# On apx-Biharmonic Kirchhoff Problem with Navier Boundary Conditions

Authors: Hafid Lebrimchi; Mohamed Talbi; Mohammed Massar; Najib Tsouli
Journal: Abstract and Applied Analysis
 (2021)

---

## Abstract

In this article, we study the existence of solutions for nonlocalpx-biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.


---

## Body

## 1. Introduction

We consider the problem with Navier boundary conditions.(1)PλM∫ΩΔuxpxpxdxΔp2u=λmxuxqx−2uxinΩ,u=Δu=0on∂Ωwhere Ω is a bounded domain in ℝNN≥3 with smooth boundary ∂Ω, λ>0, and Δpx2 is the px-biharmonic operator defined by Δpx2u=ΔΔupx−2Δu.p,q∈C+Ω¯=h∈CΩ¯,h−>1, where h−=minx∈Ω¯hx. We denoted by p+=maxx∈Ω¯px.We assume that the weightm and the Kirchhoff function M satisfy the following conditions:(m). m∈LβxΩ,mx>0a.einΩ, with β∈C+Ω¯ such that
(2)qx<βx−1βxp2∗x,where
(3)p2∗x=NpxN−2pxifpx<N2,+∞ifpx≥N2.(M1). M:ℝ+⟶ℝ+ is a continuous function verifying
(4)m1tα−1≤Mt≤m2tα−1∀t>0,where m1,m2,α are real numbers such that 0<m1≤m2 and α>1.Example 1.
A typical example of (Pλ) satisfying the conditions (m)-(M1) is given by
(5)m1+m22∫ΩΔupxpxα−1Δpx2u=λx−sxuqx−2uinΩ,u=Δu=0on∂Ω,where s∈CΩ¯ such that 0<s−≤s+<N/2, q−>1, and
(6)qx<N−2sxNp2∗x,∀x∈Ω¯.Putβx=N/2sx. Then, we have qβ′x<p2∗x, where β′ is the conjugate of β.Furthermore,x−sx∈LβxΩ. Indeed,
(7)∫Ωx−sxβxdx=∫Ωx−N/2dx<∞becauseN2<N.Problem (Pλ) is related to the stationary problem of a model introduced by Kirchhoff [1]. To be more precise, Kirchhoff established a model given by the equation
(8)ρ∂2u∂t2−ρ0h+E2L∫0L∂u∂x2dx∂2u∂x2=0,where ρ,ρ0,h,E,L are constants, which extends the classical D’Alambert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. In two dimensions, Kirchhoff equations model the oscillations of thin plates and the most usual plate operator is the biharmonic operator Δ2=Δ·Δ [2].Fourth-order equations have various applications in many domains like microelectromechanical systems, surface diffusion on solids, thin film theory, and interface dynamics; for recent contributions concerning this type of equations, we refer to [3–10]. In recent years, the study of variational problems with variable exponent has received considerable attention; these problems arises from nonlinear electrorheological fluids, elastic mechanics, image restoration, and mathematical biology (see [11–15]). The interplay between the fourth-order equation and the variable exponent equation goes to the px-biharmonic problems. The px-biharmonic operator possesses more complicated structure than the p-biharmonic operator Δp2, where p>1 is a real constant; for example, it is not homogeneous. A study on px-biharmonic problems with Navier boundary condition was treated by many authors (see, for example, [16–20]). The authors in [21, 22] proved the existence and multiplicity of weak solutions for the px-biharmonic problems under Navier boundary conditions. Their approach is of variational nature and does not require any symmetry of the nonlinearities. In [23], a similar problem to ours has been investigated in the case of px-Laplacian and with weight 1. In [17], the authors examined a px-biharmonic Kirchhoff-type problem but in the case where the weight is bounded and without parameter λ. Motivated by the above papers and the results in [24, 25], we determine by different variational methods intervals of parameters for which this problem admits at least one nontrivial solution.


## 2. Preliminaries

We state some definitions and basic properties of variable exponent Lebesgue-Sobolev spaces. We refer the reader to [26–29] for details.For anypx∈C+Ω¯, we define the variable exponent Lebesgue space by
(9)LpxΩ=u/u:Ω⟶ℝismeasurable,∫Ωupxdx<∞,with the norm
(10)upx=infλ>0:∫Ωuxλpxdx≤1.Proposition 1 (see [29]).
The spaceLpxΩ,.px is separable, uniformly convex, and reflexive and its conjugate space is Lp0xΩ, where 1/px+1/p0x=1 for all x∈Ω¯. For u∈Lpx and v∈Lp0x, we have
(11)∫Ωuvdx≤1p−+1p0−upxvp0x.Proposition 2 (see [26]).
Letρu=∫Ωupxdx be the modular of the LpxΩ space. For u,un∈LpxΩ,n=1,2,⋯, we have
(12)upx<=;>1⇒ρu<=;>1,upx>1⇒upxp−≤ρu≤upxp+,upx<1⇒upxp+≤ρu≤upxp−,limn⟶+∞unpx=0⇔limn⟶+∞ρun=0,limn⟶+∞unpx=+∞⇔limn⟶+∞ρun=+∞.Fork≥1, we define the variable exponent Sobolev space
(13)Wk,pxΩ=u∈LpxΩ:Dαu∈LpxΩ,α≤k,where Dαu=∂αu/∂α1x1⋯∂αNxN, where α=α1,⋯,αN is a multi-index and α=∑i=1Nαi. The space Wk,pxΩ equipped with the norm
(14)uk,px=∑α≤kDαupxbecomes a separable, reflexive, and uniformly convex Banach space.Proposition 3 ([26]).
Forp,r∈C+Ω¯ such that rx≤pk∗x for all x∈Ω, there is a continuous embedding
(15)Wk,pxΩ↪LrxΩ.
If we replace≤ with <, the embedding is compact.We denote(16)X=W2,pxΩ∩W01,pxΩ,where W0k,pxΩ is the closure of C0∞Ω in Wk,pxΩ.Foru∈X, we define
(17)u=infλ>0:∫ΩΔuxλpxdx≤1.X endowed with the above norm is a separable and reflexive Banach space.Remark 4.
From [30], the norms u2,px and u are equivalent in X.Letd:Ω⟶ℝ be a measurable real function dx>0 a.e. x∈Ω. We define the weighted variable exponent Lebesgue space
(18)LdxpxΩ=u/u:Ω⟶ℝismeasurable,∫Ωdxupxdx<∞.LdxpxΩ equipped with the norm udx,px=infλ>0:∫Ωdxux/λdx≤1 is a Banach space which has similar properties with the usual variable exponent Lebesgue spaces. The modular of this space is ρdx,px:LdxpxΩ⟶ℝ defined by
(19)ρdx,px=∫Ωdxuxpxdx.Proposition 5 ([31]).
Foru,un∈LdxpxΩ,n=1,2,⋯, we have
(1)
udx,px<=;>1⇒ρdx,pxu<=;>1(2)
udx,px>1⇒udx,pxp−≤ρdx,pxu≤udx,pxp+(3)
udx,px<1⇒udx,pxp+≤ρdx,pxu≤udx,pxp−(4)
limn⟶+∞undx,px=0⇔limn⟶+∞ρdx,pxun=0(5)
limn⟶+∞undx,px=∞⇔limn⟶∞ρdx,pxun=∞In the same way as in [32], we show the following proposition.Proposition 6.
Assume that the boundary ofΩ possesses the cone property and p∈CΩ¯. Suppose that c∈LβxΩ,dx>0fora.ex∈Ω,
β∈C+Ω¯. If q∈C+Ω¯ and
(20)qx<βx−1βxp2∗x,∀x∈Ω¯,then there is a compact embedding X↪LdxqxΩ.DenoteI:X⟶X∗ the operator defined by Iu,v=∫ΩΔupx−2ΔuΔvdx for all u,v∈X.Proposition 7 ([20]).
The operatorI satisfies the following assertions:
(i)
I is continuous, bounded, and strictly monotone(ii)
I is a mapping of S+ type, namely, unu and limsupn⟶+∞Iun,un−u≤0, which imply un⟶u(iii)
I is a homeomorphism


## 3. The Main Result

We say thatu∈X is a weak solution of (Pλ) if
(21)M∫ΩΔupxpxdx∫ΩΔup−2ΔuΔvdx=λ∫Ωmxuqx−2uvdxfor every v∈X.For anyλ>0, the energy functional corresponding to problem (Pλ) is defined as Jλ:X⟶ℝ,(22)Jλu=M^∫ΩΔupxpxdx−λ∫Ωmxuqxqxdx,where
(23)M^t=∫0tMsds.Standard arguments imply thatJλ∈C1X,ℝ and
(24)Jλ′u,v=M∫ΩΔupxpxdx∫ΩΔup−2ΔuΔvdx−λ∫Ωmxuqx−2uvdx,for any u,v∈X. Hence, we can infer that critical points of functional Jλ are the weak solutions for problem (Pλ).In the sequel, we use fountain theorem to study the existence of multiple solutions of (Pλ). We obtain the following result.Theorem 8.
Assume that (M1), (m), and αp+α<q−; then, for every λ>0, problem (Pλ) has a sequence of weak solutions ±un such that Jλ±un⟶+∞ as n⟶+∞.Before proving Theorem8, we give some preliminary results.SinceX is a reflexive and separable Banach space, then X∗ is too. There exist (see [33]) en:n=1,2,⋯⊂X and en∗:n=1,2,⋯⊂X∗ such that
(25)X=spanej:j=1,2,⋯¯,X∗=spanej∗:j=1,2,⋯¯,ei,ej∗=1ifi=j,0ifi≠j,where .,. denote the duality product between X and X∗. We define
(26)Xj=spanej,Yn=⊕j=1nXj,Zn=⊕j=n∞Xj¯.Theorem 9 (fountain theorem, see [34]).
X is a Banach space; J∈C1X,ℝ is an even functional. If for every n=1,2,⋯, there exist ρn>rn>0 such that
(a)
infJu:u∈Zn,u=rn⟶+∞asn⟶+∞(b)
maxJu:u∈Yn,u=ρn<0(c)
J satisfies the PS condition for every c>0
Then,J has a sequence of critical values tending to +∞.Lemma 10.
Ifθn=sup∫Ωmxuqxdx:u=1,u∈Zn, then under condition (m),limn⟶+∞θn=0Proof.
For everyn≥1, there exist un∈Zn,un=1 such that
(27)θn−∫Ωmxunqxdx<1n.
There exists a subsequence ofun such that unu. For every j≥1 and for every n>j, we obtain ej∗un=0 and we conclude that ej∗u=0∀j>1; hence, u=0. On the other hand, by condition (m) and Proposition 5, there is a compact embedding X↪LmxqxΩ; hence, un⟶u=0 in LmxqxΩ. By relation (27) we conclude that limn⟶+∞θn=0.☐Proof of Theorem 8.
Jλ∈C1X,ℝ, and Jλ is even. According to Lemmas 17 and 18, Jλ satisfies PSc condition for every c>0. We will prove that if n is large enough, then there exist ρn>rn>0 such that a and b hold.
(a) By condition (M1), we have for any u∈Zn such that u>1(28)Jλu=M^∫ΩΔupxpxdx−λ∫Ωmxuqxqxdx≥m1αp+αup−α−λq−∫Ωmx∣u∣uqxdxuq+≥m1αp+αup−α−λq−θnuq+.
Forn large enough, choose u such that
(29)λq−θnuq+=m1q−uαp−,and we take
(30)rn=u=m1λθn1/q+−αp−.
Then, we have(31)Jλu≥m11αp+α−1q−rnαp−.
By Lemma10 and the fact that αp+α<q−, the assertion a is verified.
(b) For any ω∈Yn with ω=1 and t>1, using M1, we have
(32)Jλtω=M^∫ΩtΔωpxpxdx−λ∫Ωmxtωqxqxdx≤m2αp−αtαp+−λq+tq−∫Ωmxωqxdx.
Byq−>αp+ and dim Yn<∞, it is easy to see that Jλu⟶−∞ as u⟶+∞ for u∈Yn.Now, a nontrivial solution of (Pλ) is given by using the coercivity and the weakly lower semicontinuity of Jλ.Theorem 11.
If we assume that (M1), (m), and q+<αp− hold, then there exists λ∗>0 such that for any λ>λ∗, problem (Pλ) possesses a nontrivial weak solution.
We start with the following auxiliary resultLemma 12.
Assume (M1), (m), and q+<αp−. Then, the functional Jλ is coercive on X.Proof.
By Proposition 5 and the compact imbeddingX↪LmxpxΩ, there exists C0 such that
(33)∫Ωmxuqxdx≤umx,qxq−+umx,qxq+≤C0q−uq−+C0q+uq+.
Therefore, foru>1 and under condition (M1), we obtain
(34)Jλu≥m1αp+αuαp−−λq−C0q−uq−+C0q+uq+.
Sinceq+<αp−, we infer that Jλu⟶+∞ as u⟶+∞.☐Proof of Theorem 11.
Jλ is a coercive functional and weakly lower semicontinuous on X. Then, there exists vλ∈X a global minimizer of Jλ (cf. Theorem 1.2 [35]). Thus, vλ is a weak solution of (Pλ). Now, we prove that vλ is a nontrivial solution for λ large enough. Letting t0>1 be a constant and Ω1 be an open subset of Ω with Ω1>0, we assume that φ∈C0∞Ω¯ is such that φx=t0 for any x∈Ω¯1 and 0≤φx≤t0 in Ω\Ω1. There exists λ∗>0 such that ∀λ∈λ∗,+∞,
(35)Jλφ≤M^∫ΩΔφpxpxdx−λq+t0q−∫Ω1mxdx<0.
Hence, forλ>λ∗, vλ is a nontrivial weak solution of (Pλ).In the following theorem, we apply Ekeland variational principle [36] to get a nontrivial solution to problem (Pλ).Theorem 13.
Assume that the conditions (M1), (m), and q−<αp−<q+ are satisfied. Then, there exists λ∗∗>0 such that for any λ∈0,λ∗∗, problem (Pλ) has at last one nontrivial weak solution.We start with two auxiliary results.Lemma 14.
Assume (M1), (m), and q−<αp−. There exists λ∗∗>0 and two positive real numbers r,a such that for any λ∈0,λ∗∗,Jλu≥a>0 for any u∈X with u=r.Proof.
By using the condition (m) and Proposition 6, the embedding from X to LmxqxΩ is compact. Then, there exists C1>0 such that for all u∈X,
(36)uqx,mx≤C1u.☐Considerr∈0,1 such that r<1/C1. Then, we get
(37)uqx,mx<1,∀u∈Xwithu=r.Furthermore, Proposition 5 yields(38)∫Ωmxuqxdx≤uqx,mxq−,and we conclude that
(39)∫Ωmxuqxdx≤C1q−uq−.Forλ>0, using (M1), (39), and Proposition 2, we get
(40)Jλu≥m1αp+uαp+−λC1q−q−uq−=rq−m1αp+rαp+−q−−λC1q−q−.By the above inequality, we remark that if we define(41)λ∗∗=m12αp+rαp+−q−q−C1q−,then for any λ∈0,λ∗∗ and any u∈X with u=r, there exists a=m1/2αp+rαp+>0 such that Jλu≥a>0, which end the proof of Lemma 14.Lemma 15.
Assume (M1) and q−<αp−. There exists e∈X such that e≥0,e≠0, and Jλte<0 for t>0 small enough.Proof.
Sinceq−<αp−, there exists ε0>0 such that q−+ε0<αp−. On the other hand, we have q∈CΩ¯; then, there exists an open ball B⊂Ω such that ∣qx−q−∣≤ε0 for all x∈B. Thus, we conclude that qx≤q−+ε0<αp− for all x∈B. Let ϕ∈C0∞Ω be such B⊂ supp(ϕ), ϕx=1 for all x∈B¯ and 0≤ϕx≤1 for all x∈Ω. Then, using the above information and M1, for all 0<t<1, we obtain
(42)Jλtϕ≤m2αp−αtαp−∫ΩΔϕpxdxα−λq+∫Ωmxtϕqxdx≤m2αp−αtαp−∫ΩΔϕpxdxα−λq+∫Bmxtϕqxdx≤m2αp−αtαp−∫ΩΔϕpxdxα−λq+tq−+ε0∫Bmxϕqxdx<0,for all t<δ with
(43)0<δ<min1,λαp−αm2q+∫Bmxϕqxdx∫ΩΔϕpxdxα1/αp−−q−−ε0.☐Proof of Theorem 13.
Letλ∗∗>0 be defined as in (41) and λ∈0,λ∗∗. By Lemma 14, it follows that on the boundary of the ball centered at the origin and of radius r in X, denoted by Br0, we have
(44)inf∂Br0Jλ>0.
On the other hand, by Lemma15, there exists ϕ∈X such that Jλtϕ<0 for all t>0 small enough. Moreover, by relations (53), we have for any u∈Br0(45)Jλu≥m1αp+uαp+−λCq−q−uq−.
It follows that(46)−∞<c¯=infu∈Br0¯Jλu<0.
Choose0<ε<infu∈∂Br0Jλu−infu∈Br0Jλu. Applying the Ekeland variational principle [36] to the functional Jλ:Br0¯⟶ℝ, we find uε∈Br0¯ such that
(47)Jλuε<c¯+ε,Jλ′uε≤ε.
The fact thatJλuε≤c¯+ε≤infBr0Jλ+ε<inf∂Br0Jλ implies that uε∈Br0. We deduce that there exists a sequence un⊂Br0 such that
(48)limn⟶∞Jλun=c¯,(49)limn⟶∞Jλ′un=0.
It is clear thatun is bounded in X. Thus, there exists v1∈X and a subsequence still denoted by un such that unv1 in X. Moreover, by the Hölder inequality, we get
(50)∫Ωmxunqx−2unun−v1dx=∫Ωmx1/q0xunqx−2unmx1/qxun−v1dx≤unqx−1mx,q0xun−v1mx,qx,where q0x is the conjugate exponent of qx, i.e., 1/qx+1/q0x=1 for all x∈Ω¯.
By the compact embeddingX↪LmxqxΩ, un⟶v1 in LmxqxΩ. Using Proposition 5, we obtain ρmx,q0xunqx−1=ρmx,qxun⟶ρmx,qxv1=ρmx,q0xv1qx−1 and we deduce that
(51)limn⟶∞unqx−1mx,q0x=v1qx−1mx,q0x.
Hence, (50) and (51) imply that ∫Ωmxunqx−2unun−v1dx⟶0 as n⟶0.
Using (48), we infer that
(52)M∫ΩΔunpxpx∫ΩΔunpx−2ΔunΔun−Δv1dx⟶0.
From (M1) and assertion (ii) of Proposition 7, it follows that un⟶v1 in X. Thus, Jλv1=c¯<0 and Jλ′v1=0.If in addition we have the following condition on(M2). There exists 0<γ<1 such that M^t≥γtMt∀t>1.Then by help of the Mountain Pass theorem, we obtain the following.Theorem 16.
Assume that (M1), (M2), (m), and αp+<γq− hold. For every λ>0, problem (Pλ) has a nontrivial weak solution.To prove Theorem16, we need the two following lemmas.Lemma 17.
Forλ>0 and under conditions (M1), (m), and αp+<q−, there exist r>0 and a>0 such that Jλu>a for any u∈X with u=r.Proof.
Let0<r<1. Under conditions (M1), (m), and as in Lemma 14, we show the existence of a constant C2>0 such that for all u∈X with u=r, we have
(53)Jλu≥m1αp+uαp+−λC2q−q−uq−=m1αp+α−λC2q−q−uq−−αp+uαp+.
Sinceq−>p+α, then we can choose 0<r<min1,1/C2 such that
(54)a=m1αp+α−λC2q−q−rq−−αp+rαp+>0,and we have Jλu≥a>0 for every u∈X,u=r.☐Lemma 18.
Forλ>0 and under conditions (M1) and αp+<q−, there exists e∈X with e>r where r is given in Lemma 17, such that Jλe<0.Proof.
Letϕ∈C0+∞Ω such that ϕ≥0 and ϕ≢0 and t>1. By (M1), we have
(55)Jλtϕ≤m2αp−αtp+α∫ΩΔϕpxdxα−λq+tq−∫Ωmxϕqxdx.
Sinceαp+<q−, we obtain limt⟶+∞Jλtϕ=−∞. Then, for t>1 large enough, we can take e=tϕ such that e>r and Jλe<0.☐Proof of Theorem 16.
By Lemmas17 and 18 and the mountain pass theorem of Ambrosetti and Rabinowitz [37], we deduce the existence of a sequence un⊂X and positive real number c such that
(56)Jλun⟶c>0,Jλ′un⟶0asn⟶+∞.
Firstly, we prove thatun is bounded in X. Arguing by contradiction and passing to a subsequence, we have un⟶+∞ as n⟶∞. Considering un>1, for n large enough and using (M1), (M2), (48), and Proposition 2, we have
(57)1+c+un≥Jλun−1q−Jλ′un,un≥γM∫ΩΔunpxpxdx∫ΩΔunpxpxdx−λ∫Ωmxunqxqxdx−1q−M∫ΩΔunpxpxdx∫ΩΔunpxdx+λq−∫Ωmxunqxdx≥m1p+α−1γp+−1q−unαp−+λ∫Ω1q−−1qxmxunqxdx≥m1p+α−1γp+−1q−unαp−.
But, this cannot hold true sinceαp−>1 and γ/p+>1/q−. Hence, un is bounded in X. The fact that X is reflexive implies that there exists a subsequence, still denoted by un, and v2∈X such that unv1 in X. Actually, with similar arguments as those used in the proof of Theorem 13, we can show that un⟶v2 in X. Thus, Jλv2=c>0 and Jλ′v2=0.



---

*Source: 4514044-2021-11-30.xml*