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
[1] | B. Ravit, K. Cooper, B. Buckley, I. Yang, A. Deshpande . Organic compounds associated with microplastic pollutants in New Jersey, U.S.A. surface waters. AIMS Environmental Science, 2019, 6(6): 445-459. doi: 10.3934/environsci.2019.6.445 |
[2] | Gina M. Moreno, Keith R. Cooper . Morphometric effects of various weathered and virgin/pure microplastics on sac fry zebrafish (Danio rerio). AIMS Environmental Science, 2021, 8(3): 204-220. doi: 10.3934/environsci.2021014 |
[3] | Fabiana Carriera, Cristina Di Fiore, Pasquale Avino . Trojan horse effects of microplastics: A mini-review about their role as a vector of organic and inorganic compounds in several matrices. AIMS Environmental Science, 2023, 10(5): 732-742. doi: 10.3934/environsci.2023040 |
[4] | Katharina Meixner, Mona Kubiczek, Ines Fritz . Microplastic in soil–current status in Europe with special focus on method tests with Austrian samples. AIMS Environmental Science, 2020, 7(2): 174-191. doi: 10.3934/environsci.2020011 |
[5] | PA Stapleton . Toxicological considerations of nano-sized plastics. AIMS Environmental Science, 2019, 6(5): 367-378. doi: 10.3934/environsci.2019.5.367 |
[6] | Ebere Enyoh Christian, Qingyue Wang, Wirnkor Verla Andrew, Chowdhury Tanzin . Index models for ecological and health risks assessment of environmental micro-and nano-sized plastics. AIMS Environmental Science, 2022, 9(1): 51-65. doi: 10.3934/environsci.2022004 |
[7] | Abigail W. Porter, Sarah J. Wolfson, Lily. Young . Pharmaceutical transforming microbes from wastewater and natural environments can colonize microplastics. AIMS Environmental Science, 2020, 7(1): 99-116. doi: 10.3934/environsci.2020006 |
[8] | Alessio Russo, Francisco J Escobedo, Stefan Zerbe . Quantifying the local-scale ecosystem services provided by urban treed streetscapes in Bolzano, Italy. AIMS Environmental Science, 2016, 3(1): 58-76. doi: 10.3934/environsci.2016.1.58 |
[9] | Michael R. Templeton, Acile S. Hammoud, Adrian P. Butler, Laura Braun, Julie-Anne Foucher, Johanna Grossmann, Moussa Boukari, Serigne Faye, Jean Patrice Jourda . Nitrate pollution of groundwater by pit latrines in developing countries. AIMS Environmental Science, 2015, 2(2): 302-313. doi: 10.3934/environsci.2015.2.302 |
[10] | K. Wayne Forsythe, Chris H. Marvin, Danielle E. Mitchell, Joseph M. Aversa, Stephen J. Swales, Debbie A. Burniston, James P. Watt, Daniel J. Jakubek, Meghan H. McHenry, Richard R. Shaker . Utilization of bathymetry data to examine lead sediment contamination distributions in Lake Ontario. AIMS Environmental Science, 2016, 3(3): 347-361. doi: 10.3934/environsci.2016.3.347 |
In the present paper, we study the following Kirchhoff system with a coupled critical nonlinearity
{−(1+∫RN|∇u|2dx)Δu=λ1u+μ1|u|p−2u+βr1|u|r1−2u|v|r2,−(1+∫RN|∇v|2dx)Δv=λ2v+μ2|v|q−2v+βr2|u|r1|v|r2−2v, | (1.1) |
having prescribed mass
∫RN|u|2dx=m1and∫RN|v|2dx=m2, | (1.2) |
where m1,m2>0, λ1,λ2,β>0 and N≥3, λ1,λ2 are unknown parameters that will appear as Lagrange multipliers.
Problem (1.1) originates from the steady-state analogy of the equation:
ρ∂2u∂t2−(P0h+E2L∫L0|∂u∂x|2dx)∂2u∂x2=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+b1∫RN|∇u|2dx)Δu=λ1u+μ1|u|p−2u+βr1|u|r1−2u|v|r2,−(a2+b2∫RN|∇v|2dx)Δv=λ2v+μ2|v|q−2v+βr2|u|r1|v|r2−2v, | (1.4) |
where ai,bi>0(i=1,2) and 2≤N≤4. 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 N≤3. Eq (1.1) can also be formally transformed into the following fractional Kirchhoff equation
{(a1+b1∫R3|(−△)s2u|2dx)(−△)su+λu=f(u)+γv,inR3,(a2+b2∫R3|(−△)s2v|2dx)(−△)sv+μv=g(v)+γu,inR3,u,v∈Hs(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 γ→0+. 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 N≥5 for the Kirchhoff-type system. This motivates us to consider the solution of the Kirchhoff system (1.1) for N≥3 and with a coupled critical nonlinearity, where 2≤p,q<2+8N and r1+r2=2∗=2NN−2.
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∗,q≤2∗ of the following equation:
{−Δu+λ1u=μ1|u|p−2u+βr1|u|r1−2|v|r2uinRN,−Δv+λ2v=μ2|v|q−2v+βr2|u|r1|v|r2−2vinRN,∫RNu2dx=a21and∫RNv2dx=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 N≥3, 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)=12∫RN(|∇u|2+|∇v|2)dx+14[∫RN(|∇u|2+|∇v|2)dx]2−μ1p∫RN|u|pdx −μ2q∫RN|v|qdx−β∫RN|u|r1|v|r2dx, | (1.7) |
on the constraint S(m1,m2)=S(m1)×S(m2), where S(m)={u∈H1(RN):‖u‖22=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]2−μ1δp∫RN|u|pdx −μ2δq∫RN|v|qdx−β2∗∫RN|u|r1|v|r2dx, |
where δt=N(t−2)2t. To accommodate the constraint S(m), it becomes crucial to define dilation
(t∗u)(x)=eNt2u(etx),fora.e. x∈RN. |
Consider the following functionals I(u,v) and Lu,v(t)
Lu,v(t)=I(t∗u,t∗v)=12e2t∫RN(|∇u|2+|∇v|2)dx+14e4t[∫RN(|∇u|2+|∇v|2)dx]2 −μ1pepδpt∫RN|u|pdx−μ2qeqδqt∫RN|v|qdx−βe2∗t∫RN|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 (t∗u,t∗v)∈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) N≥5, 2<p,q<2+2N−2, 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):‖∇u‖22+‖∇v‖22<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∗:=2NN−2 for N≥3 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=‖u‖2+‖v‖2. |
Let Lt(RN;R2) be the space Lt(RN×RN) with the norm
‖(u,v)‖tLt=‖u‖tt+‖v‖tt. |
D1,2(RN) represents the closure of the C∞c(RN) with norm
‖u‖D1,2=‖∇u‖2. |
For N≥3, the best Sobolev constant is given by
S=infu∈D1,2(RN)∖{0}‖∇u‖22‖u‖22∗. | (2.1) |
For all u∈H1(RN), we consider the Gagliardo-Nirenberg-Sobolev inequality:
‖u‖pp≤Cpp‖u‖p(1−δp)2‖∇u‖pδp2, where δp=N(p−2)2p. | (2.2) |
For any u,v∈H1(RN), by the Young's inequality, we can prove:
∫RN|u|r1|v|r2dx≤∫RNr12∗|u|2∗dx+∫RNr22∗|v|2∗dx≤S−2∗2(r12∗‖∇u‖2∗2+r22∗‖∇v‖2∗2)≤S−2∗2(‖∇u‖22+‖∇v‖22)2∗2. | (2.3) |
Furthermore, taking into consideration the existing results of the Kirchhoff equation as follows:
{−(1+∫RN|∇u|2)△u=λu+μ|u|p−2u, 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)=12∫RN|∇u|2dx+14(∫RN|∇u|2dx)2−μp∫RN|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):=infu∈S(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 m↦ζμp(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)>(‖∇γ1‖22+‖∇γ2‖22)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=(‖∇u‖22+‖∇v‖22)12. From (2.2) and (2.3), we derive that
I(u,v)≥12(‖∇u‖22+‖∇v‖22)+14(‖∇u‖22+‖∇v‖22)2−μ1pCpp‖u‖p(1−δp)2‖∇u‖pδp2 −μ2qCqq‖v‖q(1−δq)2‖∇v‖qδq2−βS−2∗2(‖∇u‖22+‖∇v‖22)2∗2≥12ρ2+14ρ4−μ1pCppmp(1−δp)21ρpδp−μ2qCqqmq(1−δq)22ρqδq−βS−2∗2ρ2∗=ρ2[12+14ρ2−μ1pCppmp(1−δp)21ρpδp−2−μ2qCqqmq(1−δq)22ρqδq−2−βS−2∗2ρ2∗−2]. | (3.1) |
Recalling that pδq<2 and qδq<2, we can take a large enough
ρτ>max{‖∇γ1‖2,‖∇γ2‖2}, |
such that
μ1pCppmp(1−δp)21ρpδp−2τ+μ2qCqqmq(1−δq)22ρqδq−2τ≤14. | (3.2) |
Due to the fact that 2∗−2>0, there exists a βτ>0 such that
βτS−2∗2(2ρτ)2∗−2≤18. | (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)−β∫RN|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,v∈S(mα1,mα2)∩V(ρτ) such that
I(u,v)≤M(mα1,mα2)+ϵ2. |
Define a cut-off function: ω∈C∞m(RN) such that
0≤ω(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 ı→0+. As a consequence, for η>0 small enough, there exists a sufficiently small ı such that
I(uı,vı)≤I(u,v)+ε4 and (‖∇uı‖22+‖∇vı‖22)12<ρτ−η. | (3.5) |
Let χ(t)∈C∞m(RN) such that supp(χ)⊂{t∈RN:4ı≤|t|≤1+4ı} and set
(um1,vm2)=(√m1−‖uı‖2‖χ‖2χ,√m2−‖vı‖2‖χ‖2χ). |
And observe that
supp(uı)∩supp(t∗um1)=∅ and supp(vı)∩supp(t∗vm2)=∅ |
for any t≤0, hence,
(uı+t∗um1,vı+t∗vm2)∈Sm. |
Next, since
I(t∗um1,t∗vm2)→0and(‖∇t∗um1‖22+‖∇t∗vm2‖22)12→0, |
as t→−∞, we can obtain
I(t∗um1,t∗vm2)≤ε4 and (‖∇t∗um1‖22+‖∇t∗vm2‖22)1/2≤η2, for t≪0. | (3.6) |
It follows that
(∇‖(uı+t∗um1)‖22+∇‖(vı+t∗vm2)||22)1/2<ρτ. |
Using (3.5) and (3.6), we conclude
M(m1,m2)≤I(uı+t∗um1,vı+t∗vm2)=I(uı,vı)+I(t∗um1,t∗vm2)≤I(u,v)+ε2≤M(mα1,mα2)+ε |
for t≪0.
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<τu2v2∈R and two zero points φ1<φ2 with τu1v1<φ1<τu2v2<φ2. Moreover,
(i) if (t∗u,t∗v)∈P(m1,m2), then t=τu1v1 or t=τu2v2;
(ii) (‖∇t∗u‖22+‖∇t∗v‖22)12≤ρτ for all t≤φ1 and
I(τu1v1∗u,τu1v1∗v)=min{I(t∗u,t∗v):t∈Rand(‖∇t∗u‖22+‖∇t∗v‖22)12≤ρτ}<0, |
where ρτ is given in Lemma 3.1;
(iii) I(τu2v2∗u,τu2v2∗v)=max{I(t∗u,t∗v):t∈R}.
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,
L′uv(t)=e2t(‖∇u‖22+‖∇v‖22)+e4t(‖∇u‖22+‖∇v‖22)2−epδptμ1δp‖u‖pp−eqδqtμ2δq‖v‖qq−e2∗t2∗β‖|u|r1|v|r2‖1. |
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<p≤NN−2 when N≥3 and 0<p<∞ when N=1,2. Let u∈Lp(RN) be a smooth, nonnegative function and satisfy −Δu≥0 in RN. Then u≡0 holds.
Lemma 3.6. Let (u,v)∈S(m1,m2), u,v≥0, 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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