# Multiple Solutions for a Nonlocal Elliptic Problem Involvingpx,qx-Biharmonic Operator

Authors: Qi Zhang; Qing Miao
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, using the variational principle, the existence and multiplicity of solutions forpx,qx-Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.


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## Body

## 1. Introduction

In this paper, we will discuss the nonlocal elliptic problem involvingpx,qx-biharmonic operator:(1)Δpx2ux−M1∫Ω∇uxpxpxdxΔpxux+ρ1xupx−2ux=λFux,u,v,inΩ,Δqx2vx−M2∫Ω∇vxqxqxdxΔqxvx+ρ2xvqx−2vx=λFvx,u,v,inΩ,u=Δu=v=Δv=0,on∂Ω,where Ω⊂ℝNN≥2 is a smooth bounded region. λ>0, Δpx2ux=ΔΔupx−2Δu. ρ1x,ρ2x∈L∞Ω, px,qx∈CΩ¯, N/2<p−≔essinfx∈Ω¯px≤p+≔esssupx∈Ω¯px<+∞, and N/2<q−≔essinfx∈Ω¯qx≤q+≔esssupx∈Ω¯qx<+∞. Furthermore, the function F:Ω¯×ℝ2⟶ℝ satisfies that F⋅,s,t is continuous in Ω¯, for all s,t∈ℝ2, and Fx,⋅,⋅ is C1 in ℝ2, for everyx∈Ω, and sups≤τ,t≤τFu⋅,s,t+Fv⋅,s,t∈L1Ω, for all τ>0, where Fu and Fv are partial derivatives of u and v, respectively. Besides, the functions Mi:0,+∞⟶ℝare continuous and satisfy m0≤Mit≤m1,i=1,2, for every t≥0.As we all know, the harmonic equation and biharmonic equation in partial differential equation are the most widely used equations in the field of theoretical research and engineering application. Among them, the focus and difficulty in science and engineering is to solve the boundary value problem of biharmonic equation. Therefore, many scholars have carried out a lot of in-depth research on biharmonic problems, see [1–6]. In particular, in [3], when the nonlinear functions F and G satisfy certain conditions, Li and Tang studied the p,q-biharmonic problem,(2)ΔΔup−2Δu=λFux,u,v+μGux,u,v,x∈Ω,ΔΔvq−2Δv=λFvx,u,v+μGvx,u,v,x∈Ω,u=Δu=v=Δv=0,x∈∂Ω,by applying the three critical point theorem by Ricceri [7]. Two years later, Masser et al. [8] supposed that the assumption in problem (2) μ was 0 and imposed appropriate conditions on F. Based on the critical point theorem introduced by Bonanno and Molica Bisci, infinitely many solutions were obtained. It well known that the variable exponent case possess more complicated properties than the constant exponent case, and some methods used in the p,q-biharmonic case cannot be applied to the px,qx-biharmonic case. Therefore, Allaoui et al. [9] have made a great contribution to such problems, and they continued to extend p,q-biharmonic operator in [8] to px,qx-biharmonic case, on the basis of Ricceri’s variational principle [10] and the basic theory of Sobolev space, and the following system is solved:(3)ΔΔupx−2Δu=λFux,u,v,x∈Ω,ΔΔvqx−2Δv=λFvx,u,v,x∈Ω,u=Δu=v=Δv=0,x∈∂Ω.With the deepening of the investigation, scholars are increasingly paying attention to the solution of nonlocal elliptic equation, see [11–15], for details.In [16], Ferrara et al. researched the problem:(4)ΔΔup−2Δu−M∫Ω∇updxp−1Δpu+ρup−2u=λfx,u,x∈Ω,u=Δu=0,x∈∂Ω.If the nonlinear term satisfies the certain conditions, then we obtain that there are at least two nontrivial solutions to problem (4) by using the variational method and mountain pass lemma. In the same year, Hssini et al. [17] studied the same problem as in [16] according to Ricceri’s variational principle and obtained multiple solutions of problem (4). In addition, in [18], Xiao and Miao extended problem (4) and continued to study the multiplicity of solutions of p1,p2,…,pn-Kirchhoff type problems. In [19], the author extended the conclusion of the problem (4) to the px-biharmonic operator.So far, there are few results on the existence and multiplicity of solutions forpx,qx-Kirchhoff type problems under Navier boundary condition. Therefore, inspired by the above research, in the present paper, our target is to show the existence and multiplicity of solutions of problem (1), according to the variational principle proposed by Ricceri and the basic results of Lebesgue or Sobolev spaces with a variable exponent.This article consists of three sections. In Section2, some basic conclusions and certain essential theorems or propositions of generalized Lebesuge–Sobolev spaces are obtained, which provides the significant framework for the research of variational problems and nonlocal elliptic problems with px,qx-biharmonic operator. In Section 3, we have the main results of this article and gain the corresponding proof mainly according to the variational principle by Ricceri. Besides, at the end of Section 3, we give an example involving px,qx-Kirchhoff type equations with Ω = [(−1, 1)]2, which is actually an application of Theorem 2, that is to say, this example explains our main results very well.


## 2. Preliminaries

Define the space as follows:(5)LpxΩ=u:Ω⟶ℝmeasurable,∫Ωuxpxdx<∞,which has the norm(6)upx=infμ>0:∫Ωuxμpxdx≤1.The Sobolev space with variable exponents is defined as(7)Wk,pxΩ=u∈LpxΩ:Dσu∈LpxΩ,σ≤k,k∈Z+,which has the norm(8)uk,px=∑σ≤kDσupx,where σ=σ1,σ2,…,σn is a multi-index; moreover, σ=∑j=1nσj was established.The closure ofC0∞Ω in Wk,pxΩ is the W0k,pxΩ. Besides, from [20], we know that LpxΩ and Wk,pxΩ are Banach space and satisfy separability, uniform convexity, and reflexivity.Proposition 1 (see [20]).
Assume1/p1x+1/p2x=1; then, Lp1xΩ and Lp2xΩ are conjugate spaces and satisfy the Holder inequality:(9)∫Ωuvdx≤1p1−+1p2−up1xvp2x,where u∈Lp1xΩ and v∈Lp2xΩ.DenoteX=W01,pxΩ∩W2,pxΩ×W01,qxΩ∩W2,qxΩ, which is a separable and reflexive Banach space endowed with the norm:(10)u,v=upx+vqx,where(11)upx=infω>0:∫ΩΔuxωpx+∇uxωpx+uxωpxdx≤1,vqx=infω>0:∫ΩΔvxωqx+∇vxωqx+vxωqxdx≤1.From [21], we know that ⋅2,p⋅, ⋅p⋅, and Δ⋅p⋅ are equivalent norms of X.Proposition 2 (see [9]).
Sincep−,q−>N/2 and Ω⊂ℝN is a bounded region, hence, the space X to CΩ¯×CΩ¯ is a compact embedding, and there exists a positive constant K such that(12)K=maxsupu∈W2,pxΩ∩W01,pxΩ\0maxx∈Ω¯uxupx,supv∈W2,qxΩ∩W01,qxΩ\0maxx∈Ω¯vxvqx<+∞.Proposition 3 (see [9, 19]).
LetJu=∫ΩΔupxdx, for u∈W2,pxΩ∩W01,pxΩ; then, we deduce(h1)upx<1=1,>1⇔Ju<1=1,>1(h2)upx≥1⇒upxp−≤Ju≤upxp+(h3)upx≤1⇒upxp+≤Ju≤upxp−(h4)upx⟶0⇔Ju⟶0Definition 1.
If the following equality(13)∫ΩΔupx−2ΔuΔϕdx+∫ΩΔvqx−2ΔvΔψdx+M1∫Ω∇upxpxdx×∫Ω∇upx−2∇u∇ϕdx+M2∫Ω∇vqxqxdx×∫Ω∇vqx−2∇v∇ψdx+∫Ωρ1xupx−2uϕdx+∫Ωρ2xvqx−2vψdx−λ∫ΩFux,u,vϕdx−λ∫ΩFvx,u,vψdx=0,holds, for all ϕ,ψ∈X, then u,v∈X is a weak solution of problem (1).
Define the functionalIλ:X⟶ℝ:(14)Iλu,v=Φu,v−λΨu,v,Φu,v=∫ΩΔupxpxdx+M^1∫Ω∇upxpxdx+∫Ωρ1xupxpxdx+∫ΩΔvqxqxdx+M^2∫Ω∇vqxqxdx+∫Ωρ2xvqxqxdx,Ψu,v=∫ΩFx,u,vdx,where M^it=∫0tMisds,i=1,2.
The functionalΦ,Ψ:X⟶ℝ are well defined and Gateaux differentiable functions for∀u,v∈X; we have(15)Φ′u,v,ϕ,ψ=∫ΩΔupx−2ΔuΔϕdx+M1∫Ω∇upxpxdx×∫Ω∇upx−2∇u∇ϕdx+∫ΩΔvqx−2ΔvΔψdx+M2∫Ω∇vqxqxdx×∫Ω∇vqx−2∇v∇ψdx+∫Ωρ1xupx−2uϕdx+∫Ωρ2xvqx−2vψdx,Ψ′u,v,ϕ,ψ=∫ΩFux,u,vϕdx+∫ΩFvx,u,vψdx.
Hence, according to Definition1, we know that if u,v∈X is the weak solution of problem (1), it is equivalent to that u,v is the critical point of Iλ. In addition, since the space X to CΩ¯×CΩ¯ is a compact embedding, so Φ,Ψ:X⟶ℝ are sequentially weakly lower semicontinuous, and it is clear that Φ is coercive.Theorem 1 (see [10]).
LetX be a reflexive real Banach space. Φ,Ψ:X⟶ℝ are Gateaux differential functionals; Φ satisfies coercive and sequentially weakly lower semicontinuity and Ψ satisfies sequentially weakly upper semicontinuity. If r>infXΦ, denotes(16)φr≔infu∈Φ−1−∞,rsupv∈Φ−1−∞,rΨv−Ψur−Φu,γ≔limr⟶+∞infφr,δ≔limr⟶infXΦ+infφr,then one has(a)
For everyr>infXΦ and every λ∈0,1/φr, the functional Iλ=Φ−λΨ has a global minimum in Φ−1−∞,r, which is a critical point (local minimum) of Iλ in X.(b)
Ifγ<+∞, then ∀λ∈0,1/γ; the following alternative holds: either(b1)
Iλ has a global minimum or(b2)
Iλ has a series of critical points (local minimum) defined as un satisfying limn⟶+∞Φun=+∞.(c)
Ifδ<+∞, then ∀λ∈0,1/δ; the following alternative holds: either(c1)
Φ has a global minimum which is a local minimum of Iλ or(c2)
Iλ has a series of pairwise distinct critical points (local minimum) which weakly converges to global minimum of Φ.


## 3. Main Results and Proof

Theorem 2.
SupposeA1∀x,s,t∈Ω×0,+∞2, Fx,s,t≥0.A2∃x0∈Ω and two real numbers O2>O1>0 satisfy Rx0,O2⊆Ω. If we put(17)α≔limb⟶+∞inf∫Ωsups+t<bFx,s,tdxbminp−,q−,β≔lims,t⟶+∞sup∫Rx0,O1Fx,s,tdxsp+/p−+tq+/q−,one has(18)α<Lβ,where(19)L≔minL1,L2,L1=1Kp+/M−1/p−+q+/M−1/q−minp−,q−⋅1M+θ1+θ2+ρ1x∞θ3,L2=1Kp+/M−1/p−+q+/M−1/q−minp−,q−⋅1M+θ1+θ2+ρ2x∞θ3,θ1=2πN/2NΓN/22NO22−O12p+O2N−O1N,θ2=2πN/2ΓN/22O22−O12p+O2p++N−O1p++Np++N,θ3=2πN/2ΓN/2O1NN+1O22−O12p+∫O1O2O22−r2p+rN−1dr,M+=max1,m1,M−=min1,m0.
Then, for every(20)λ∈Λ≔1Kp+/M−1/p−+q+/M−1/q−minp−,q−1Lβ,1α.
Problem (1) has a sequence of solutions un,un such that limn⟶+∞Φun,un=+∞.Proof.
SinceΦ0,0=0 and Ψ0,0≥0, if r>0, then(21)φr=infw,z∈Φ−1−∞,rsupu,v∈Φ−1−∞,rΨu,v−Ψw,zr−Φw,z≤supΦu,v<r∫ΩFx,u,vdxr.
FromA2, we know that there exists a sequence ξn satisfying limn⟶+∞ξn=+∞ and(22)limn⟶+∞∫Ωsups+t<ξnFx,s,tdxξnminp−,q−=α<+∞.
Put(23)rn=ξnminp−,q−Kp+/M−1/p−+q+/M−1/q−minp−,q−,for all n∈ℕ. When u,v∈Φ−1−∞,rn, we have Φu,v<rn.
Since(24)Φu,v≥1p+∫ΩΔupxdx+m0∫Ω∇upxdx+∫Ωρ1xupxdx+1q+∫ΩΔvqxdx+m0∫Ω∇vqxdx+∫Ωρ2xvqxdx≥M−p+∫ΩΔupx+∇upx+ρ1xupxdx+M−q+∫ΩΔvqx+∇vqx+ρ2xvqxdx,according to Proposition 3, for u∈X, we infer that(25)ifupx≥1,upxp−≤∫ΩΔupx+∇upx+ρ1xupxdx≤upxp+,(26)ifupx≤1,upxp+≤∫ΩΔupx+∇upx+ρ1xupxdx≤upxp−.
Then,(27)M−p+minupxp−,upxp++M−q+minvqxq−,vqxq+<rn.
Thus,(28)M−p+minupxp−,upxp+<rn,M−q+minvqxq−,vqxq+<rn.
Ifupx≥1, we obtain M−/p+upxp−<rn; then, upx<rnp+/M−1/p−; if upx≤1, we obtain M−/p+upxp+<rn; then, upx<rnp+/M−1/p+.
Hence, forn large enough rn>M−,(29)upx<rnp+M−1/p−,vqx<rnq+M−1/q−,from (12)(30)ux<Krnp+M−1/p−,vx<Krnq+M−1/q−,thus(31)ux+vx<Kp+M−1/p−+q+M−1/q−rn1/minp−,q−=ξn.
Then, the inclusion of sets is valid:(32)Φ−1−∞,rn⊆u,v∈X|ux+vx<ξn.
From (21)–(32), we have(33)φrn≤supΦu,v<rn∫ΩFx,u,vdxrn≤Kp+M−1/p−+q+M−1/q−minp−,q−sups+t<ξn∫ΩFx,s,tdxξnminp−,q−.
Put(34)γ≔limr⟶+∞infφr.
From (22) and (33), we can deduce(35)γ≤limn⟶+∞infφrn≤Kp+M−1/p−+q+M−1/q−minp−,q−⋅limn⟶+∞∫Ωsups+t<ξnFx,s,tdxξnminp−,q−=αKp+M−1/p−+q+M−1/q−minp−,q−<+∞.
Thus,Λ⊂0,1/γ.
Next, it will prove whenλ∈Λ; Iλ is unbounded from below. Indeed, on account of(36)1λ<Kp+M−1/p−+q+M−1/q−minp−,q−⋅Lβ.
There exists a column of positive real numbersηn and θ>0 such that limn⟶+∞ηn=+∞ and(37)1λ<θ<LKp+M−1/p−+q+M−1/q−minp−,q−⋅∫Rx0,O1Fx,ηn,ηndxηnp+/p−+ηnq+/q−.
Now, we defineun∈X as follows:(38)unx=0,x∈Ω\Rx0,O2,ηn,x∈Rx0,O1,ηnO22−O12O22−x−x02,x∈Rx0,O2\Rx0,O1.
Then,un,un∈X and(39)∫ΩΔunpxdx=∫Rx0,O2\Rx0,O1Δunpxdx≤θ1ηnp+,∫Ω∇unpxdx=∫Rx0,O2\Rx0,O1∇unpxdx≤θ2ηnp+,∫Ωunpxdx=∫Rx0,O1ηnpxdx+∫Rx0,O2\Rx0,O1unpxdx≤θ3ηnp+.
Thus,(40)Φun,un≤1p−∫ΩΔunpxdx+m1∫Ω∇unpxdx+∫Ωρ1xunpxdx+1q−∫ΩΔunqxdx+m1∫Ω∇unqxdx+∫Ωρ2xunqxdx≤M+p−θ1+θ2+ρ1x∞θ3ηnp++M+q−θ1+θ2+ρ2x∞θ3ηnq+=1Kp+/M−1/p−+q+/M−1/q−minp−,q−⋅ηnp+p−L1+ηnq+q−L2.
FromA1, we have(41)Ψun,un=∫ΩFx,un,undx≥∫Rx0,O1Fx,ηn,ηndx.
Whenn is large enough, from (37), (40), and (41), we can obtain(42)Iλun,un=Φun,un−λΨun,un≤1Kp+/M−1/p−+q+/M−1/q−minp−,q−⋅ηnp+p−L1+ηnq+q−L2−λ∫Rx0,O1Fx,ηn,ηndx≤1L1Kp+/M−1/p−+q+/M−1/q−minp−,q−⋅ηnp+p−+ηnq+q−−λ∫Rx0,O1Fx,ηn,ηndx≤1−λθL⋅ηnp+p−+ηnq+q−⋅1Kp+/M−1/p−+q+/M−1/q−minp−,q−,so(43)Iλun,un=−∞,asn⟶+∞.
According to Theorem1 (b), we know that Theorem 2 is established.Theorem 3.
PresumeA1 holds andB1, ∀x∈Ω,Fx,0,0=0B2, ∃x0∈Ω,O2>O1>0(44)α0≔limb⟶0+inf∫Ωsups+t<bFx,s,tdxbmaxp+,q+,β0≔lims,t⟶0+sup∫Rx0,O1Fx,s,tdxsp−/p−+tq−/q−,and one has(45)α0<L0β0,where(46)L0≔minL3,L4,L3=1Kp+/M−1/p++q+/M−1/q+maxp+,q+⋅1M+θ¯1+θ¯2+ρ1x∞θ¯3,L4=1Kp+/M−1/p++q+/M−1/q+maxp+,q+⋅1M+θ¯1+θ¯2+ρ2x∞θ¯3,θ¯1=2πN/2NΓN/22NO22−O12p−O2N−O1N,θ¯2=2πN/2ΓN/22O22−O12p−O2p−+N−O1p−+Np−+N,θ¯3=2πN/2ΓN/2O1NN+1O22−O12p−∫O1O2O22−r2p−rN−1dr,M+=max1,m1,M−=min1,m0.
Then, for every(47)λ∈Λ¯≔1Kp+/M−1/p++q+/M−1/q+maxp+,q+1L0β0,1α0,problem (1) has infinity solutions converging to 0.Proof.
Obviously,minXΦ=Φ0,0=0.
There is a real sequenceξn satisfying ξn⟶0+ as n⟶+∞ and(48)limn⟶+∞∫Ωsups+t<ξnFx,s,tdxξnmaxp+,q+=α0<+∞.
Putrn=ξnmaxp+,q+/Kp+/M−1/p++q+/M−1/q+maxp+,q+, for all n∈ℕ and δ=limr⟶0+infφr; then, by assumption B2, we have(49)δ≤limn⟶+∞infφrn≤Kp+M−1/p++q+M−1/q+maxp+,q+⋅limn⟶+∞∫Ωsups+t<ξnFx,s,tdxξnmaxp+,q+=α0Kp+M−1/p++q+M−1/q+maxp+,q+<+∞.
Hence,Λ¯⊆0,1/δ.
In the next step, we prove thatIλ does not have the local minimum at 0. For λ∈Λ¯, we have(50)1λ<Kp+M−1/p++q+M−1/q+maxp+,q+⋅L0β0,and there exists a column of positive real numbers ηn⟶0+ as n⟶+∞ and θ>0 such that(51)1λ<θ<L0Kp+M−1/p++q+M−1/q+maxp+,q+⋅∫Rx0,O1Fx,ηn,ηndxηnp−/p−+ηnq−/q−.
Letunx be defined by (38); then, we can obtain(52)Φun,un≤ηnp−p−L3+ηnq−q−L41Kp+/M−1/p++q+/M−1/q+maxp+,q+.
Combining (41), (51), and (52), we have(53)Iλun,un=Φun,un−λΨun,un≤1L0ηnp−p−+ηnq−q−1Kp+/M−1/p++q+/M−1/q+maxp+,q+−λ∫Rx0,O1Fx,ηn,ηndx≤1−λθL0⋅ηnp−p−+ηnq−q−1Kp+/M−1/p++q+/M−1/q+maxp+,q+≤0=Iλ0,0.
Obviously, according to Theorem1 (c), we have completed the proof of Theorem 3. Moreover,un,un is the solution satisfying the conditions and un,un⟶0.Example 1.
LetΩ=−1,12; for all t>0, M1t=a+bt and M2t=c+dt, where a, b, c, and d are positive numbers and m0=mina,c. On Ω, pxandqx are defined as(54)px=x2+y2+2,qx=x2+y2+3.
Define the functionF:ℝ2⟶ℝby(55)Fs,t=an+16e−1/1−s−an+12−t−an+12,ifs,t∈∪n≥1Ran+1,an+1,1,0,otherwise,where a1=3 and an+1=nan3+3n≥1 and Ran+1,an+1,1 is an open unit ball with its center point at an+1,an+1.
It is clear thatF≥0 and F∈C1ℝ2, for all n∈ℕ+. The maximum of F on Ran+1,an+1,1 is Fan+1,an+1=an+16e−1.
Hence,(56)limn⟶+∞supFan+1,an+1an+14/2+an+15/3=+∞.
Therefore,(57)β≔lims,t⟶+∞sup∫Rx0,O1Fs,tdxs4/2+t5/3=Rx0,O1⋅lims,t⟶+∞Fs,ts4/2+t5/3=+∞,where Rx0,O1 is the measure of Rx0,O1.
At the time,∀n∈ℕ+, and we deduce(58)sups+t<an+1−1Fs,t=an6e−1,so(59)limn⟶+∞sups+t<an+1−1Fs,tan+1−12=0,that is to say(60)limb⟶+∞sups+t<bFs,tb2=0.
Thus,(61)α≔limb⟶+∞inf∫Ωsups+t<bFs,tdxb2=Ωlimb⟶+∞infsups+t<bFs,tb2=0<Lβ=+∞,∀λ>0; from Theorem 2, we can infer that the equations,(62)ΔΔux2+y2Δu−a+b∫Ω∇uxx2+y2+2x2+y2+2dxdiv∇ux2+y2∇u+ρ1xux2+y2ux=λFux,u,v,inΩ,ΔΔvx2+y2+1Δv−c+d∫Ω∇vxx2+y2+3x2+y2+3dxdiv∇vx2+y2+1∇v+ρ2xvx2+y2+1vx=λFvx,u,v,inΩ,u=Δu=v=Δv=0,on∂Ω,have a sequence of unbounded weak solutions.



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