# On Fourth-Order Elliptic Equations of Kirchhoff Type with Dependence on the Gradient and the Laplacian

Authors: Yuanfang Ru; Fanglei Wang; Yunhai Wang; Tianqing An
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We consider a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and LaplacianΔ2u-a+b∫Ω∇u2dxΔu=fx,u,∇u,Δu, in Ω, u=0, Δu=0, on ∂Ω, where a, b are positive constants. We will show that there exists b⁎>0 such that the problem has a nontrivial solution for 0<b<b⁎ through an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al., 2004.


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## Body

## 1. Introduction

This paper concerns with the existence of solutions of the fourth-order Kirchhoff type elliptic equation as follows:(1)Δ2u-a+b∫Ω∇u2dxΔu=fx,u,∇u,Δu,inΩ,u=0,Δu=0,on∂Ω,where Ω is a bounded and smooth domain in RN (N≥5), a, b are positive constants, and f:Ω×R×RN×R→R is locally Lipschitz continuous.The fourth-order elliptic equation(2)Δ2u-aΔu=fx,u,inΩ,u=0,Δu=0,on∂Ωarises in the study of traveling waves in suspension bridges, which has been extensively investigated in recent years, such as [1–6]. To our attention, some authors paid more attention to a more general biharmonic elliptic problem (3)Δ2u+qΔu+αxu=fx,u,Δu,∇u,inΩ,ux=0,Δux=0,on∂Ω.For this problem, due to the presence of Δu and ∇u in f, it is not variational. To overcome this difficulty, in [5], Wang deals with this problem via the upper and lower solutions and monotone iterative methods; in [7], the authors apply a technique developed by De Figueiredo et al. [8, 9] in studying a second-order elliptic problem involving the gradient, which “freezes" the gradient, and use truncation on the nonlinearity f. Thus the new problem becomes variational and an iterative scheme of the mountain pass “approximated” solutions is built.In addition, the nonlocal fourth-order equation(4)Δ2u-M∫Ω∇u2dxΔu=fx,u,inΩ,u=0,Δu=0,on∂Ωhas also been studied by many authors. We refer the readers to [10–20]. Particularly, Wang et al. studied the following fourth-order equation of Kirchhoff type equation (5)Δ2u-λa+b∫Ω∇u2dxΔu=fx,u,inΩ,u=0,Δu=0,on∂Ω,where λ is a positive parameter. The authors showed that there exists λ∗ such that the fourth-order elliptic equation has a nontrivial solution for 0<λ<λ∗ by using the mountain pass iterative techniques and the truncation method.Motivated by these works, to study problem (1), we combine the famous mountain pass lemma with a technique developed by De Figueiredo et al. [8], which “freezes” the gradient and the Laplacian variable and use truncation on the nonlinearity of f. For convenience, we recall a definition and restate the mountain pass theorem.Definition 1.
LetX be a real Banach space and I:X→R a C1-functional. A sequence {un} in E is a (PS)-sequence for I if I(un)→C for some constant C≥0 as n→∞, while 〈I′(un),un〉→0 as n→∞. We say that the functional I satisfies the (PS)-condition if any (PS)-sequence for I has a convergent subsequence.Theorem A (mountain pass lemma).
LetE be a real Banach space; I∈C1(E,R) satisfying (PS)-condition. Suppose the following: (1)
There existρ>0, α>0 such that (6)I∂Bρ≥I0+α,whereBρ=u∈E∣u≤ρ.(2)
There ise∈E and e>ρ such that (7)Ie≤I0.Then I(u) has a critical value c which can be characterized as (8)C=infγ∈Γmaxu∈γ0,1Iu,where Γ=γ∈C0,1,E∣γ0=0,γ1=e.


## 2. The Main Result

Theorem 2.
Assume that the functionf satisfies the following conditions:
(
f
1
)
f
:
Ω
×
R
×
R
N
×
R
→
R is locally Lipschitz continuous, and there exist d1>0, 1<p<N+4/(N-4)  which satisfy r≔r1+r2<2 such that (9)fx,t,ξ1,ξ2≤d11+tp1+ξ1r11+ξ2r2for all (x,t,ξ1,ξ2)∈Ω×RN+2.
(
f
2
)
l
i
m
t
→
0
f
(
x
,
t
,
ξ
1
,
ξ
2
)
/
t
=
0 uniformly with respect to x∈Ω, ξ1∈RN and ξ2∈R.
(
f
3
) There exist Θ>2 and t1>0 such that (10)0<ΘFx,t,ξ1,ξ2≤tfx,t,ξ1,ξ2,∀t≥t1,x∈Ω,ξ1,ξ2∈RN+1,where F(x,t,ξ1,ξ2)=∫0t‍f(x,s,ξ1,ξ2)ds.
(
f
4
) There exist positive constants ρi>0 (i=1,2,3) depending on a, b, Θ, and d1 such that Lρi (i=1,2,3) satisfy (11)τ12Lρ1+τ1τ2Lρ2+τ1τ3Lρ3<min1,a,where (12)Lρ1=supfx,t′,ξ1,ξ2-fx,t′′,ξ1,ξ2t′-t′′:x∈Ω,t′,t′′≤ρ1,ξ1≤ρ2,ξ2≤ρ3,Lρ2=supfx,t,ξ1′,ξ2-fx,t,ξ1′′,ξ2ξ1′-ξ1′′:x∈Ω,t≤ρ1,ξ1′,ξ1′′≤ρ2,ξ2≤ρ3,Lρ3=supfx,t,ξ1,ξ2′-fx,t,ξ1,ξ2′′ξ2′-ξ2′′:x∈Ω,t≤ρ1,ξ1≤ρ2,ξ2′,ξ2′′≤ρ3,and τi (i=1,2,3) are the optimal constants of the following inequalities:(13)∫Ωu2dx1/2≤τ1u,∫Ω∇u2dx1/2≤τ2u,∫ΩΔu2dx1/2≤τ3u,where · is the norm of the Hilbert space X=H2Ω∩H01(Ω) defined by (14)u2=∫ΩΔu2dx+∫Ω∇u2dx.
Then there existsb∗>0 such that (1) has at least a nontrivial solution for 0<b<b∗.For eachω∈X and R>0, we study the following “truncate" and “freezed” problem(15)Δ2uωR-ΦR∇ωΔuωR=fRx,uωR,∇ω,Δω,inΩ,uωR=0,ΔuωR=0,on∂Ω,where (16)fRx,t,∇ω,Δω=fx,t,∇ωφR∇ω,ΔωφRΔω,ΦR∇ω=a+b∫Ω∇ωφR∇ω2dx,φR∈C1(R,R) satisfies |φR|≤1, and (17)φRξ=1,ifξ≤R,0,ifξ≥R+1.The associated functional JωR:X→R is defined as (18)JωRuωR=12∫ΩΔuωR2dx+12ΦR∇ω∫Ω∇uωR2dx-∫ΩFRx,uωR,∇ω,Δωdx,where (19)FRx,t,∇ω,Δω=∫0tfRx,s,∇ω,Δωds.Lemma 3.
LetR>0 and ω∈X be fixed. Then (1)
there exist constantsρ>0,α>0 such that JωRv∂Bρ≥α with Bρ=v∈X:v≤ρ;(2)
for fixedv with v=1, JωR(tv)→-∞ as t→+∞.Proof.
On one hand, by(f2), for any ε>0, there exists a constant σ>0 such that, for |t|<σ, one has(20)FRx,t,∇ω,Δω≤12εt2.On the other hand, if |t|>σ, from (f1) it follows that there exists C1>0 such that (21)FRx,t,∇ω,Δω≤C1tp+1R+2r.Then, from (20), (21), and the Sobolev inequality, we have (22)JωRv=12∫ΩΔv2dx+12ΦR∇ω∫Ω∇v2dx-∫ΩFRx,v∇ω,Δωdx≥12∫ΩΔv2dx+12ΦR∇ω∫Ω∇v2dx-ϵ2∫Ωv2dx-C1R+2r∫Ωvp+1dx≥min⁡1,a2-ϵC2v2-CR+2rvp+1for some positive constant C. Therefore, for sufficiently small ϵ>0, we can choose ρ>0 and α>0 such that the first result of Lemma 3 holds.
Now, we show that(f3) implies that there exist a2,a3>0 such that(23)FRx,t,ξ1,ξ2≥a2tθ-a3,∀x∈Ω,t∈R,ξ1∈RN,ξ2∈R.In fact, from (f3), we have f(x,t,ξ1,ξ2)/F(x,t,ξ1,ξ2)≥θ/t, for any |t|≥t1. Being integral from t1 to t, we get (24)ln⁡Fx,t,ξ1,ξ2-ln⁡Fx,t1,ξ1,ξ2≥θln⁡t-ln⁡t1;namely, (25)Fx,t,ξ1,ξ2≥Fx,t1,ξ1,ξ2t1θtθ,∀t≥t1.Then (26)FRx,t,ξ1,ξ2=Fx,t,ξ1φRξ1,ξ2φRξ2≥tθt1θFx,t1,ξ1φRξ1,ξ2φRξ2.Let (27)a2=1t1θminx∈Ω,ξ1∈RN,ξ2∈RFx,t1,ξ1φRξ1,ξ2φRξ2=1t1θminx∈Ω,ξ1≤R+1,ξ2≤R+1Fx,t1,ξ1,ξ2>0,a3=maxx∈Ω,t≤t1,ξ1≤R+1,ξ2≤R+1Fx,t1,ξ1,ξ2>0,and then inequality (23) holds.
Taking an arbitraryv∈X with v=1, then from (23), we get (28)JωRtv=12∫ΩΔtv2dx+12ΦR∇ω∫Ω∇tv2dx-∫ΩFRx,tv,∇ω,Δωdx≤12∫ΩΔtv2dx+12ΦR∇v∫Ω∇tv2dx-a2tΘ∫ΩvΘdx+a3Ω≤max1,ΦR∇ω2t2v2-a2tΘ∫ΩvΘdx+a3Ω⟶-∞,ast⟶+∞,which implies that the second result of Lemma 3 holds.Lemma 4.
LetR>0 and ω∈X be fixed. Then the functional JωR(·) satisfies the (PS)-condition.Proof.
Let{un}⊂X be a (PS)-sequence; namely,(29)JωRun⟶C,Jω′Run,un⟶0asn⟶∞.From the standard processes, we only need to prove that {un} is bounded in X. On a contradiction, suppose that un→+∞; then, from (f3), we obtain (30)JωRun-1ΘJω′Run,un=12∫ΩΔun2dx+12ΦR∇ω∫Ω∇un2dx-∫ΩFRx,un,∇ω,Δωdx-1Θ∫ΩΔun2dx-1ΘΦR∇ω∫Ω∇un2dx+1Θ∫ΩfRx,un,∇ω,Δωundx=12-1Θ∫ΩΔun2dx+ΦR∇ω∫Ω∇un2dx+∫Ω1ΘfRx,un,∇ω,Δωun-FRx,un,∇ω,Δωdx≥12-1Θmin1,au2+∫Ω1ΘfRx,un,∇ω,Δωun-FRx,un,∇ω,Δωdx≥12-1Θmin1,au2.On the other hand, from (29) we know that (31)JωRun-1ΘJω′Run,un≤C+Cun.Then, from the above inequalities, we get (32)12-1θmin⁡1,aun2≤C+Cun,which contradicts with un→+∞. Therefore the sequence {un} is bounded in X.Lemma 5.
For anyR>0 and ω∈X, problem (15) has a nontrivial weak solution.Proof.
By Theorem A, Lemmas3, and 4, the result holds.Lemma 6.
LetR>0 be fixed. Then there exist positive constants ν1 and ν2≔ν2(R), independent of ω, such that (33)ν1≤uωR≤ν2for every solution uωR obtained in Lemma 5.Proof.
Firstly, sinceJωR(uωR)≤maxt≥0JωR(tv), from (23) it follows that (34)JωRtv≤max1,ΦR2t2v2-a2tΘ∫ΩvΘdx+a3Ω≤max1,a+bR+12Ω2t2-a2tΘ∫ΩvΘdx+a3Ω.As Θ>2, we can get a C>0 such that JωR(uωR)≤C; that is, (35)12∫ΩΔuωR2dx+12ΦR∇ω∫Ω∇uωR2dx≤C+∫ΩFRx,uωR,∇ω,Δωdx.
DefineD=x∈Ω:uωR>t1, where t1 is defined in (f3). Then we get (36)12∫ΩΔuωR2dx+12ΦR∇ω∫Ω∇uωR2dx≤C+∫ΩFRx,uωR,∇ω,Δωdx=C+∫Ω∖DFRx,uωR,∇ω,Δωdx+∫DFRx,uωR,∇ω,Δωdx≤C+a1t1+t1p+1p+1R+2rΩ∖D+1Θ∫ΩΔuωR2dx+1ΘΦR∇ω∫Ω∇uωR2dx.Furthermore, we have (37)12-1Θmin1,auωR2≤C+a1t1+t1p+1p+1R+2rΩ∖D≤CR+2r,where C is independence of b, R>0, and ω∈H. Therefore, uωR≤ν2, for some ν2≔ν2(R)>0.
Secondly, from(f1) and (f2), given ϵ>0, there exists Cϵ>0 such that (38)fRx,uωR,∇ω,Δω≤ϵuωR+CϵR+2ruωRp.Since 〈Jω′R(uωR),uωR〉=0, it is easy to obtain that (39)min⁡1,auωR2≤C2ϵuωR2+C3CϵuωRp+1for some constants C2,C3≥0. Therefore, there exists ν1>0 such that uωR≥ν1.Lemma 7 (see [7]).
LetR be fixed, and choose ω∈C4,α(Ω¯) for α∈(0,1). If uωR∈X is a weak solution of problem (15), then uωR∈C4,β(Ω¯) for some β∈(0,1), and Δ(uωR)(x)=0 if x∈∂Ω.Lemma 8.
There exist three constantsηi>0 (i=1,2,3), independent of b, ω, and R, such that (40)uωRC0≤η1R+2r/2,∇uωRC0≤η2R+2r/2,ΔuωRC0≤η3R+2r/2.In addition, there exists R¯ such that (41)ηiR¯+2r/2≤R¯i=1,2,3.Proof.
From (37) and the proof of Lemma 6, there exists C>0, independent of R>0 and ω∈X, such that (42)uωR≤CR+2r/2.Then by Lemma 7 and the Sobolev embedding theorem, the inequalities in the lemma are as follows. In addition, since r/2<1 and lims→∞(s+2)r/2/s=0, there exists a sufficiently large R¯>0 such that θi(R¯+2)r/2≤R¯.Now letunR (n=1,2,…) be the weak solution of the following problem:(43)Δ2unR-ΦR∇un-1RΔunR=fRx,unR,∇un-1R,Δun-1R,inΩunR=0,ΔunR=0,on∂Ωwith ω=un-1R, where un-1R was found in Lemma 5 and R=R¯ obtained in Lemma 8. From Lemmas 6–8, we have unR∈C4(Ω¯) satisfying unR≥ν1 and (44)unRC0,∇unRC0,ΔunRC0≤R¯.Thus (45)ΦR∇un-1R=a+b∫Ω∇un-1RφR∇un-1R2dx=a+b∫Ω∇un-1R2dx,fRx,unR,∇un-1R,Δun-1R=fx,unR,∇un-1R,Δun-1R.Lemma 9.
Assume that(f4) holds. Let (46)ρ1=infσ1:unRC0≤σ1,∀n∈N+>0,ρ2=infσ2:∇unRC0≤σ2,∀n∈N+>0,ρ3=infσ3:ΔunRC0≤σ3,∀n∈N+>0.Then {unR} strongly converges in X.Proof.
Letun+1R and unR be the weak solutions of (43) with ω=unR and ω=un-1R, respectively. Then, multiplying (43) by (un+1R-unR), we obtain (47)min1,aun+1R-unR2≤∫Ωfx,un+1R,∇unR,ΔunR-fx,unR,∇un-1R,Δun-1Run+1R-unRdx+b∫Ω∇unR2dx∫ΩΔun+1Run+1R-unRdx-b∫Ω∇un-1R2dx∫ΩΔunRun+1R-unRdx.Furthermore, by (f1), (f4), and the Ho¨lder inequality, we have(48)∫Ωfx,un+1R,∇unR,ΔunR-fx,unR,∇un-1R,Δun-1Run+1R-unRdx=∫Ωfx,un+1R,∇unR,ΔunR-fx,unR,∇unR,ΔunRun+1R-unRdx+∫Ωfx,unR,∇unR,ΔunR-fx,unR,∇un-1R,ΔunRun+1R-unRdx+∫Ωfx,unR,∇un-1R,ΔunR-fx,unR,∇un-1R,Δun-1Run+1R-unRdx≤τ12Lρ1un+1R-unR2+τ1τ2Lρ2unR-un-1R·un+1R-unR+τ1τ3Lρ3unR-un-1R·un+1R-unR,b∫Ω∇unR2dx∫ΩΔun+1Run+1R-unRdx-b∫Ω∇un-1R2dx∫ΩΔunRun+1R-unRdx≤bρ3Ω1/2un+1R-unRL2·∫Ω∇unR+∇un-1R·∇unR-un-1Rdx≤2bρ2ρ3Ωτ1τ2un+1R-unR·unR-un-1R.Hence, by (47) and (48), we get (49)min⁡1,aun+1R-unR2≤τ12Lρ1un+1R-unR2+τ1τ2Lρ2unR-un-1R·un+1R-unR+τ1τ3Lρ3unR-un-1R·un+1R-unR+2bρ2ρ3τ1τ2Ωun+1R-unR·unR-un-1R;namely, (50)un+1R-unR≤τ1τ2Lρ2+τ1τ3Lρ3+2bρ2ρ3τ1τ2Ωmin⁡1,a-Lρ1τ12unR-un-1R.Now, choosing b∗=min{1,a}-Lρ1τ12-Lρ2τ1τ2-Lρ3τ1τ3/2ρ2ρ3τ1τ2|Ω|, then (51)τ1τ2Lρ2+τ1τ3Lρ3+2bρ2ρ3τ1τ2Ωmin1,a-Lρ1τ12<1,for0<b<b∗.
Therefore,{unR} converges strongly in X.Proof of Theorem2.
Firstly, from Lemma6, we get unR≥ν1>0, and unRC0, ∇unRC0, ΔunRC0 are uniformly bounded. Secondly, set vn=ΔunR; then (52)Δvn=hx=fx,unR,∇un-1R,Δun-1R+a+b∫Ω∇un-1R2dxΔunR.Since hCβ≤C, for some positive constant C, by the Schauder theorem, there exists a constant C0>0 such that vnC2,β≤C0; that is, unRC4,β≤C0. Furthermore, by the Arzela-Ascoli theorem and Lemma 9, the sequence {unR} satisfies (53)∂j∂xijunRx⟶∂j∂xijuRx,asn⟶∞uniformly in Ω¯ for j=0,1,2,3,4 and i=1,…,N. Finally, passing to the limit in (43), we obtain that uR(x) is a classical solution of (1).Example 10.
Consider the following problem:(54)Δ2u-a+b∫Ω∇u2dxΔu=fx,u,∇u,Δu,inΩ,u=0,Δu=0,on∂Ω,where f(x,t,ξ1,ξ2)=α(x)|t|t(1+|ξ1|)1/2(1+|ξ2|)3/4+β(x)t3 and α(x) and β(x) are positive and continuous functions. It is easy to verify that f(x,t,ξ1,ξ2) satisfies all the conditions of (f1)–(f4).


## 3. Conclusion

The paper considers a class of fourth-order elliptic equations of Kirchhoff type with dependence on the gradient and Laplacian. The existence of a nontrivial solution of (1) is established when we choose appropriate b∗ such that 0<b<b∗. The paper generalized the conclusions in [7, 14] and weakened the condition in [7]. In the following research work, we will also consider problem (1), but we just truncate the right side of the equation, and the left of the equation remains the same.



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