 2:=2NN2 for N3 and 2= for N=1,2, a new L2-critical exponent PN:=2+8N arises that plays a pivotal role in the study of normalized solutions to (1.1). This threshold determines whether the constrained functional I(u,v) remains bounded from below on S(m1,m2).

    Definition 1.5. We say that (˜u,˜v) is a couple of ground state solutions to (1.1) on S(m1,m2) if it is a couple of solutions to (1.1) having minimal energy among all the solutions, i.e., dI|S(m1,m2)(˜u,˜v)=0 and

    I(˜u,˜v)=inf{I(u,v):dI|S(m1,m2)(u,v)=0 and (u,v)S(m1,m2)}.

    In this section, we recall some preliminary results that will be used later. Throughout this paper, we represent the norms on Lt(RN) and H1(RN) with t and , respectively. Denote H1(RN)×H1(RN) by V with the norm

    (u,v)2V=u2+v2.

    Let Lt(RN;R2) be the space Lt(RN×RN) with the norm

    (u,v)tLt=utt+vtt.

    D1,2(RN) represents the closure of the Cc(RN) with norm

    uD1,2=u2.

    For N3, the best Sobolev constant is given by

    S=infuD1,2(RN){0}u22u22. (2.1)

    For all uH1(RN), we consider the Gagliardo-Nirenberg-Sobolev inequality:

    uppCppup(1δp)2upδp2, where  δp=N(p2)2p. (2.2)

    For any u,vH1(RN), by the Young's inequality, we can prove:

    RN|u|r1|v|r2dxRNr12|u|2dx+RNr22|v|2dxS22(r12u22+r22v22)S22(u22+v22)22. (2.3)

    Furthermore, taking into consideration the existing results of the Kirchhoff equation as follows:

    {(1+RN|u|2)u=λu+μ|u|p2u, in RN;RN|u|2=m>0. (Pm)

    Solution u of (Pm) can be found as critical points of the functional Iμ(u) defined by

    Iμ(u)=12RN|u|2dx+14(RN|u|2dx)2μpRN|u|pdx

    constrained to the L2-sphere S(m).

    Similar to [14] and [6], we can get the following lemma.

    Lemma 2.1 ([6]). Assume that p(2,2+8N), m>0, and μ>0. Set

    ζμp(m):=infuS(m)Iμ(u).

    Then,

    (i) there exists a unique couple (um,μ,λm)R+×H1(RN) satisfying (Pm);

    (ii) Iμ(um,μ)=ζμp(m)<0;

    (iii) the map mζμp(m) is strictly decreasing with respect to m, and ζμp(m) as m+.

    To begin with, we set

    γ1=um1,μ1,  γ2=um2,μ2

    and

    ζ1=Iμ(γ1), ζ2=Iμ(γ2).

    Lemma 3.1. Let m1,m2,μ1,μ2>0 be given and assume (H1) or (H2) holds. Then there exists βτ = βτ(m1,m2)>0 and ρτ=ρτ(m1,m2)>(γ122+γ222)12 such that

    I(u,v)>0onS(m1,m2)V(2ρτ)V(ρτ)for any  0<β<βτ.

    Proof. For (u,v)V, let ρ=(u22+v22)12. From (2.2) and (2.3), we derive that

    I(u,v)12(u22+v22)+14(u22+v22)2μ1pCppup(1δp)2upδp2  μ2qCqqvq(1δq)2vqδq2βS22(u22+v22)2212ρ2+14ρ4μ1pCppmp(1δp)21ρpδpμ2qCqqmq(1δq)22ρqδqβS22ρ2=ρ2[12+14ρ2μ1pCppmp(1δp)21ρpδp2μ2qCqqmq(1δq)22ρqδq2βS22ρ22]. (3.1)

    Recalling that pδq<2 and qδq<2, we can take a large enough

    ρτ>max{γ12,γ22},

    such that

    μ1pCppmp(1δp)21ρpδp2τ+μ2qCqqmq(1δq)22ρqδq2τ14. (3.2)

    Due to the fact that 22>0, there exists a βτ>0 such that

    βτS22(2ρτ)2218. (3.3)

    We conclude that I(u,v)>0 follows from (3.1)–(3.3).

    Define

    M(m1,m2):=inf(u,v)S(m1,m2)V(2ρτ)I(u,v),

    where ρτ is defined in Lemma 3.1.

    Lemma 3.2. Let m1,m2,μ1,μ2>0 be given, and (H1) or (H2) is true. Then for arbitrary 0<β<βτ, the following statements are true:

    (i) M(m1,m2)<ζ1+ζ2<0;

    (ii) M(m1,m2)M(mα1,mα2), for any 0<mα1<m1 and 0<mα2<m2.

    Proof. (i) From Lemma 3.1, we know that (γ1,γ2)V(ρτ). Moreover, we deduce that

    M(m1,m2)I(γ1,γ2)=Iμ1(γ1)+Iμ2(γ2)βRN|γ1|r1|γ2|r2dx<ζ1+ζ2<0.

    (ii) The proof is similar to that of [15]. We just need to prove that for arbitrary ϵ>0,

    M(m1,m2)M(mα1,mα2)+ϵ

    for any 0<mα1<m1 and 0<mα2<m2.

    By Lemma 3.1 and the definition of M(mα1,mα2), there exist u,vS(mα1,mα2)V(ρτ) such that

    I(u,v)M(mα1,mα2)+ϵ2.

    Define a cut-off function: ωCm(RN) such that

    0ω(t)1  and ω(t)={1, |t|1;0, |t|2. (3.4)

    For any ı>0, we define (uı(t),vı(t)) = (uω(ıt),vω(ıt)). Clearly, (uı,vı)(u,v) in V as ı0+. As a consequence, for η>0 small enough, there exists a sufficiently small ı such that

    I(uı,vı)I(u,v)+ε4  and   (uı22+vı22)12<ρτη. (3.5)

    Let χ(t)Cm(RN) such that supp(χ){tRN:4ı|t|1+4ı} and set

    (um1,vm2)=(m1uı2χ2χ,m2vı2χ2χ).

    And observe that

    supp(uı)supp(tum1)=  and  supp(vı)supp(tvm2)=

    for any t0, hence,

    (uı+tum1,vı+tvm2)Sm.

    Next, since

    I(tum1,tvm2)0and(tum122+tvm222)120,

    as t, we can obtain

    I(tum1,tvm2)ε4 and (tum122+tvm222)1/2η2,  for t0. (3.6)

    It follows that

    ((uı+tum1)22+(vı+tvm2)||22)1/2<ρτ.

    Using (3.5) and (3.6), we conclude

    M(m1,m2)I(uı+tum1,vı+tvm2)=I(uı,vı)+I(tum1,tvm2)I(u,v)+ε2M(mα1,mα2)+ε

    for t0.

    Lemma 3.3. Let m1,m2,μ1,μ2>0, and assume that either (H1) is true or (H2) is true. Then, for arbitrary 0<β<βτ and (u,v)S(m1,m2), Lu,v(t) has two critical points τu1v1<τu2v2R and two zero points φ1<φ2 with τu1v1<φ1<τu2v2<φ2. Moreover,

    (i) if (tu,tv)P(m1,m2), then t=τu1v1 or t=τu2v2;

    (ii) (tu22+tv22)12ρτ for all tφ1 and

    I(τu1v1u,τu1v1v)=min{I(tu,tv):tRand(tu22+tv22)12ρτ}<0,

    where ρτ is given in Lemma 3.1;

    (iii) I(τu2v2u,τu2v2v)=max{I(tu,tv):tR}.

    Proof. (i) Since qδq,pδp<2<2, it can be seen that Lu,v()=0 and Lu,v(+)=. According to Lemma 3.1, we obtain that Lu,v(t) has at least two critical points τu1v1<τu2v2, with τu1v1 local minimum point of Lu,v(t) at a negative level and τu2v2 global maximum point at a positive level. Secondly, similar to [5], it is not difficult to check that there are no other critical points. On the other hand,

    Luv(t)=e2t(u22+v22)+e4t(u22+v22)2epδptμ1δpuppeqδqtμ2δqvqqe2t2β|u|r1|v|r21.

    Putting together all the considerations mentioned above, we conclude that Lu,v has exactly two critical points. By monotonicity and recalling the behavior at infinity, Lu,v has moreover exactly two zeros points φ1<φ2 with τu1v1<φ1<τu2v2<φ2. From Lemma 3.1 and (i), we can deduce the (ii) and (iii).

    Corollary 3.4. Let m1,m2,μ1,μ2>0, and assume that either (H1) is true or (H2) is true. Then, for arbitrary 0<β<βτ, the following inequality holds:

    <M(m1,m2)=infP(m1,m2)I(u,v)<0.

    Next, we establish a necessary condition for the existence of a non-negative solution to (1.1). This Liouville-type result will be used to prove the existence of a positive solution.

    Lemma 3.5.([16]) Suppose 0<pNN2 when N3 and 0<p< when N=1,2. Let uLp(RN) be a smooth, nonnegative function and satisfy Δu0 in RN. Then u0 holds.

    Lemma 3.6. Let (u,v)S(m1,m2), u,v0, and u,v, if (u, v) satisfies

    \begin{align} \begin{cases}-(1+\int_{ \mathbb{R}^N}|\nabla u|^2dx)\Delta u = \lambda_{1}u+\mu_{1}|u|^{p-2}u+\beta r_{1}|v|^{r_{2}}|u|^{r_{1}-2}u, \\-(1+\int_{ \mathbb{R}^N}|\nabla v|^2dx)\Delta v = \lambda_{2}v+\mu_{2}|v|^{q-2}v+\beta r_{2}|u|^{r_{1}}|v|^{r_{2}-2}v, \end{cases} \end{align} (3.7)

    then \lambda_1, \lambda_2 < 0 .

    Proof. Arguing by contradiction, we assume that \lambda_1\geq0 . Since u\geq0 , we have that all components on the right-hand side of

    \begin{equation} -(1+\int_{ \mathbb{R}^N}|\nabla u|^2dx)\Delta u = \lambda_{1}u+\mu_{1}|u|^{p-2}u+\beta r_{1}|v|^{r_{2}}|u|^{r_{1}-2}u \notag \end{equation}

    are nonnegative. Hence,

    \begin{equation} -(1+\int_{ \mathbb{R}^N}|\nabla u|^2dx)\Delta u\geq0\notag , \end{equation}

    it is easy to see that

    \begin{equation} -\Delta u\geq0 \notag. \end{equation}

    Moreover, modifying the standard elliptic regularity theorems, we can ensure that the smoothness of ( u, v ) is up to C^2 . Hence, it follows from Lemma 3.5 that u = 0 . This contradicts with u\not\equiv0 ; thus, \lambda_1 < 0 . The proof of \lambda_2 < 0 is the same as that of \lambda_1 < 0 .

    Lemma 3.7.([17]) Let (u_n)_{n \geq 0} \subset H^1(\mathbb{R}^N) be a bounded sequence of spherically symmetric functions. If N \geq 2 or if u_n(x) is a nonincreasing function of |x| for every n \geq 0, then there exist a subsequence (u_{n_k})_{k \geq 0} and u \in H^1(\mathbb{R}^N) such that u_{n_k} \rightarrow u as k \rightarrow \infty in L^p(\mathbb{R}^N) for every 2 < p < \frac{2N}{N-2}.

    Proof of Theorem 1.2. Let us consider a minimizing sequence \{(u_n, v_n)\} for \mathcal{I}|_{S(m_1, m_2)\cap V(2\rho_\tau)} and \{(u_n, v_n)\}\subset \mathcal{V}\cap S(m_1, m_2) . Without loss of generality, we can assume that (u_n, v_n)\subset \mathcal{V} are nonnegative and radially decreasing for every n [Otherwise, we replace ( u_n, v_n ) with ( |u_n|^*, |v_n|^* ), which is the Schwarz rearrangement of ( |u_n|, |v_n| )]. Furthermore, by Lemma 3.3 (ii) , (\|\nabla s\ast u\|_{2}^{2}+\|\nabla s \ast v\|_{2}^{2})^\frac{1}{2}\leq \rho_\tau , and \{\tau_{u_{n}v_{n}}\ast u, \tau_{u_{n}v_{n}}\ast v\} is still a minimizing sequence for \mathcal{I}|_{S(m_1, m_2)\cap V(2\rho_\tau)} . And hence, by the Ekeland variational principle [18], it yields that there exists a new minimizing sequence \{(\tilde{u}_n, \tilde{v}_n)\} satisfying

    \begin{align} \begin{cases}\|\tilde{u}_n-\tau_{u_{n}v_{n}}\ast \tilde{u}_n\|+\|\tilde{v}_n-\tau_{u_{n}v_{n}}\ast \tilde{u}_n\|\to0, &\text{as }n\to\infty, \\ \mathcal{I}(\tilde{u}_n, \tilde{v}_n)\to \mathcal{M}(m_1, m_2), &\text{as }n\to\infty, \\ \mathcal{\vartheta}(\tilde{u}_n, \tilde{v}_n)\to0, &\text{as }n\to\infty, \\ \mathcal{I}'|_{S(m_1, m_2)}(\tilde{u}_n, \tilde{v}_n)\to0, &\text{as }n\to\infty.\end{cases} \end{align} (3.8)

    In the sequel, we divide the proof into three steps.

    Step 1: (\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v) in L^t(\mathbb{R}^N; \mathbb{R}^2) for arbitrarily t \in (2, 2^*) .

    In fact, from (3.8), we can know that \mathcal{I}'|_{S(m_1, m_2)}(\tilde{u}_n, \tilde{v}_n)\to0 . By the Lagrange multipliers theorem, there exist two sequences \{\lambda_{1, n}\} \subset \mathbb{R} and \{\lambda_{2, n}\} \subset \mathbb{R} satisfying the following equation

    \begin{align} &\int_{\mathbb{R}^N}\left(\nabla \tilde{u}_n\nabla\phi+\nabla \tilde{v}_n\nabla\psi\right)dx+\left(\int_{\mathbb{R}^N}\left(\nabla \tilde{u}_n\nabla\phi+\nabla \tilde{v}_n\nabla\psi\right)dx\right)^2\\ &-\int_{\mathbb{R}^N}\left(\mu_1|\tilde{u}_n|^{p-2}\tilde{u}_n\phi+\mu_2|\tilde{v}_n|^{p-2}\tilde{u}_n\psi\right)dx -\beta r_1\int_{\mathbb{R}^N}|\tilde{u}_n|^{r_1-2}|\tilde{v}_n|^{r_2}\tilde{u}_n\phi dx\\ &-\beta r_2\int_{\mathbb{R}^N}|\tilde{u}_n|^{r_1}|\tilde{v}_n|^{r_2-2}\tilde{v}_n\psi dx\\ = &\int_{\mathbb{R}^N}\left(\lambda_{1, n}\tilde{u}_n\phi+\lambda_{2, n}\tilde{v}_n\psi\right)dx+o_n(1)\big(\|\phi\|+\|\psi\|\big), \end{align} (3.9)

    for arbitrarily (\phi, \psi)\in \mathcal{V} . By substituting (\tilde{u}_n, 0) and (0, \tilde{v}_n) into (3.9), we can derive

    \begin{align} & \lambda_{1, n}m_1 = \|\nabla \tilde{u}_n\|_2^2+\|\nabla \tilde{u}_n\|_2^4-\mu_1\|\tilde{u}_n\|_p^p \end{align}

    and

    \begin{align} &\lambda_{2, n}m_2 = \|\nabla \tilde{v}_n\|_2^2+\|\nabla \tilde{v}_n\|_2^4-\mu_2\|\tilde{v}_n\|_q^q. \end{align}

    Since \{\tilde{u}_n, \tilde{v}_n\}\subset{S(m_1, m_2)\cap V(2\rho_\tau)} , up to a subsequence, (\lambda_{1, n}, \lambda_{2, n}) \rightarrow (\lambda_1, \lambda_2) \in \mathbb{R}^2 and (\tilde{u}_n, \tilde{v}_n) \rightharpoonup (u, v) \in \mathcal{V} , where both u and v are non-negative. Combined with that, \mathcal{\vartheta}(u, v) = 0 , then ( u, v ) is a weak solution of (1.1). By Lemma 3.7, we obtain that (\tilde{u}_n, \tilde{v}_n) \rightarrow (u, v) in L^t(\mathbb{R}^N, \mathbb{R}^2) for any t \in (2, 2^*) .

    Step 2: (\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v) in D^{1, 2}(\mathbb{R}^N; \mathbb{R}^2) .

    Let (u_n, v_n) = (\tilde{u}_n-u, \tilde{v}_n-v) . Then u_n\rightarrow0 in L^p(\mathbb{R}^N) and v_n\rightarrow0 in L^q(\mathbb{R}^N) . Moreover, from the Br \acute{e} zis-Lieb Lemma, we have

    \begin{align} \int_{\mathbb{R}^N}[|\tilde{u}_n|^{r_1}|\tilde{v}_n|^{r_2}-|u|^{r_1}|v|^{r_2}]dx = \int_{\mathbb{R}^N}|{u}_n|^{r_1}|{v}_n|^{r_2}dx+o_n(1). \end{align} (3.10)

    Since \mathcal{\vartheta}(\tilde{u}_n, \tilde{v}_n)-\mathcal{\vartheta}(u, v)\rightarrow 0 , we can infer from (2.3) and (3.10) that

    \begin{align} &\|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}+\left(\|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\right)^2 \\ = &\beta2^{*}\int_{\mathbb{R}^{N}}|{u}_{n}|^{r_{1}}|{v}_{n}|^{r_{2}}dx +o_{n}(1) \\ \leq&\beta2^{*}S^{-\frac{2^{*}}{2}}\left(\|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\right)^{\frac{2^{*}}{2}}+o_{n}(1). \end{align} (3.11)

    Up to a subsequence, we assume that \|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\rightarrow R\geq0 . Then R = 0 or R\geq \left(\frac{1}{\beta 2^*}\right)^{\frac{N-2}{2}}S^{\frac{N}{2}} . If R\geq \left(\frac{1}{\beta 2^*}\right)^{\frac{N-2}{2}}S^{\frac{N}{2}} , from (3.8), (3.10), and (3.11), we have

    \begin{align} \mathcal{M}(m_1, m_2) = &\lim\limits_{n\to\infty}\mathcal{I}(\tilde{u}_n, \tilde{v}_n) = \mathcal{I}(u, v)+\lim\limits_{n\to\infty}\mathcal{I}({u}_{n}, {v}_{n}) \\ \geq & \mathcal{M}(\|u\|_{2}^{2}, \|v\|_{2}^{2})+\lim\limits_{n\to\infty}\big[\frac{1}{2}\left(\|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\right) \\ &+\frac{1}{4}\left( \|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\right)^2 -\beta\int_{\mathbb{R}^{N}}|{u}_{n}|^{r_{1}}|{v}_{n}|^{r_{2}}\big] \\ \geq & m(\|u\|_{2}^{2}, \|v\|_{2}^{2})+\frac{1}{N}\lim\limits_{n\to\infty}\left(\|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\right) \\ = &m(\|u\|_{2}^{2}, \|v\|_{2}^{2})+\frac{1}{N}(\frac{1}{\beta2^{*}})^{\frac{N-2}{2}}S^{\frac{N}{2}}. \end{align}

    This contradicts with Lemma 3.2 (ii) . Then \|\nabla{u}_{n}\|_{2}^{2}+\|\nabla{v}_{n}\|_{2}^{2}\rightarrow 0 . Thus, we conclude (\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v) in D^{1, 2}(\mathbb{R}^N; \mathbb{R}^2) .

    Step 3: (\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v) in \mathcal{V} .

    From Step 1, then, as in [19], we know that there exists (u, v) \in \mathcal{V} that is a weak solution of

    \begin{align} \begin{cases}-(1+\int_{ \mathbb{R}^N}|\nabla u|^2dx)\Delta u = \lambda_{1}u+\mu_{1}|u|^{p-2}u+\beta r_{1}|v|^{r_{2}}|u|^{r_{1}-2}u, \\-(1+\int_{ \mathbb{R}^N}|\nabla v|^2dx)\Delta v = \lambda_{2}v+\mu_{2}|v|^{q-2}v+\beta r_{2}|u|^{r_{1}}|v|^{r_{2}-2}v, \end{cases} \end{align} (3.12)

    with

    \begin{equation} \|u\|_2^2\leq \lim \inf\|\tilde{u}_n\|_2^2 = m_1 \ \quad \mbox{and} \ \quad \|v\|_2^2\leq \lim \inf\|\tilde{v}_n\|_2^2 = m_2\notag . \end{equation}

    We claim that u\neq0 and v\neq0 . Indeed, if v = 0 , then u satisfies

    \begin{equation} \left\{ \begin{array}{lr} -(1+\int_{\mathbb{R}^N}|\nabla u|^2dx)\triangle{u} = {\lambda}u +\mu |u|^{p-2}u, \ \mbox{in} \ {\mathbb{R}^N}, \\\ \|u\|_2^2\leq m_1. \end{array} \right.\notag \end{equation}

    By applying Lemma 2.1, we know that \zeta_p^\mu(m) is strictly decreasing with respect to m . So

    \begin{equation} \zeta_p^{\mu_{1}}(m_1)\leq\zeta_p^{\mu_{1}}(\|u\|_2^2) = \frac{1}{2}\|\nabla u\|_2^2+\frac{1}{4}\|\nabla u\|_2^4-\frac{\mu_{1}}{p}\|u\|_p^p.\notag \end{equation}

    However,

    \begin{align} \mathcal{M}(m_1, m_2)& = \lim\limits_{n\rightarrow \infty}\mathcal{I}(\tilde{u}_n, \tilde{v}_n)\\ & = \lim\limits_{n\rightarrow \infty}\frac{1}{2}(\|\nabla \tilde{u}_n \|_2^2+\|\nabla \tilde{v}_n \|_2^2)+\frac{1}{4}(\|\nabla \tilde{u}_n \|_2^2+\|\nabla \tilde{v}_n \|_2^2)^2\\ \quad &\; \ \ -\frac{\mu_{1}}{p}\|\tilde{u}_n\|_p^p-\frac{\mu_{2}}{q}\|\tilde{v}_n\|_q^q-\beta \int_{ \mathbb{R}^N}|\tilde{u}_n|^{r_1}|\tilde{v}_n|^{r_2}\\ &\geq \frac{1}{2}(\|\nabla u \|_2^2+\|\nabla v \|_2^2)+\frac{1}{4}(\|\nabla u \|_2^2+\|\nabla v \|_2^2)^2\\ \quad &\; \ \ -\frac{\mu_{1}}{p}\|u\|_p^p-\frac{\mu_{2}}{q}\|v\|_q^q\\ &\geq \zeta_p^{\mu_{1}}(m_1)+\zeta_p^{\mu_{2}}(m_2), \end{align}

    which contradicts to Lemma 3.2 (i) . Hence, v\neq0 . Similarly, we have u\neq0 . Thus, from Lemma 3.6, we know \lambda_1, \lambda_2 < 0 . Then, by substituting ( \tilde{u}_n, 0 ) and ( u, 0 ) into (3.9), we can derive

    \begin{align} &\|\nabla \tilde{u}_n\|_2^2+\|\nabla \tilde{u}_n\|_2^4+\mu_1\|\tilde{u}_n\|_p^p = \lambda_1\|\tilde{u}_n\|_2^2+o_n(1) \end{align}

    and

    \begin{align} &\|\nabla u\|_2^2+\|\nabla u\|_2^4+\mu_1\|u\|_p^p = \lambda_1\|u\|_2^2, \end{align}

    which implies that \tilde{u}_n \rightarrow u in H^1(\mathbb{R}^N) as \lambda_1 < 0 . Similarly, we obtain \tilde{v}_n \rightarrow v in H^1(\mathbb{R}^N) .

    Therefore, we have (\tilde{u}_n, \tilde{v}_n)\rightarrow (u, v) in \mathcal{V} and by Corollary 3.4, we have

    \begin{equation} \mathcal{I}(u, v) = \inf\limits_{(u, v)\in P(a, b)}\mathcal{I}(u, v) = \inf\limits_{(u, v)\in S(m_1, m_2)\cap V(\rho_\tau)}\mathcal{I}(u, v) < 0. \notag \end{equation}

    Therefore, we deduce that ( u, v ) is a normalized solution. By the maximum principle, we conclude that ( u, v ) is a positive solution.

    In this paper, we establish the existence of a ground state solution for a nonlinear Kirchhoff-type system using the minimization of the energy functional over a combination of the mass-constrained and the closed balls. To the best of our knowledge, there are few articles that deal with a coupled critical nonlinearity of the Kirchhoff system. Especially, our assumptions on the parameters are different from the previous related works. Therefore, we need to use some new analytical tricks to estimate the critical value. Our results in this article improve and generalize the related ones in the literature. In addition, condition 2\leq p, q < 2+\frac{8}{N} means that our results are established in a critical setting. Therefore, a new research direction closely related to problem (1.1) is to replace 2\leq p, q < 2+\frac{8}{N} with the following L^2 -supercritical condition: 2+\frac{8}{N}\leq p, q < 2^* $.

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

    This work is supported by the National Natural Science Foundation of China (No. 11961014) and Guangxi Natural Science Foundation (2021GXNSFAA196040).

    The authors declare there is no conflicts of interest.

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