# Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth inℝN

Authors: Li Zhou; Chuanxi Zhu
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the formΔ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,inℝN,u∈H2ℝN, where a>0, b≥0, λ is a positive parameter, α∈N−2,N, 5≤N≤8, V:ℝN⟶ℝ is a potential function, and Iα is a Riesz potential of order α. Here, 2∗∗=2N/n−4 with N≥5 is the Sobolev critical exponent, and Δ2u=ΔΔu is the biharmonic operator. Under certain assumptions on Vx and fu, we prove that the equation has ground state solutions by variational methods.


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## Body

## 1. Introduction

In this article, we study the following fourth-order elliptic Kirchhoff-type equation involving the critical growth of the form:(1)Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,inℝN,u∈H2ℝN,where a>0, b≥0, λ is a positive parameter, 5≤N≤8, V:ℝN⟶ℝ is a potential function, and Iα is a Riesz potential of order α∈N−2,N defined by Iα=ΓN−α/2/Γα/2πN/22αxN−α. Here, 2∗∗=2N/N−4 with N≥5 is the Sobolev critical exponent, and Δ2u=ΔΔu is the biharmonic operator, that is, Δ2u=∑i=1N∂4/∂xi4u+∑i≠jN∂4/∂xi2∂xj2u. Besides, Vx:ℝN⟶ℝ is a potential function satisfying(V1)V∈CℝN,ℝ and infx∈ℝNVx≔V0>0(V2)measx∈ℝN:Vx≤M<∞, where meas denotes the Lebesegue measure in ℝN and V0 and M are positive constants.Furthermore, we suppose that the functionf∈C1ℝ,ℝ satisfies(f1) ft=otα/N as t⟶0(f2) limt⟶+∞ft/tα+2/N−2=0(f3) ft/t is increasing on 0,+∞ and decreasing on −∞,0(f4) ft is increasing on ℝOn the one hand, in 2012, by the variational methods, Wang and An [1] studied the following fourth-order equation of Kirchhoff type:
(2)Δ2u−M∫Ω∇u2dxΔu=fx,u,x∈Ω,u=∇u=0,x∈∂Ω,and obtained the existence and multiplicity of solutions. Later, Cabada and Figueiredo [2] considered a class of generalized extensible beam equations with critical growth in ℝN as follows:
(3)Δ2u−M∫ℝN∇u2dxΔu+u=λfu+u2∗∗−2u,xinℝN,u∈H2ℝN,where M:ℝ+⟶ℝ+ is continuous, f∈Cℝ,ℝ, N≥5, and λ>0 is a parameter. With the help of the minimax theorem and the truncation technique, the existence of nontrivial solutions of equation (3) is proved for λ sufficiently large. Recently, Song and Shi [3] proved the multiplicity of solutions for the following fourth-order elliptic equation with critical exponent
(4)Δ2u−g∫Ω∇u2dxΔu=λhx,u+u2∗∗−2u,x∈Ω,u=∇u=0,x∈∂Ω,where Ω⊂ℝN is an open bounded domain with smooth boundary, g:ℝ+⟶ℝ+ is continuous, h∈CΩ¯×ℝ,ℝ, N≥5, and λ is a positive parameter. Soon after that, Liang et al. [4] obtained multiplicity of solutions to the following generalized extensible beam equation with critical growth:
(5)Δ2u−M∫ℝN∇u2dxΔu+Vxu=kxuq−2u+λu2∗∗−2u,xinℝN.In fact, in the earlier time, Ma has already applied the variational methods to study the existence and multiplicity of solutions for the following fourth-order boundary value problem of Kirchhoff type:(6)u″″−M∫01u′2dxu′=fx,u,x∈0,L,u″″−M∫01u′2dxu″=qxfx,u,u′x∈0,1,u0=u1=u″0=u″1=0.For more details, readers can refer to [5, 6] and the references therein.Actually, for the special case of problem (2) with Mt=a+bt, then problem (2) is reduced to the following fourth-order elliptic equations of Kirchhoff type:
(7)Δ2u−a+b∫Ω∇u2dxΔu=fx,u,x∈Ω,u=∇u=0,x∈∂Ω.This problem is related to the stationary analog of the evolution equation of Kirchhoff type:(8)utt+Δ2u−a+b∫ℝN∇u2dxΔu=fx,t.Dimensions one and two are relevant from the point of view of physics, engineering, and other sciences, because in those situations model (8) is considered a good approximation for describing nonlinear vibrations of beams or plates (see [7, 8]). Different approaches have been taken to attack this problem under various hypotheses on the nonlinearity. For example, very recently, Wang et al. [9] concentrated on the following Navier BVPs:
(9)Δ2u−λa+b∫Ω∇u2dxΔu=fx,u,x∈Ω,u=∇u=0,x∈∂Ω,where Ω⊂ℝN is a smooth bounded domain and λ, a, b>0. Applying mountain pass techniques and the truncation method, they obtained the existence of nontrivial solution to equation (9) for λ small enough when fx,u satisfies some superlinear assumptions. For whole space ℝN, Song and Chen [10] studied the class of Schrödinger-Kirchhoff-type biharmonic problems:
(10)Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=fx,u,x∈ℝN,u∈H2ℝN,where f satisfies the Ambrosetti-Rabinowitz type conditions. Under appropriate assumptions on V and f, the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.On the other hand, in the past decades, many scholars have studied the following problem:(11)−Δu+Vxu=Iα∗upup−2u,which is called nonlinear Choquard type equation. For the physical background, we refer to [11–13] and the references therein. Mathematically, the existence and qualitative properties of solutions to equation (5) have been studied for decades by variational methods. See [11, 14, 15] for earlier results and [16–26] for a recent work.Motivated by the works we mentioned above, especially by [4, 10], we consider the combination of equations (7) and (11) and extend to the general convolution case in ℝN. In our paper, we get the ground state solution of problem (1).Our main results are as follows:Theorem 1.
IfV satisfies (V1)-(V2) and f∈C1ℝ,ℝ verifies (f1)-(f4), then problem (1) has a ground state solution.
For the convenience of expression, hereafter, we use the following notations:
H≔H2ℝN=u∈L2ℝN:∇u,Δu∈L2ℝN equipped with scalar product u,vH=∫ℝNΔuΔv+∇u∇v+uvdx
E≔u∈H:∫ℝNVxu2dx<∞ equipped with scalar product u,v=∫ℝNΔuΔv+∇u∇v+Vxuvdx, therefore, u=∫ℝNΔu2+∇u2+Vxu2dx1/2(i)
LsℝN1≤s≤∞ denotes the Lebesgue space with the norm us=∫ℝNusdx1/s(ii)
For anyu∈H2ℝN\0,ut is denoted as
(12)ut=0,t=0,tuxt,t>0(iii)
C,C1,C2,⋯ denote positive constants possibly different in different linesRemark 2.
By the assumptions ofV, it is obvious that the embedding E°H2ℝN is continuous. Furthermore, the embedding E°LrℝN is continuous for r∈1,2∗∗, and compact for r∈1,2∗, where 2∗=2N/N−2 if N≥3 and 2∗=∞ if N=1 or 2. Thus, for each s∈2,2∗∗, there is a Cε>0 such that us≤Cεu for all u∈E.


## 2. Preliminaries

In this section, we will give some very important lemmas.Lemma 3.
Assume (f1)-(f4) hold, then we have the following:(1)
For allε>0, there is aCε>0 such that ft≤εtα/N+Cεtα+2/N−2 and Ft≤εtN+α/N+CεtN+α/N−2(2)
For allε>0, there is a Cε>0 such that for every p∈2,2∗,Ft≤εtN+α/N+tN+α/N−2+CεtpN+α/2N, and Ft2N/N+α≤εt2+t2N/N−α+Cεtp(3)
For anys≠0,sfs>2Fs and Fs>0Proof.
One can easily obtain the results by elementary calculation.Lemma 4 (Hardy-Littlewood-Sobolev inequality [27]).
Let0<α<N,p,q>1 and 1≤r<s<∞ be such that
(13)1p+1q=1+αN,1r−1s=αN.(1)
For anyf∈LpℝN and g∈LqℝN, one has
(14)∫ℝN∫ℝNfxgyx−yN−αdxdy≤CN,α,pfLpℝNgLqℝN(2)
For anyf∈LrℝN, one has
1/·N−α∗fLsℝN≤CN,α,rfLrℝNRemark 5.
By Lemma3(1), Lemma 4(1) and Sobolev imbedding theorem, we can get
(15)∫ℝNIα∗FuFudx≤CFu2N/N+α2≤C∫ℝNuN+α/N+uN+α/N−22N/N+αdxN+α/N≤C∫ℝNu2+u2N/N−2dxN+α/N≤Cu2N+2α/N+u2N+2α/N−2.


## 3. Variational Formulation

The associated energy function of problem (1) is given by
(16)Iu=12∫ℝNΔu2dx+12∫ℝNa∇u2+Vxu2dx+b4∫ℝN∇u2dx2−12∫ℝNIα∗FuFudx−λ2∗∗∫ℝNu2∗∗dxi.e. the critical points of the functionalIu are weak solutions of problem (1). Under the assumptions, I∈C1E,ℝ, and for all u,v∈E, it holds that
(17)I′u,v=∫ℝNΔuΔvdx+a∫ℝN∇u∇vdx+∫ℝNVxuvdx+b∫ℝN∇u2dx∫ℝN∇u∇vdx−∫ℝNIα∗Fufuvdx−λ∫ℝNu2∗∗−2uvdx.Thus,(18)I′u,u=∫ℝNΔu2dx+∫ℝNa∇u2+Vxu2dx+b∫ℝN∇u2dx2−∫ℝNIα∗Fufuudx−λ∫ℝNu2∗∗dx.In this section, we prove the following results.Lemma 6.
The functionalIu possesses the mountain-pass geometry, i.e.,
(1)
There existsρ,δ>0 such that Iu≥δ for all u=ρ(2)
There existse∈H2ℝN such that e>ρ and Ie<0Proof.
(1) By Lemma3(1) and Lemma 4, we have
(19)Iu=12∫ℝNΔu2dx+12∫ℝNa∇u2+Vxu2dx+b4∫ℝN∇u2dx2−12∫ℝNIα∗FuFudx−λ2∗∗∫ℝNu2∗∗dx≥C1u2−C2u2N+2α/N+u2N+2α/N−2−C3u2∗∗.
Thus, there existsρ,δ>0 such that I∞≥δ for all u=ρ>0 small enough.
(2) For anyu∈E\0,
(20)Iut=tN−32∫ℝNΔu2dx+atN−12∫ℝN∇u2dx+tN+12∫ℝNVxu2dx+bt2N−24∫ℝN∇u2dx2−tN+α2∫ℝNIα∗FuFudx−λ2∗∗tN2+3N/N−4∫ℝNu2∗∗dx⟶−∞,as t⟶∞, since α>N−2; thus, we see I∞ut<0 for t>0 large. Note that
(21)ut2=tN−3∫ℝNΔu2dx+tN−1∫ℝN∇u2dx+tN+1∫ℝNVxu2dx.
Takinge=ut0, with t0>0 large, we have e>ρ and I∞e<0.Hence, we define the mountain-pass level ofI:
(22)c=infγ∈Γmaxt∈0,1Iγt>0,where Γ=γ∈C0,1,E:γ0=0,Iγ1<0.Lemma 7.
Ifun⊂E is a PSc sequence of I, then un is bounded in E.Proof.
Letun⊂E be PSc sequence, i.e., Iun⟶c and I′un⟶0; then, we have
(23)c=Iun−14I′un,un+o1=14∫ℝNΔun2dx+14∫ℝNa∇un2+Vxun2dx+14∫ℝNIα∗Funfunun−2Fundx+λ4−λ2∗∗∫ℝNun2∗∗dx+o1≥C4un2.
Consequently,un is bounded in E.Remark 8.
By Lemma7, we can assume that there exists a u such that
(24)unuinE,un⟶uinLlocsℝN,∀s∈2,2∗∗,un⟶ua.e.onℝN.
Then, by the similar method as Lemma 3.3 in [4], we can obtain limn⟶∞∫ℝNun2∗∗dx=∫ℝNu2∗∗dx.Lemma 9.
Iu satisfies PS condition.Proof.
Letun⊂E be PSc sequence, i.e., Iun⟶c and I′un⟶0; by Lemma 7, un is bounded in E. Hence, up to a subsequence, we may assume that there exists a u such that
(25)unuinE,un⟶uinLlocsℝN,∀s∈2,2∗∗,un⟶ua.e.onℝN.
Then, using the lower semicontinuity of the norm, Brezis-Lieb Lemma [28] and Remark 8, we have
(26)o1un=I′un,un=∫ℝNΔun2dx+∫ℝNa∇un2+Vxun2dx+b∫ℝN∇un2dx2−∫ℝNIα∗Funfunundx−λ∫ℝNun2∗∗dx=∫ℝNΔun−Δu2dx+∫ℝNa∇un−u2+Vxun−u2dx+∫ℝNΔu2dx+∫ℝNa∇u2+Vxu2dx+b∫ℝN∇un2dx2−∫ℝNIα∗Funfunundx−λ∫ℝNun2∗∗dx+o1≥C5un−u2+∫ℝNΔu2dx+∫ℝNa∇u2+Vxu2dx+b∫ℝN∇u2dx2−∫ℝNIα∗Fufuudx−λ∫ℝNu2∗∗dx+o1.
Thus, we can haveun−u2⟶0, which implies that un strongly converges to u in E. This completes the proof of Lemma 9.Remark 10.
Now, we recall the Nehari manifold(27)M≔u∈E\0:I′u,u=0.
Letm=infu∈MIu; then for any u∈M, we have
(28)Iu=Iu−14I′u,u≥C6un2≥0.
Hence,m is well defined. Moreover, by the similar argument as Chapter 4 [29], we have the following characterization:
(29)c=infγ∈Γmaxt∈0,1Iγt=m=infu∈MIu.


## 4. Ground State Solution for Problem (1)

In this section, we prove the main theorem.Proof of Theorem 11.
From Lemmas6 and 7, we know that there exists a bounded PSc sequence un, that is, Iun⟶c=m, I′un⟶0. Next, let δ≔limsupn⟶∞supy∈ℝN∫B1yun2dx. We claim δ>0. On the contrary, by similar argument as Lions’ concentration compactness principle, we can proof un⟶0 in LpℝN for 2<p<2∗∗. By Lemma 3(2), for any ε>0, there exists a constant Cε>0 such that
(30)limsupn⟶∞∫ℝNIα∗Funfunundx≤Climsupn⟶∞ε∫ℝNun2dx+∫ℝNun2N/N−2dx+Cε∫ℝNunpdxN+α/N≤CεC7+Cεlimsupn⟶∞∫ℝNunpdxN+α/N=CεC8N+α/N.
Note thatε is arbitrary; we get
(31)∫ℝNIα∗Funfunundx=o1.
Combining withI′un⟶0 and Remark 8, we can get
(32)o1=I′un,un=∫ℝNΔun2dx+a∫ℝN∇un2dx+b∫ℝN∇un2dx2+∫ℝNVxun2dx−∫ℝNIα∗Funfunundx−λ∫ℝNun2∗∗dx≥C9un2−∫ℝNIα∗Funfunundx−λ∫ℝNun2∗∗dx,which implies that
(33)C9un2≤∫ℝNIα∗Funfunundx+λ∫ℝNun2∗∗dx+o1.
Then, we haveun⟶0, which implies un⟶0 in E. We deduce that c=0, which contradicts to the fact that c>0. Hence, δ>0, and there exists yn⊂ℝN such that ∫B1ynunpdx≥δ/2>0. We set vnx=unx+yn; then, un=vn,∫B10vnpdx>δ/2 and Ivn⟶c=m,I′vn⟶0. Thus, there exists a v0≠0 such that
(34)vnv0inE,vn⟶v0inLlocsℝN,∀s∈2,2∗∗vn⟶v0a.e.onℝN.
Then, for anyφ∈C0∞ℝN, we have 0=I′vn,φ+o1=I′v0,φ, which means v0 is a solution of equation (1).
On the other hand, combining with the Fatou Lemma, we can obtain(35)m=Ivn−14I′vn,vn+o1=14∫ℝNΔvn2dx+14∫ℝNa∇vn2+Vxvn2dx+14∫ℝNIα∗Fvnfvnvn−2Fvndx+λ4−λ2∗∗∫ℝNvn2∗∗dx+o1≥14∫ℝNΔv02dx+14∫ℝNa∇v02+Vxv02dx+14∫ℝNIα∗Fv0fv0v0−2Fv0dx+λ4−λ2∗∗∫ℝNv02∗∗dx+o1=Iv0−14I′v0,v0+o1=Iv0+o1.
At the same time, we knowm=c≤Iv0 by the definition of m. Then, we can deduce that v0 is a ground state solution of equation (1). Thus, we complete the proof of Theorem 11.



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