# Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

Authors: Junjun Zhou; Xiangyun Hu; Tiaojie Xiao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we consider the following Kirchhoff-type problems involving critical exponent−a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u,x∈Ωu>0,x∈Ωu=0,x∈∂Ω. The existence and multiplicity of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth are studied under some suitable assumptions on Vx and gx,u. By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions are obtained.


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## Body

## 1. Introduction

We consider the following nonlinear boundary value problem for second-order impulsive differential equations:(1)−a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u,x∈Ω,u>0,x∈Ω,u=0,x∈∂Ω,where Ω⊂ℝNN≥3 is a smooth bounded domain and ∂Ω is a smooth boundary of Ω. λ,μ>0,a,b≥0,a+b>0. V∈CΩ,ℝ and g∈CΩ¯×ℝ,ℝ. 2∗=2N/N−2 is the critical Sobolev exponent for the embedding H01Ω ↪ LpΩ. Moreover, Vx and gx,u satisfy some conditions which will be given later.Over the past decades, the following Kirchhoff equation:(2)−a+b∫Ω∇u2dxΔu=gx,u,x∈Ω,u=0,x∈∂Ω,has been extensively considered. With various assumptions about the nonlinearity gx,u, the existence and multiplicity of solutions for system (2) are obtained by variational methods, see [1–5] and the references therein.To our best knowledge, system (2) is related to the stationary analogue of the equation(3)ρ∂2u∂t2−P0h+E2L∫0L∂u∂x2dx∂2u∂x2=0,which was proposed by Kirchhoff in [6]. In fact, (3) is an extension and generalization of the classical D’Alembert wave equation in some ways. This model is widely used in many fields, such as non-Newtonian mechanics, cosmophysics, elastic theory, and electromagnetics. It is worth noting that equation (2) has a nonlocal term ∫0L∂u/∂x2dx. Only after Lions [7] proposed an abstract functional analysis framework about the following equation:(4)utt−a+b∫ΩDu2Δu=gx,u,problem (4) received much attention, and we refer the readers to [8–15] for more details and the references therein. More precisely, Bisci and Pizzimenti [13] studied the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian by using variational methods. In [15], Bisci considered the existence of (weak) solutions for some Kirchhoff-type problems on a geodesic ball of the hyperbolic space and the main technical approach is based on variational and topological methods. In [16], the authors firstly used the variation method to study the existence of positive solution of the Kirchhoff-type problems with the Sobolev critical exponent. After that, there are many works on the existence and multiplicity of solutions for Kirchhoff-type problems with the Sobolev critical exponent (one can see [17–21] and the references therein).In [17], the author considered the following Kirchhoff-type elliptic equation:(5)−a+b∫Ω∇u2dxΔu=μgx,u+u5,u>0,x∈Ω,u=0,x∈∂Ω,and by using the variation method, the existence of positive solutions of system (5) is obtained. To our best knowledge, a nonlinear elliptic boundary value problem has a critical term, which is a difficulty to prove the existence of solutions for the problem. The difficulty is caused by the lack of compactness of the embedding H01Ω ↪ L2∗Ω, which makes the PS condition cannot be checked directly. In [16], the authors make the parameter μ large enough to make a critical value below a certain level. In [17], under the AR condition, the authors restored the compactness of the embedding by using the second concentration compactness lemma, which is an extension of the work in [8].In [18], the authors used the variational method to consider system (5) with μ=1 and the existence and multiplicity of solutions for the system are obtained.Moreover, problems on the unbounded domainRN have also been widely studied by some researchers, for example, [22–26]. More precisely, in [25], Liu and He used the variant version of fountain theorem to get the existence of infinitely many high energy solutions of the system. In [26], the authors studied the concentration behavior of positive solutions. For more information about this problem, we refer the readers to [11, 20, 27, 28] and the reference therein.Motivated by the above facts, we want to consider the positive solutions of system (1). By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions of system (1) are obtained.To show our main results, we introduce some conditions on nonlinearitygx,u and Vx.V
1 Vx is 1-periodic in each of xii=1,2,…,N, and there exists a positive constant V0 such that(6)Vx≥V0>0,x∈Ω.F1 If s≤0, then gx,s≡0; if s≥0, then gx,s≥0.F2(7)lims⟶0+gx,ss=0,lims⟶+∞gx,ss5=0,x∈Ω¯.F3∀x,s∈Ω×R+, there is(8)gx,ss−4Gx,s≥−aλ1s2,whereGx,s=∫0sgx,tdt and λ1>0 is the first eigenvalue of −Δ in H01Ω.G1gx,u=fxuq−12≤q<2∗. fx∈L2∗−q/2∗Ω and fx≥0,fx≡0.G2Vx∈L2Ω. In addition, Vx>0 is bounded in Ω.In the next section, we will present our main results.Theorem 1.
LetN=3 and λ=μ=1. If V1 and F1−F3 hold, then system (1) has a positive ground state solution.Remark 1.
In [17], the author used a condition which is stronger than (F3), that is,
F
3
′ There exists a constant θ∈4,6, such that(9)gx,ss−θGx,s≥0,∀x,s∈Ω×R+.Theorem 2.
LetN=4 and 0<μ<bS2. If G1 and G2 hold, then there exists a constant λ∗>0 (we will give in the proof of Theorem 2 in Section 3) such that ∀λ>λ∗, and system (1) has at least two positive solutions.Remark 2.
In reference [21], the authors considered system (1) as fx≡1, 1<q<2, Vx=0, and μ=1. Underlying the condition λ<λ0 (a constant the authors given in their paper), two positive solutions are obtained. However, our results are very different from those in [21]. In our paper, 2≤q<2∗, and if λ>λ∗ (a constant we give in the proof of Theorem 2) is sufficiently large, two positive solutions are obtained. Besides, in [21], N=3; in our paper, for N=4, the multiple solutions of higher dimensional space are obtained.Remark 3.
Whena=1,b=0, and Vx=0, system (1) degenerates to a classical semilinear elliptic problem. Theorem 2 can be the generalization of the corresponding results in [8] of Kirchhoff-type problems.
The reminder of this paper is organized as follows. In Section2, some preliminary results are presented. The proof of main results will be given in Section 3.


## 2. Preliminaries

In this paper, we make some notations as follows:(i)
The spaceH01Ω (denoted by E) is equipped with the norm u=∫Ω∇u2dx1/2, and we also define the equivalent norm by uE=∫Ω∇u2+Vxu2dx1/2. In addition, LpΩ1≤p<+∞ is a Lebesgue space with the norm up=∫Ωupds1/p.(ii)
The sequencexn in H01Ω is a PSc sequence if Ixn⟶c and I′xn⟶0 as n⟶∞. We say that if the functional satisfies PCc condition for any PSc sequence, it has a convergent subsequence.(iii)
C,C1,C2,…, denote various positive constants.(iv)
u+x=maxux,0 and u−x=max−ux,0.(v)
o1 shows when n⟶∞,o1⟶0.(vi)
LetS be the best Sobolev constant, that is,(10)S≔infu∈H01Ω\0u2u2∗2.Now, we give the energy functional corresponding to problem (1), that is,(11)Iu=a2u2+b4u4+12∫ΩVxu2dx−λ∫ΩGx,udx−μ2∗∫Ωu+2∗dx.It is obvious thatI∈C1E,R and has the following derivative:(12)I′u,v=a+bu2∫Ω∇u⋅∇vdx+∫ΩVxuvdx−λ∫Ωgx,uvdx−μ∫Ωu+2∗−1vdx.Using the continuity ofgx,u and Vx, it shows that u∈E is a critical point of I, if it is a solution of problem (1).Lemma 1.
LetN=3 and λ=μ=1. If (V1), (F1), and (F2) are satisfied, then the following hold:(1)
There exist constantsρ,α>0, such that(13)Iu≥α,∀u∈H01Ω,u=ρ.(2)
There existsu∈E such that(14)Iu<0u>ρ.Proof.
From (F1) and (F2), it shows that there has a constant C1>0 such that(15)Gx,s≤aλ14s2+C1s6,∀x,s∈Ω¯×R.
By (10), (15), V1, and Sobolev inequality, it follows that(16)Iu≥a2u2+b4u4+12∫ΩVxu2−∫ΩGx,udx−16∫Ωu6dx,≥a2u2−a4u2−C1u6−16u6≥a4u2−C2u6.(1)
Takingρ>0 small enough, there exists a constant α>0 such that(17)Iu≥α,∀u∈H01Ω,u=ρ.(2)
If we takev0∈H01Ω and v0≡0, then one gets the following:(18)Itv0=a2t2v02+b4t4v04+12t2∫ΩVxv02−∫ΩGx,tv0dx−16t6∫Ωv0+6dx≤t22maxa,1v0E2+b4t4v0E4−16t6∫Ωv0+6dx.
Since⋅ and the ⋅E are equivalent, then(19)Itv0≤C3t22maxa,1v02+C4b4t4v04−16t6∫Ωv0+6dx.
It is obvious that(20)Itv0⟶−∞t⟶+∞.
Therefore, we can find a positive constantt0, and t0v0>ρ, such that(21)It0v0<0.
Letu=t0v0 and the conclusion is satisfied.Lemma 2.
LetN=3andλ=μ=1. Vx satisfies (V1) and gx,u satisfies (F1)—(F3). Suppose(22)Λ=ab4S3+b324S6+aS6b2S4+4aS+b224S4b2S4+4aS,c∈0,Λ,then I satisfies the PSc condition.Proof.
By (F1) and (F2), there exists a constant C5>0, such that ∀x,s∈Ω¯×R, and it has(23)15gx,ss−Gx,s≤130s6+C5.
Supposeun is a PSc sequence, c∈0,Λ,(24)Iun⟶c,I′un⟶0n⟶∞.
The next work is to prove the boundness ofun. Clearly, one has(25)1+c+o1un≥Iun−15I′un,un=a2−a5un2+b4−b5un4+12−15∫ΩVxun2dx+15−16∫Ωun+6dx+∫Ω15gx,un+un+−Gx,un+dx≥310aun2+b20un4−C5Ω.
That is to sayun is bounded in E. Going necessary to a subsequence, it has(26)un⇀u,u∈H01Ω,un⟶u,u∈LpΩ1≤p<2∗=6,unx⟶ux,a.e.x∈Ω.
By (V1) and (F2), one has(27)∫Ωgx,uuundx⟶∫Ωgx,uudxn⟶∞,∫ΩGx,uudx⟶∫ΩGx,udxn⟶∞,∫ΩVxun2dx⟶∫ΩVxu2dxn⟶∞.
Letvn=un−u, then we can claim vn⟶0 as n⟶∞. Otherwise, there exist a subsequence (for convenience, we still denote it by vn) such that(28)limn⟶∞vn2=l,where l>0; then(29)un2=vn2+u2+o1.
By using Brézis–Lieb lemma in [29], it has(30)∫Ωun+6=∫Ωvn+6+∫Ωu+6+o1.
BecauseI′un⟶0 in E∗, one has(31)I′un,un=aun2+bun4+∫ΩVxun2dx−∫Ωgx,unundx−∫Ωun+6dx=o1,which shows that(32)al+au2+bl2+bu4+2blu2+∫ΩVxu2dx−∫Ωgx,uudx−∫Ωvn+6dx−∫Ωu+6dx=o1.
It also has(33)limn⟶∞I′un,u=au2+blu2+bu4+∫ΩVxu2dx−∫Ωgx,uudx−∫Ωu+6dx=0.
Combining (32), (33), and (10), one gets(34)al+bl2+blu2=∫Ωvn+6dx+o1=vn6S3+o1≤l3S3+o1,which implies that(35)al+bl2+blu2≤l3S3.
By (35),(36)l≥bS3+b2S6+4a+bu2S32≥bS3+b2S6+4aS32.
AsIun⟶cn⟶∞, we have(37)c=a2un2+b4un4+12∫ΩVxun2dx−∫ΩGx,undx−16∫Ωun+6dx+o1=a2l+a2u2+b4u4+b4l2+bl2u2+12∫ΩVxu2dx−∫ΩGx,udx−16∫Ωvn+6dx−16∫Ωu+6dx+o1.
From (34) and (37), it has(38)Iu=a2u2+b4u4+12∫ΩVxu2dx−∫ΩGx,udx−16∫Ωu+6dx=c−a3l+b12l2+bl3u2≤c−a3bS3+b2S6+4aS32−b12bS3+b2S6+4aS322−bl3u2=c−Λ−bl3u2<−bl3u2.
From the above inequality, it has(39)Iu+bl3u2<0.
On the other hand, from (F3) and (33), which deduce(40)bl3u2+Iu=bl3u2+a2u2+b4u4+12∫ΩVxu2dx−∫ΩGx,udx−16∫Ωu+6dx=bl3u2+a4u2−bl4u2+14∫ΩVxu2dx+∫Ω14gx,uu−Gx,udx+112∫Ωu+6dx≥bl12u2+a4u2−aλ14∫Ωu2dx+112∫Ωu+6dx≥bl12u2+112∫Ωu+6dx≥0,which is a contradiction with (39).
Sol=0, that is to say, un⟶u in E as n⟶∞. Thus, I satisfies the PSc condition.Lemma 3.
LetN=4 and a,b>0. If G1 and G2 are satisfied, then there exists a positive constant μ∗=bS2>0, such that for every μ∈0,μ∗, the functional Iu satisfies the PS condition in H01Ω.Proof.
Ifun⊂H01Ω is a PSc sequence of I, that is,(41)Iun⟶c,I′un⟶0n⟶∞.
AsN=4, and by (4) and Hölder inequality, one has(42)Iu=a2u2+b4u4+12∫ΩVxu2dx−μ4∫Ωu+4dx−λq∫Ωfxu+qdx≥a2u2+b4u4−μ4S−2u4−λqf4/4−qS−q/2uq.
Chooseμ∗=bS2 and 2≤q<2∗=4. For every μ∈0,μ∗, (42) implies that the functional I is coercive and bounded in H01Ω for all λ>0. Therefore, the sequence un is bounded in H01Ω. It means that there exists a subsequence and we still denote it by un for simplicity, such that(43)un⇀u,u∈H01Ω,un⟶u,u∈LpΩ1≤p<2∗=4,unx⟶ux,a.e.x∈Ω.
The following is to proveun⟶un⟶∞ in H01Ω. Let wn=un−u. By (43), one has(44)un2=wn2+u2+o1,(45)un4=wn4+2wn2u2+wn4+o1.And by Brézis–Lieb’s lemma in [29], one obtains(46)∫Ωun+4dx=∫Ωwn+4dx+∫Ωu+4dx+o1,and we can claim that(47)limn⟶∞∫ΩVxun2dx=∫ΩVxu2dx,(48)limn⟶∞∫Ωfxun+qdx=∫Ωfxu+qdx.
In fact, by the Sobolev imbedding theorem, there exists a constantC>0, such that un4≤C<∞. ∀ε>0,∃δ>0, for any S⊂Ω with measS<δ and by Hölder inequality, one has(49)∫SVxun2dx≤V2un42≤εC2,(50)∫Sfxun+qdx≤f4/4−qun4q<εCq.
In view of the absolute continuity of the integrals∫SVx2dx and ∫Sfx4/4−qdx, it means that(51)limmeasS⟶0∫SVx2dx=0,(52)limmeasS⟶0∫Sfx4/4−qdx=0.
By (49)–(52), it follows that ∫SVxun2dx≤εC2 and ∫Sfxun+qdx≤εCq. Thus, by Vitali’s theorem [30], (47) and (48) hold. Similarly, one obtains(53)limn⟶∞∫Ωun+3udx=∫Ωu+4dx,(54)limn⟶∞∫ΩVxunudx=∫ΩVxu2dx,(55)limn⟶∞∫Ωfxun+q−1udx=∫Ωfxu+qdx.
By (41) and (53)–(55), one gets(56)I′un,un=aun2+bun4+∫ΩVxun2dx−μ∫Ωun+4dx−λ∫Ωfxun+qdx=o1(57)limn⟶∞I′un,u=au2+bwn2u2+bu4+∫ΩVxu2dx−μ∫Ωu+4dx−λ∫Ωfxu+qdx=0.
By (44)–(48) and (57), it follows that(58)awn2+au2+bwn4+bu4+2bwn2u2+∫ΩVxu2dx−μ∫Ωwn+4dx−μ∫Ωu+4dx−λ∫Ωfxu+qdx=o1.
Combining (57) and (58), one can get(59)awn2+bwn4+bwn2u2−μ∫Ωwn+4dx=o1.
From (10), it can be deduced that(60)∫Ωwn+4dx≤∫Ωwn4dx≤S−2wn4.
Letwn=l. Consequently, from (59) and (60), one gets(61)al2+bl2u2+bl4≤μS−2l4.
Choosingμ∗=bS2>0, ∀μ∈0,μ∗, inequality (61) implies l=0. Thus, un⟶u in H01Ω. This completes the proof of Lemma 3.


## 3. Proof of Main Results

Now we will prove Theorem1 and Theorem 2.Proof of Theorem 1.
Combining Lemma1 with Lemma 2, we can say that it exists u∈E, such that(62)Iu=c,I′u=0.
LetW=u∈E\0,I′u=0 and m=infu∈WIu. Then, W≠∅,m≤c. By (F1) and (F2), there exists a constant C6>0, such that(63)gx,ss≤aλ12s2+C6s6,∀x,s∈Ω¯×R.
By Sobolev inequality,∀u∈W, one has(64)au2+bu4+∫ΩVxu2dx=∫Ωgx,uudx+∫Ωu+6dx≤aλ12∫Ωu2dx+C6∫Ωu6dx+∫Ωu+6dx≤a2u2+C7u6.
It is easy to know(65)a2u2+bu4≤C7u6.
The above inequality can deduce that there exists a constantC>0, such that(66)u≥C,∀u∈M.
We claim that ifC8>0, such that(67)∫Ωu+6dx≥C8,∀u∈M.
Otherwise, we assume thatun∈M, such that(68)limn⟶∞∫Ωun+6dx=0.
Then,(69)limn⟶∞∫Ωun+2dx=0.
In addition, we can calculate(70)aC2≤aun2+bun4+∫ΩVxun2=∫Ωgx,un+un+dx+∫Ωun+6dx≤aλ12∫Ωun+2dx+C6+1∫Ωun+6⟶0,which is a contradiction. Therefore, the assertion is established. Therefore, ∀u∈M, we have(71)Iu=Iu−14I′u,u=a4u2+14∫ΩVxu2dx+∫Ω14gx,uu−Gx,udx+112∫Ωu+6dx≥aλ14∫Ωu2dx−aλ14∫Ωu2dx+112∫Ωu+6dx≥112C8,which implies m>0; by the definition of m, we can get a PSm sequence. By using Lemma 1 and Lemma 2, there exists u∈E, such that(72)Iu=m,I′u=0.
ByI′u,u−=0, where u−=max−u,0, we can get u−=0, so u=u+. Then, by strong maximum principle, it implies u>0.Proof of Theorem 2.
It is divided into two steps to prove Theorem2. Firstly, we claim that system (1) has a positive global minimizer solution u1. In fact, from the proof of Lemma 3, it can be known that the functional I is coercive and bounded, so m=infu∈H01ΩIu is defined. By (G2), Vx is bounded in Ω, so there exists a constant M0>0, such that(73)Vx≤M0.
By Hölder inequality, (10), and the above inequality, one gets(74)Iu=a2u2+b4u4+12∫ΩVxu2dx−μ4∫Ωu+4dx−λq∫Ωfxu+qdx≤a2u2+b4u4+12M0Ω1/2S−1u2−λq∫Ωfxu+qdx.
Choosingu0=1, from (74), one has(75)Iu0≤2a+b+2M0Ω1/2S−14−λq∫Ωfxu+qdx<0,for all λ>λ∗=q2a+b+2M0Ω1/2S−1/4∫Ωfxu+qdx. Therefore, m<0. By Lemma 2.3 and Theorem 4.4 in [31], there exists u1∈H01Ω such that Iu1=m<0. Letting v=u1− in (12), it follows that u1>0. Thus, u1 is a nonzero and nonnegative solution of system (1). Moreover, by the strong maximum principle, it has u1>0. That is to say, u1 is a positive global minimizer solution of system (1), such that Iu1=m<0.
Secondly, we will prove that system (1) has another positive solution. As 0<q<2∗=4, it is easy to know 0 is a local minimum point of functional I in H01Ω. Defining c as follows: c=infγ∈Γmaxt∈0,1Iγt, where(76)Γ=γ∈C0,1,H01Ωγ0=0,γ1=u1.
It is obvious thatc>0. By the mountain pass lemma in [32], there exists u2∈H01Ω, such that Iu2=c>0 and I′u2=0. Similarly, taking v=u2− in (12), we can get u2 is also a nonzero and nonnegative solution of system (1). By the strong maximum principle, it has u2>0, such that Iu2=c>0. This proves Theorem 2.



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