In this article, we study the following bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth:
{−(∫Ω|∇u|2dx)rΔu+ϕu=u5+λ(∫ΩF(x,u)dx)sf(x,u),in Ω,−Δϕ=u2,u>0,in Ω,u=ϕ=0,on ∂Ω,
where Ω⊂R3 is a smooth bounded domain, λ>0, 0≤r<1, 0<s<1−r3(r+1) and f(x,u) satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.
Citation: Guaiqi Tian, Hongmin Suo, Yucheng An. Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth[J]. Electronic Research Archive, 2022, 30(12): 4493-4506. doi: 10.3934/era.2022228
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In this article, we study the following bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth:
{−(∫Ω|∇u|2dx)rΔu+ϕu=u5+λ(∫ΩF(x,u)dx)sf(x,u),in Ω,−Δϕ=u2,u>0,in Ω,u=ϕ=0,on ∂Ω,
where Ω⊂R3 is a smooth bounded domain, λ>0, 0≤r<1, 0<s<1−r3(r+1) and f(x,u) satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.
This paper is concerned with the following Kirchhoff-Schr¨odinger-Poisson system:
{−(∫Ω|∇u|2dx)rΔu+ϕu=u5+λ(∫ΩF(x,u)dx)sf(x,u),in Ω,−Δϕ=u2,u>0,in Ω,u=ϕ=0,on ∂Ω, | (1.1) |
where Ω⊂R3 is a smooth bounded domain, λ>0, 0≤r<1, 0<s<1−r3(r+1) and F(x,u)=∫u0f(x,ξ)dξ. We assume that f∈C1(¯Ω×R,R) and there exist constants a1,a2>0 and 6(r+1)r+2<q<4s+1,0<s<1−r3(r+1), such that
a1tq−1≤f(x,t)≤a2tq−1forany(x,t)∈¯Ω×R. | (1.2) |
When (∫Ω|∇u|2dx)r=1 and s=0, the system (1.1) reduces to the boundary value problem
{−Δu+Vu+ϕu=f(x,u),in Ω,−Δϕ=u2,u>0,in Ω,u=ϕ=0,on ∂Ω. | (1.3) |
Problem (1.3) has been extensively studied, by using variational methods and critical point theory under suitable assumptions on V, f; see [1,2,3,4,5,6,7] and the references therein.
On the other hand, considering just the first equation in (1.1) with the potential equal to zero, we have the problem
{−(a+b∫Ω|∇u|2dx)Δu=f(x,u),in Ω,u=0,on ∂Ω, | (1.4) |
where a,b>0, which was proposed by Kirchhoff in [8] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. The appearance of the nonlocal term ∫Ω|∇u|2dx in the equations makes them important in many physical applications. We have to point out that such nonlocal problems appear in other fields like biological systems, such as population density, where u describes a process which depends on the average of itself (see [9]). The Kirchhoff type problem (1.4) with critical growth began to call the attention of researchers; we can see [10,11,12,13,14,15,16,17,18] and the references therein.
In particular, Che et al. in [19] considered the following Kirchhoff-Schr¨odinger-Poisson system with critical growth:
{−(a+b∫R3|∇u|2dx)Δu+V(x)u+ϕu=λg(x)|u|q−1+h(x)u5,inR3,−Δϕ=u2,u>0,inR3, |
where a>0,b≥0,q∈[4,6), and λ>0 is a parameter. Under some suitable conditions on V(x), g(x) and h(x), by using the Nehari manifold technique and Ljusternik-Schnirelmann category theory, they established the number of positive solutions with the topology of the global maximum set of h when λ is small enough. Furthermore, with the aid of the mountain pass theorem, they obtained an existence result for λ sufficiently large.
In [20], Chabrowski investigated the bi-nonlocal problem for the nonlinear elliptic equation of the form
{−(∫Ω|∇u|2dx)sΔu=Q(x)|u|p−2u+(∫Ω|u|qdx)r|u|q−2u,in Ω,∂u∂ν=0,on ∂Ω, |
where 2<p≤2∗,2<q<2∗,s>0,r>0, and 2∗=2NN−2(N≥3) denotes the critical Sobolev exponent. The existence of solutions in critical and subcritical cases is obtained by using the variational method. A similar problem with Dirichlet boundary conditions has been considered in [21]. Motivated by the above references, we study the existence of multiple positive solutions for system (1.1). Our main difficulties are as follows: The critical growth of system (1.1) leads to the lack of compactness of the embedding H10(Ω)↪L6(Ω), and it is difficult to prove the energy functional belongs to the range where the (PS) condition holds. We overcome this difficulty by using the concentration compactness principle.
Throughout this paper, we make use of the following notations:
● The space H10(Ω) is equipped with the norm ‖u‖=(∫Ω|∇u|2dx)12, and the norm in Lp(Ω) is denoted by |u|p=(∫Ω|u|pdx)1p;
● C,C1,C2,… denote various positive constants, which may vary from line to line;
● We denote by Sρ (respectively, Bρ) the sphere (respectively, the closed ball) of center zero and radius ρ, i.e., Sρ={u∈H10(Ω):‖u‖=ρ}, Bρ={u∈H10(Ω):‖u‖≤ρ};
● → (respectively, ⇀) denotes strong (respectively, weak) convergence;
● Let S be the best Sobolev constant, namely,
S=infu∈H10(Ω)∖{0}∫Ω|∇u|2dx(∫Ω|u|6dx)1/3. | (1.5) |
Our main result is the following:
Theorem 1.1. Assume that q(s+1)<4≤2(r+1)<6, λ>0, and 0<s<1−r3(r+1). Then, there exists Λ∗>0 such that for any λ∈(0,Λ∗), system (1.1) has at least two positive solutions.
Remark 1.1. As we shall see, in the system (1.1), when f(x,u)=|u|q−2u, F(x,u)=|u|q, Chabrowski established the existence of solutions. In this paper, due to the nonlocal term ϕu in (1.1), new treatments are needed for our problem. Therefore, in this article, we extend the relevant results of [20].
First, by using the Lax-Milgram theorem, for each u∈H10(Ω), there exists a unique solution ϕu∈H10(Ω) which satisfies the second equation of system (1.1). We substitute ϕu into the first equation of system (1.1), and then system (1.1) is transformed into the following problem:
{−(∫Ω|∇u|2dx)rΔu+ϕuu=u5+λ(∫ΩF(x,u)dx)sf(x,u),in Ω,u>0,in Ω,u=0,on ∂Ω. | (2.1) |
We define the energy functional corresponding to problem (2.1) by
Iλ(u)=12(r+1)(∫Ω|∇u|2dx)r+1+14∫Ωϕuu2dx−16∫Ω|u|6dx−λs+1(∫ΩF(x,u)dx)s+1. |
We say that a function u∈H10(Ω) is called a weak solution of problem (2.1) if for every φ∈H10(Ω), there holds
(∫Ω|∇u|2dx)r∫Ω∇u∇φdx+∫Ωϕuuφdx−∫Ω|u|4uφdx−λ(∫ΩF(x,u)dx)s∫Ωf(x,u)φdx=0. |
Before proving Theorem 1.1, we give the following important Lemma.
Lemma 2.1. (See [22,23]) For all u∈H10(Ω), there exists a unique solution ϕu∈H10(Ω) of
{−Δϕ=u2,in Ω,ϕ=0,on ∂Ω, |
and
(1) ‖ϕu‖2=∫Ωϕuu2dx;
(2) ϕu≥0. Moreover, ϕu>0 when u≠0;
(3) For each t≠0, ϕtu=t2ϕu;
(4)
∫Ω|∇ϕu|2dx=∫Ωϕuu2dx≤S−1|u|412/5≤C‖u‖4; |
(5) If un⇀u in H10(Ω), then ϕun→ϕu in H10(Ω), and
∫Ωϕununvdx→∫Ωϕuuvdx,∀v∈H10(Ω). |
Lemma 2.2. There exist constants δ,ρ,Λ0>0, for all λ∈(0,Λ0) such that the functional Iλ satisfies the following conditions:
(i) Iλ|u∈Sρ≥δ>0, infu∈BρIλ(u)<0;
(ii) There exists e∈H10(Ω) with ‖e‖>ρ such that Iλ(e)<0.
Proof. (i) According to H¨older's inequality and (1.5), one has
(∫Ω|u|qdx)s+1≤[(∫Ω|u|6dx)q6(∫Ω166−qdx)6−q6]s+1≤|Ω|(6−q)(s+1)6S−q(s+1)2‖u‖q(s+1). | (2.2) |
By (1.2), we have
a1|t|q≤f(x,t)t≤a2|t|qforany(x,t)∈¯Ω×R, | (2.3) |
and
a1q|t|q≤F(x,t)≤a2q|t|qforany(x,t)∈¯Ω×R. | (2.4) |
Therefore, it follows from (1.5), (2.2) and (2.4) that
Iλ(u)≥12(r+1)‖u‖2(r+1)−16S−3‖u‖6−λs+1(a2q)s+1(∫Ω|u|qdx)s+1≥12(r+1)‖u‖2(r+1)−16S−3‖u‖6−λs+1(a2q|Ω|6−q6S−q2‖u‖q)s+1=‖u‖q(s+1)[‖u‖2(r+1)−q(s+1)2(r+1)−S−3‖u‖6−q(s+1)6−λs+1(a2q|Ω|6−q6S−q2)s+1]. |
Let H(t)=12(r+1)t2(r+1)−q(s+1)−16S−3t6−q(s+1) for t>0, and then there exists
ρ=[3S3[2(r+1)−q(s+1)](r+1)[6−q(s+1)]]16−2(r+1)>0, |
such that maxt>0H(t)=H(ρ)>0. Setting
Λ0=(s+1)qs+1Sq(s+1)2as+12|Ω|(6−q)(s+1)6H(ρ), |
there exists a constant δ>0, such that Iλ|u∈Sρ≥δ for each λ∈(0,Λ0). Moreover, for every u∈H10(Ω)∖{0}, we get
limτ→0+Iλ(τu)τq(s+1)=limτ→0+−λs+1(∫ΩF(x,τu)dx)s+1≤−λs+1(a1q)s+1(∫Ω|u|qdx)s+1<0. |
So, we obtain Iλ(τu)<0 for all u≠0 and τ small enough. Hence, for ‖u‖ small enough, we have
m=infu∈BρIλ(u)<0. |
(ii) Set u∈H10(Ω), and we get
Iλ(τu)≤τ2(r+1)2(r+1)(∫Ω|∇u|2dx)r+1+τ44∫Ωϕuu2dx−τ66∫Ω|u|6dx−λτq(s+1)s+1(a1q)s+1(∫Ω|u|qdx)s+1→−∞, |
as τ→+∞, which implies that Iλ(τu)<0 for τ>0 large enough. Consequently, we can find e∈H10(Ω) with ‖e‖>ρ such that Iλ(e)<0. The proof is complete.
Lemma 2.3. Assume that λ>0, q(s+1)<4≤2(r+1)<6, and 0<s<1−r3(r+1). Then, the functional Iλ satisfies the (PS)cλ condition for each
cλ<c∗=2−r6(r+1)S3(r+1)2−r−Dλ2(r+1)2(r+1)−q(s+1), |
where
D=(2−r)[2(r+1)−q(s+1)]6q(r+1)(s+1)[6as+12−q(s+1)as+112qs(2−r)|Ω|(6−q)(s+1)6S−q(s+1)2]2(r+1)2(r+1)−q(s+1). |
Proof. Let {un}⊂H10(Ω) be a (PS) sequence for Iλ at the level cλ, that is,
Iλ(un)→cλ,andI′λ(un)→0asn→∞. | (2.5) |
Combining with (2.2)–(2.4), we get
cλ+1+o(‖un‖)≥Iλ(un)−16⟨I′λ(un),un⟩=(12(r+1)−16)‖un‖2(r+1)+(14−16)∫Ωϕunu2ndx−λ1s+1(∫ΩF(x,un)dx)s+1+λ6(∫ΩF(x,un)dx)s∫Ωf(x,un)undx≥(12(r+1)−16)‖un‖2(r+1)+λ[q6(a1q)s+1−1s+1(a2q)s+1](∫Ω|un|qdx)s+1≥(12(r+1)−16)‖un‖2(r+1)+λ[q6(a1q)s+1−1s+1(a2q)s+1]|Ω|(6−q)(s+1)6S−q(s+1)2‖un‖q(s+1). |
Then, this implies that {un} is bounded in H10(Ω) for all q(s+1)<4≤2(r+1)<6. Thus, we may assume up to a subsequence, still denoted by {un}, there exists u∈H10(Ω) such that
{un⇀u,weaklyinH10(Ω),un→u,stronglyinLp(Ω)(1≤p<6),un(x)→u(x),a.e.inΩ, | (2.6) |
as n→∞. Next, we prove that un→u strongly in H10(Ω). By using the concentration compactness principle (see [24]), there exist an at most countable set J, a family of points {xj}j∈J⊂ˉΩ, and positive numbers {νj}j∈J, {μj}j∈J such that
|un|6⇀dν=|u|6+∑j∈Jνjδxj, |
|∇un|2⇀dμ≥|∇u|2+∑j∈Jμjδxj. |
Moreover, we have
μj,νj≥0,μj≥Sν13j. | (2.7) |
Let φε,j(x) be a smooth cut-off function centered at xj such that 0≤φε,j≤1, |∇φε,j|≤2ε, ε>0, and
φε,j(x)={1, in B(xj,ε2),0, in Ω∖B(xj,ε). | (2.8) |
Noting that {unφε,j} is bounded in H10(Ω) uniformly for n, combining with (2.6) and (2.8), we have
limε→0limn→∞∫Ω|un|qφε,jdx≤limε→0limn→∞∫B(xj,ε)|un|qdx=0. | (2.9) |
Similarly, we can obtain
limε→0limn→∞∫Ωϕunu2nφε,jdx≤limε→0∫B(xj,ε)ϕuu2φε,jdx=0. | (2.10) |
By using the H¨older inequality and |∇φε,j|≤2ε, there exists C2>0, and we have
limε→0limn→∞∫Ω⟨∇un,∇φε,j⟩undx≤limε→0limn→∞(∫Ω|∇un|2dx)12(∫Ω|un|2|∇φε,j|2dx)12≤C1limε→0(∫B(xj,ε)|u|6dx)16(∫B(xj,ε)|∇φε,j|3dx)13≤C1limε→0(∫B(xj,ε)|u|6dx)16[∫B(xj,ε)(2ε)3dx]13≤C2limε→0(∫B(xj,ε)|u|6dx)16=0. | (2.11) |
We also derive that
limε→0limn→∞∫Ω|∇un|2φε,jdx≥limε→0∫Ω|∇u|2φε,jdx+μj=μj, | (2.12) |
and
limε→0limn→∞∫Ω|un|6φε,jdx=limε→0∫Ω|u|6φε,jdx+νj=νj. | (2.13) |
By (2.5) and (2.9)–(2.13), we get
0=limε→0limn→∞⟨I′λ(un),unφε,j⟩=limε→0limn→∞{(∫Ω|∇un|2dx)r∫Ω⟨∇un,∇(unφε,j)⟩dx+∫Ωϕunu2nφε,jdx−∫Ω|un|6φε,jdx−λ(∫ΩF(x,un)dx)s∫Ωf(x,un)unφε,jdx}=limε→0limn→∞{(∫Ω|∇un|2dx)r∫Ω|∇un|2φε,jdx+(∫Ω|∇un|2dx)r∫Ω⟨∇un,∇φε,j⟩undx−∫Ω|un|6φε,jdx}≥limε→0{(∫Ω|∇u|2dx+μj)r(∫Ω|∇u|2φε,jdx+μj)−νj}≥μr+1j−νj, |
that is, νj≥μr+1j. If νj>0, by (2.7), we obtain
νj≥S3(r+1)2−r,μj≥S32−r. | (2.14) |
Now, we show that (2.14) is impossible. Assume that there exists j0∈J, such that μj0≥S32−r and xj0∈Ω. It follows from (2.2)–(2.5) that
cλ=limn→∞{Iλ(un)−16⟨I′λ(un),un⟩}=limn→∞{(12(r+1)−16)(∫Ω|∇un|2dx)r+1+(14−16)∫Ωϕunu2ndx−λs+1(∫ΩF(x,un)dx)s+1+λ6(∫ΩF(x,un)dx)s∫Ωf(x,un)undx}≥2−r6(r+1)(∫Ω|∇u|2dx+∑j∈Jμj)r(∫Ω|∇u|2dx+∑j∈Jμj)−λ[1s+1(a2q)s+1−q6(a1q)s+1](∫Ω|u|qdx)s+1≥2−r6(r+1)μr+1j0+2−r6(r+1)‖u‖2(r+1)−λ6as+12−q(s+1)as+116(s+1)qs+1|Ω|(6−q)(s+1)6S−q(s+1)2‖u‖q(s+1). |
Let
G(t)=2−r6(r+1)t2(r+1)−λ6as+12−q(s+1)as+116(s+1)qs+1|Ω|(6−q)(s+1)6S−q(s+1)2tq(s+1). |
It is clear that limt→0G(t)=0, and limt→+∞G(t)=+∞. Therefore, there exists T>0 such that G(T)=mint≥0G(t), that is,
G′(t)|T=2−r3T2r+1−λ6as+12−q(s+1)as+116qs|Ω|(6−q)(s+1)6S−q(s+1)2Tq(s+1)−1=0. | (2.15) |
From (2.15) we obtain
T=(λ(6as+12−q(s+1)as+11)2qs(2−r)|Ω|(6−q)(s+1)6S−q(s+1)2)12(r+1)−q(s+1), |
and by simple calculation, we have
G(T)=2−r6(r+1)[λ6as+12−q(s+1)as+112qs(2−r)|Ω|(6−q)(s+1)6S−q(s+1)2]2(r+1)2(r+1)−q(s+1)−λ6as+12−q(s+1)as+116(s+1)qs+1(|Ω|6−q6S−q2)s+1[λ6as+12−q(s+1)as+112qs(2−r)|Ω|(6−q)(s+1)6S−q(s+1)2]q(s+1)2(r+1)−q(s+1)=(2−r)[q(s+1)−2(r+1)]6q(r+1)(s+1)[λ6as+12−q(s+1)as+112qs(2−r)|Ω|(6−q)(s+1)6S−q(s+1)2]2(r+1)2(r+1)−q(s+1). |
Hence, we can see that
cλ≥2−r6(r+1)S3(r+1)2−r−Dλ2(r+1)2(r+1)−q(s+1), |
where
D=(2−r)[2(r+1)−q(s+1)]6q(r+1)(s+1)[6as+12−q(s+1)as+112qs(2−r)|Ω|(6−q)(s+1)6S−q(s+1)2]2(r+1)2(r+1)−q(s+1). |
We obtain that cλ≥c∗. This is a contradiction, which indicates that νj=μj=0 for every j∈J, which implies that un→u in L6(Ω). We may assume that
∫Ω|∇un|2dx→A2,∫Ω|∇u|2dx≤A2. |
Combining with (2.5) and (2.6), we have
0=limn→∞⟨I′λ(un),un−u⟩=limn→∞[(∫Ω|∇un|2dx)r(∫Ω|∇un|2dx−∫Ω∇un∇udx)+∫Ωϕunun(un−u)dx−∫Ω|un|4un(un−u)dx−λ(∫ΩF(x,un)dx)s∫Ωf(x,un)(un−u)dx]=A2r(A2−∫Ω|∇u|2dx). |
Hence, we obtain
∫Ω|∇un|2dx→∫Ω|∇u|2dxasn→∞, |
which implies un→u in H10(Ω). The proof is complete.
Choose the extremal function
Uε(x)=(3ε2)14(ε2+|x|2)12,ε>0, |
satisfying
−ΔUε=U5εinR3. |
Let Ψ∈C1(R3) such that Ψ(x)=1 on BR2(0), Ψ(x)=0 on R3∖BR(0), and 0≤Ψ(x)≤1 on R3. Set uε(x)=Ψ(x)Uε(x). From [25], one has
{∫Ω|∇uε|2dx=S32+O(ε),∫Ω|uε|6dx=S32+O(ε3), |
and
|uε|αα={O(εα2),α∈[2,3),O(εα2|lnε|),α=3,O(ε6−α2),α∈(3,6). | (2.16) |
Then, we have the following Lemma.
Lemma 2.4. Suppose that λ>0, q(s+1)<4≤2(r+1)<6, and 0<s<1−r3(r+1). Then,
supt≥0Iλ(tuε)<2−r6(r+1)S3(r+1)2−r−Dλ2(r+1)2(r+1)−q(s+1). |
Proof. According to the definition of uε and (2.4), it holds that
λs+1(∫ΩF(x,tuε)dx)s+1≥λs+1tq(s+1)(a1q)s+1(∫Ω|uε|qdx)s+1≥C3λ(∫BR/2(0)εq2(ε2+|x|2)q2dx)s+1=C3λε(6−q)(s+1)2(∫R/2ε0y2(1+y2)q2dy)s+1≥C3λε(6−q)(s+1)2(∫10y2(1+y2)q2dy)s+1≥C4λε(6−q)(s+1)2. | (2.17) |
From Lemma 2.1 and (2.16), we have the following estimate:
∫Ωϕuεu2εdx≤S−1|uε|412/5≤O(ε2). | (2.18) |
Since Iλ(tuε)→−∞ as t→∞, by Lemma 2.2, there exists tε>0 such that
Iλ(tεuε)=supt>0Iλ(tuε)≥δ>0. |
Moreover, by the continuity of Iλ, there exist positive constants t1 and t2 such that 0<t1≤tε≤t2<+∞. As a consequence of the above fact, one has
supt≥0Iλ(tuε)=supt≥0{12(r+1)(∫Ω|∇tuε|2dx)r+1+14∫Ωϕtuε|tuε|2dx−16∫Ω|tuε|6dx−λs+1(∫ΩF(x,tuε)dx)s+1}≤supt≥0{12(r+1)(∫Ω|∇tuε|2dx)r+1−16∫Ω|tuε|6dx}+O(ε2)−λC4ε(6−q)(s+1)2≤2−r6(r+1)[(∫Ω|∇uε|2dx)r+1∫Ω|uε|6dx]2(r+1)6−2(r+1)(∫Ω|∇uε|2dx)r+1+O(ε2)−λC4ε(6−q)(s+1)2≤2−r6(r+1)S3(r+1)2−r+C5ε−λC4ε(6−q)(s+1)2<2−r6(r+1)S3(r+1)2−r−Dλ2(r+1)2(r+1)−q(s+1). |
We have used the fact that 6(r+1)r+2<q<4s+1 and let ε=λ2(r+1)2(r+1)−q(s+1).
0<λ<Λ1=min{[2−r6(r+1)DS3(r+1)2−r]2(r+1)−q(s+1)2(r+1),(C5+DC4)2(r+1)−q(s+1)(s+1)[(r+1)(6−q)−q] }, |
and then
C5ε−C4λε(6−q)(s+1)2=C5λ2(r+1)2(r+1)−q(s+1) −C4λ(r+1)(6−q)(s+1)2(r+1)−q(s+1) +1=λ2(r+1)2(r+1)−q(s+1) (C5−C4λ(s+1)[(r+1)(6−q)−q]2(r+1)−q(s+1) )<−Dλ2(r+1)2(r+1)−q(s+1). |
The proof is complete.
Lemma 2.5. Suppose that 0<λ<Λ0 (Λ0 is as in Lemma 2.2). Then, system (1.1) has a positive solution uλ satisfying Iλ(uλ)<0.
Proof. It follows from Lemma 2.2 that
m=infu∈¯Bρ(0)Iλ(u)<0. |
By the Ekeland variational principle [26], there exists a minimizing sequence {un}⊂¯Bρ(0) such that
Iλ(un)≤infu∈¯Bρ(0)Iλ(u)+1n,Iλ(v)≥Iλ(un)−1n‖v−un‖,v∈¯Bρ(0). |
Therefore, we obtain that Iλ(un)→m and I′λ(un)→0 as n→∞. Since {un} is a bounded sequence, and ¯Bρ(0) is a closed convex set, we may assume up to a subsequence, still denoted by {un}, there exists uλ∈¯Bρ(0)⊂H10(Ω) such that
{un⇀uλ,weaklyinH10(Ω),un→uλ,stronglyinLq(Ω)(1≤p<6),un(x)→uλ(x),a.e.inΩ. |
By the lower semi-continuity of the norm with respect to weak convergence, we get
m≥lim infn→∞[Iλ(un)−16⟨I′λ(un),un⟩]=lim infn→∞[(12(r+1)−16)(∫Ω|∇un|2dx)r+1+(14−16)∫Ωϕunu2ndx−λ1s+1(∫ΩF(x,un)dx)s+1+λ6(∫ΩF(x,un)dx)s∫Ωf(x,un)undx]≥2−r6(r+1)(∫Ω|∇uλ|2dx)r+1+112∫Ωϕuλu2λdx−λ1s+1(∫ΩF(x,uλ)dx)s+1+λ6(∫ΩF(x,uλ)dx)s∫Ωf(x,uλ)uλdx=Iλ(uλ)−16⟨I′λ(uλ),uλ⟩=Iλ(uλ)≥m. |
Thus, Iλ(uλ)=m<0, and we can see that uλ≢0. Iλ(|uλ|)=Iλ(uλ), which suggests that uλ≥0. Therefore, by the strong maximum principle, we obtain that uλ is a positive solution of system (1.1). The proof is complete.
Lemma 2.6. Assume that 0<λ<Λ∗ (Λ∗=min{Λ0,Λ1}). Then, the system (1.1) has a positive solution u∗∈H10(Ω) with Iλ(u∗)>0.
Proof. From the mountain pass lemma and Lemma 2.2, there exists a sequence {un}⊂H10(Ω) such that
Iλ(un)→cλ>0,andI′λ(un)→0asn→∞, |
where
cλ=infγ∈Γmaxt∈[0,1]Iλ(γ(t)), |
and
Γ={γ∈C([0,1],H10(Ω)):γ(0)=0,γ(1)=e}. |
According to Lemma 2.3, we know that {un}⊂H10(Ω) has a convergent subsequence, still denoted by {un}, such that un→u∗ in H10(Ω) as n→∞.
Iλ(u∗)=limn→∞Iλ(un)=cλ>δ>0, |
which implies that u∗≢0. It is similar to Lemma 2.5 that u∗>0, that is, u∗ is a positive solution of system (1.1) such that Iλ(u∗)>0. The proof is complete.
In this paper, we considered a class of bi-nonlocal Kirchhoff-Schr¨odinger-Poisson system with critical growth. Under some suitable assumptions, by using the concentration compactness principle, we obtained the multiplicity of positive solutions.
The authors thank the anonymous referees for careful reading and some helpful comments, which greatly improved the manuscript. This work is supported by the National Natural Science Foundation of China (No.11661021; No.11861021).
The authors declare there are no conflicts of interest.
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