# Existence and Multiplicity of Solutions for a Biharmonic Equation withp(x)-Kirchhoff Type

Authors: Qi Zhang; Qing Miao
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Based on the basic theory and critical point theory of variable exponential Lebesgue Sobolev space, this paper investigates the existence and multiplicity of solutions for a class of nonlocal elliptic equations with Navier boundary value conditions when (AR) condition does not hold and improves or generalizes the original conclusions.


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## Body

## 1. Introduction

We deal with the following system:(1)a+b∫ΩΔupxpxdxΔpx2u=fx,u,x∈Ω,u=Δu=0,x∈∂Ω,where Ω⊂ℝN is a bounded smooth region. The constants a and b satisfy a>0 and b≥0.px∈CΩ¯, and 1<p−≔infx∈Ωpx≤p+≔supx∈Ωpx<+∞. The nonlinear term fx,u∈CΩ¯×ℝ,ℝ satisfies Carathéodory condition. And, Δpx2ux=ΔΔupx−2Δu.The differential equations withp(x)-Laplacian or biharmonic operators are derived from nonlinear elastic mechanics and fluid mechanics. This model can describe the physical phenomenon of “point by point anisotropy.” Various methods have extensively studied the problem of differential operators with variable exponents, as shown in [1–6]. In [7, 8], the authors used changeable nonlinear terms to outline the boundary of the real image, eliminate the possible noise, and solve the problems in image processing.Recently, the question(2)−a+b∫Ω∇u2dxΔu=fx,u,x∈Ω,u=0,x∈∂Ω,has been widely concerned by many scholars after Lions [9] found an abstract method to solve this problem. Furthermore, many interesting results can be seen in [10–12]. In addition, in [13], Dai and Liu studied the following cases with a variable exponent:(3)−a+b∫Ω1px∇upxdxdiv∇upx−2∇u=fx,u,x∈Ω,u=0,x∈∂Ω.According to the variational method and the basis theorem of Sobolev space under suitable assumptions, the infinite positive solutions of the problem with p(x)-Laplace operator is obtained.As we all know, the Ambrosetti–Rabinowitz [14] (it is written as (AR)) condition can not only ensure that fx,s is superlinear with respect to the variable s at infinity but also ensure the boundedness of the Palais–Smale (it is written as (PS)) sequence. Therefore, (AR) condition is essential for the study of many boundary value problems, see [15–18]. In particular, when a = 1 and b = 0 in problem (1) and (AR) condition holds, the existence of solution of problem (1) is discussed in [16]. In [17], Guo and Zhao made reasonable presupposition on the nonlinear terms M and f, respectively, and obtained the existence number of solutions for p(x)-Laplace equation in the whole space by applying the basic theory of weighted function and variable exponent space. In [18], Chung, respectively, applied the minimum principle or critical point theory to discuss the problem when the perturbation term f is sublinear or superlinear at infinity. In addition, based on the critical point theory, Z2-group index theory, and operator equation theory, Shu, Lai, and Xu [19] constructed a lower semicontinuous convex function φxt,xt−τ with subdifferentiability and obtained infinite subharmonic periodic solutions of second-order neutral nonlinear functional differential equations. There are similar articles such as [20].In [21], Wang and An studied the following problem:(4)Δ2u−M∫Ω∇u2dxΔu=fx,u,x∈Ω,u=Δu=0,x∈∂Ω.According to the mountain pass lemma satisfying (AR) condition, it is proved that the above elliptic equation has at least two solutions, both of which are nontrivial, where one is positive and the other is negative.However, although many functions satisfy the superlinear growth condition, the (AR) condition does not hold. For example, in [22], Zhang considered the following problems:(5)a+b∫ΩΔupx−λupxpxdxΔpx2u−λupx−2u=fx,u,x∈Ω,u=Δu=0,x∈∂Ω.When the condition of perturbation term f is weaker than (AR), based on the fountain theorem, it is obtained that there are infinite solutions to the above problem.In [23], the author further investigated the existence of multiple solutions for formula (1) when the (AR) condition is invalid according to the same critical point theory. It should be noted that although problem (1) has been studied in [23], this paper will prove that under weaker conditions, we can still use the fountain theorem to obtain the same result as that in reference [23], which illustrates that our conclusion is more universal. Furthermore, we will obtain mountain pass solutions and infinitely many solutions converging to 0.This article is mainly composed of four sections. Section2 introduces the basic theory of Lebesgue and Sobolev spaces and gives the related propositions and theorems needed below. Section 3 first proves that the solution of problem (1) exists under reasonable assumptions and then further proves the existence of at least one nontrivial solution by quoting mountain pass lemma. In the last but most important part (Section 4), the sufficient conditions for the existence of multiple solutions of problem (1) are obtained according to the variational method and critical point theory. It is worth noting that the original results are generalized and improved in this part, and we will explain in detail that our results are better later.


## 2. Preliminaries

Let(6)C+Ω¯=qx|qx∈CΩ¯,qx>1,x∈Ω∀qx∈CΩ¯,q+≔maxΩ¯qx,q−≔minΩ¯qx.Define the following spaces:(7)LpxΩ=u:Ω⟶ℝmeasurable,∫Ωuxpxdx<∞,having the norm(8)upx=infϑ>0:∫Ωuxϑpxdx≤1.In addition, the variable exponential Sobolev space is defined as follows:(9)Wh,pxΩ=u∈LpxΩ:Dσu∈LpxΩ,σ≤h,h∈Z+,having the norm(10)uh,px=∑σ≤hDσupx,where σ=σ1,σ2,…,σn and formula σ=∑j=1nσj is true. The closure of C0∞Ω in Wh,pxΩ is the W0h,pxΩ. Besides, from [24], we know that LpxΩ and Wh,pxΩ are Banach space and satisfy separability, uniform convexity, and reflexivity.DenoteX=W01,pxΩ∩W2,pxΩ, which is a reflexive and separable Banach space endowed with the norm(11)u=uX=u1,px+u2,px.We have(12)u=infκ>0:∫ΩΔuxκpxdx≤1.By [25], we realize that ⋅2,p⋅, Δ⋅p⋅, and ⋅ are equivalent to the norm of X.Define the energy functionalΗ:X⟶ℝ as follows:(13)Ηu=a∫ΩΔupxpxdx+b2∫ΩΔupxpxdx2−∫ΩFx,udx,where Fx,u=∫0ufx,tdt and Η∈C1X,ℝ. Furthermore, ∀u,v∈X, we know that(14)Η′u,v=a+b∫ΩΔupxpxdx∫ΩΔupx−2ΔuΔvdx−∫Ωfx,uvdx.Above all, it is easy to obtain that ifu∈X is the weak solution of problem (1), it means that u is the critical point of functional Η.Proposition 1 (see [2, 23]).
Assume that∂Ω possesses the cone property; moreover, ∀x∈Ω¯,qx∈C+Ω¯ withqx<p2∗x; then, X⟶LqxΩ is a continuous and compact embedding, where(15)p2∗x=NpxN−2px,px<N2,∞,px≥N2.Proposition 2 (see [1]).
(1)
J is a homeomorphism.(2)
J:X⟶X∗ is a strictly monotone, bounded, and continuous operator.(3)
J is a mapping of type S+, namely, on X; if un⟶wu and, furthermore, limn⟶∞Jun−Ju,un−u≤0, then un⟶u in X.Remark 1.
According to Proposition2, we have that Η satisfies type S+ and weakly lower semicontinuous.Proposition 3 (see [24]).
Assume1/p1x+1/p2x=1; then, ∀u∈Lp1xΩ and ∀v∈Lp2xΩ, we have(16)∫Ωuvdx≤1p1−+1p2−up1xvp2x,where Lp1xΩ and Lp2xΩ are conjugate space.Proposition 4 (see [2, 24]).
LetJu=∫ΩΔupxdx; ∀u∈W2,pxΩ∩W01,pxΩ, we deduce(17)h1u<1=1,>1⟺Ju<1=1,>1,h2u≥1⟹up−≤Ju≤up+,h3u≤1⟹up+≤Ju≤up−,h4u⟶0⟺Ju⟶0.Theorem 1 (see [26]).
LetX be a real Banach space. If Φ∈C1X,ℝ satisfies (PS) condition and(18)Q1Φ0=0,Q2∃ζ,τ>0such thatΦu≥τasu=ζ,Q3∃v∈X,v>ζsuch thatΦv≤0,then Φ has a critical value expressed as c=infl∈Γmax0≤t≤1Φlt with c≥τ, where Γ≔l∈C0,1|l0=0,l1=v.
Owing toX as a reflexive and separable Banach space, then there exists ei⊂X such that(19)Xi=spanei,Yk=⊕i=1kXi,Zk=⊕i≥k∞Xi¯.Thus, X=Yk⊕Zk. If k¯∈ℕ, it is clear that Yk¯=⊕i=1k¯Xi is the k¯-dimensional subspace of X.Theorem 2 (see [26]).
Assume the following:(M1)X is a Banach space, Ψ∈C1X,ℝ satisfies Ψ0=0, and Ψ−u=Ψu. If for each k = 1, 2, …, there exist ρk>λk>0 such that we have the following:(M2)bk≔infu∈Zk,u=λkΨu⟶+∞ as k⟶+∞(M3)ak≔maxu∈Yk,u=ρkΨu≤0(M4) The functionalΨ satisfies Cerami condition (abbreviated as (C) condition), namely, for any un⊂X such that Ψun⟶C and Ψ′un1+un⟶0 asn⟶+∞ has a convergent subsequence, then Ψ has a series of critical values that tend to be positive infinity.Theorem 3 (see [26]).
Assume that (M1) is satisfied and there isk0>0 such that, for each k≥k0, there exists ρ¯k>λ¯k>0 such that we have the following:(N1)infu∈Zk,u=ρ¯kΨu≥0(N2)mk≔maxu∈Yk,u=λ¯kΨu<0(N3)nk≔infu∈Zk,u≤ρ¯kΨu⟶0 as k⟶+∞(N4) For everyc∈dk0,0, Ψ satisfies PSc∗ condition, i.e., if any sequence unj⊂X such that nj⟶+∞, unj∈Ynj, Ψunj⟶c, and Ψ|Ynj′unj⟶0 contain a subsequence converging to a critical point of Ψ, then Ψ has a series of critical values that converge to zero, where these critical values are negative
Next, the nonlinear termfx,t always has to verify the following assumptions.
(g0)∀x∈Ω,t∈ℝ, we get(20)fx,t≤c11+tαx−1,where αx∈C+Ω¯ and 2p+<α−≤αx≤α+<p2∗x.Remark 2.
From Proposition1, we can infer that X is compact-embedded into Lp−Ω and Lp+Ω, respectively, that is to say, for any u∈X, there are r,c¯>0 such that uLp−Ωp−≤rup− and uLp+Ωp+≤c¯up+. Besides, let ε>0 be small enough such that εc¯≤2a/3p+. We have the following:(g1) There exists0<μ<a/r such that(21)limt⟶∞inftfx,t−2p+Fx,ttp−>−μuniformly inx∈Ω.(g2)limt⟶∞infFx,t/t2p+=+∞ uniformly in x∈Ω(g3)fx,t=οtp+−1,t⟶0 uniformly in x∈Ω(g4)Fx,t=Fx,−t,∀x∈Ωandt∈ℝ(g5)fx,t≥c2tγx−1,t⟶0, where γx∈C+Ω¯,γ−≤γ+<p− for a.e. x∈Ω
The main conclusions and proofs of the article will be introduced in the following. Note thatcii=1,2,…, are positive constants and have different meanings in different places.


## 3. Existence of Solutions

Theorem 4.
Assume(22)fx,t≤c31+tβ−1,where 1≤β<p−<p2∗; then, there is a weak solution to formula (1).Proof.
By (22), we deduce that Fx,t≤c4t+tβ. If u is large enough and using Proposition 1, we get(23)Ηu=a∫ΩΔupxpxdx+b2∫ΩΔupxpxdx2−∫ΩFx,udx≥ap+∫ΩΔupxdx+b2p+2∫ΩΔupxdx2−c4∫Ωudx−c4∫Ωuβdx≥ap+up−+b2p+2u2p−−c4uL1Ω−c4uLβΩβ≥ap+up−+b2p+2u2p−−c5u−c6uβ.Thanks to u>1, we deduce that(24)Ηu≥ap+up−+b2p+2up−−c5u−c6uβ≥L1up−−c5u−c6uβ⟶+∞,where L1=mina/p+,b/2p+2. Hence, H is coercive. Since X is reflexive and H is weakly lower semicontinuous, then Η has a minimum point u in X; moreover, u is a weak solution of (1).Lemma 1.
Iff satisfies (g0) and (g1), then Η satisfies (C) condition.Proof.
Letun⊂X be a Cerami sequence, i.e.,Ηun⟶C and Η′un1+un⟶0. Next, we show that un is bounded inX. Assume un⟶+∞ as n⟶+∞; by (g1), there exists R>1 such that(25)tfx,t−2p+Fx,t≥−μtp−,∀t>R.
PutΩn=x∈Ω,u>R. By (25), there exists δ>0 such that(26)C+un≥Ηun+12p+Η′un1+un≥Ηun−12p+Η′un,un≥a2p+∫ΩΔunpxdx+∫Ω12p+fx,unun−Fx,undx=a2p+∫ΩΔunpxdx+12p+∫Ωn+∫Ω/Ωnfx,unun−2p+Fx,undx≥a2p+unp−−μ2p+∫Ωnunp−dx−δ≥a2p+unp−−μr2p+unp−−δ=a−μr2p+unp−−δ,since p−>1 and μ<a/r; so, this is contradictory, which means un is bounded. Without loss of generality, if un⟶wu, we have Η′unun−u⟶0. Then, it can be obtained un⟶u according to conditions (g0), Hölder inequality, and Remark 1, and the specific proof can be seen in [23].Remark 3.
This is the same as the proof of Lemma1, and it can be obtained that Η also satisfies the (PS) sequence.Theorem 5.
Iffx,t satisfies (g0), (g1), (g2), and (g3), then there is a nontrivial solution to problem (1).Proof.
Obviously, H satisfies (PS) sequence andΗ0=0. Next, we need to show (Q2) and (Q3).
First of all, (Q2) is established. From (g0) and (g3), we can infer that(27)Fx,t≤εtp++c7tαx,∀x,t∈Ω×ℝ.When u≤1, we have(28)Ηu≥ap+∫ΩΔupxdx+b2p+2∫ΩΔupxdx2−ε∫Ωup+dx−c7∫Ωuαxdx≥ap+up++b2p+2u2p+−εc¯up+−c8uα−≥a3p+up++b2p+2u2p+−c8uα−≥L2u2p+−c8uα−,where L2=mina/3p+,b/2p+2; therefore, there exist ζ>0andτ>0 such that Ηu≥τ>0 for everyu=ζ.
It only remains to prove (Q3). By (g2), we obtain that∀η>0, there exists c˜>0 that satisfies(29)Fx,t≥ηt2p+−c˜,∀w∈X\0 withw=1 and t>1; we gain(30)Ηtw≤atp+p−∫ΩΔwpxdx+bt2p+2p−2∫ΩΔwpxdx2−η∫Ωtw2p+dx+c˜Ω≤t2p+ap−wp++b2p−2w2p+−ηt2p+∫Ωw2p+dx+c˜Ω≤2L3−ηC˜t2p++c˜Ω,where L3=maxa/p−,b/2p−2 and C˜ is a normal number. If η is sufficiently large and t⟶+∞, then Ηtw⟶−∞. Hence, based on Theorem 1, we can get that Η has at least one nontrivial solution.


## 4. Multiple Solutions

The conclusions of this section are as follows.Theorem 6.
Iff satisfies (g0), (g1), (g2), and (g4), then equation (1) has a sequence of solutions ±ukk=1∞ such that Η±uk⟶+∞ ask⟶+∞.Theorem 7.
Iff satisfies (g0), (g1), (g3), (g4), and (g5), then equation (1) has a sequence of solutions ±vkk=1∞ such that Η±vk<0 and Η±vk⟶0 as k⟶+∞.
Next, we will use the fountain theorem and dual fountain theorem to prove Theorem6 and Theorem 7, respectively.Lemma 2 (see [2, 18]).
Ifαx∈C+Ω¯ and αx<p2∗x for any x∈Ω¯, denote(31)ξk=supvαx:v=1,v∈Zk,thenlimk⟶+∞ξk=0.Lemma 3 (see [2, 18]).
If conditions (g0) and (g1) are true, thenΗ satisfies the PSc∗ condition.Proof of Theorem6.
Thanks to Lemma1 and (g4), it is easy to gain (M1) and (M4). Now, we only need to prove that (M2) and (M3) are tenable.
Let us start with (M2).∀u∈Zk,u=λk=4c11ξkα+L1−11/p−−α+; by (g0), we have(32)Ηu≥ap+∫ΩΔupxdx+b2p+2∫ΩΔupxdx2−c9∫Ωuαxdx−c9∫Ωudx≥ap+up−+b2p+2u2p−−c9∫Ωuαxdx−c10u≥L1up−−c10u−c9,uαx≤1,c11ξkuα+,uαx>1,≥L1up−−c11ξkα+uα+−c10u−c11,and we know ξk⟶0 and λk⟶+∞ as k⟶+∞ after that(33)Ηu≥34L1up−−c10u−c11sincep−>1,so,bk≔infu∈Zk,u=λkΗu⟶+∞,ask⟶+∞.
Then, we show the correctness of (M3). By (29), we infer that for any u∈Yk, with u≥1, afterwards(34)Ηu≤ap−up++b2p−2u2p+−η∫Ωu2p+dx+c˜Ω≤2L3u2p+−η∫Ωu2p+dx+c˜Ω.
SincedimYk<+∞ and the norm in a finite-dimensional space is equivalent, when η is sufficiently large and u⟶+∞, we know that Ηu⟶−∞.
Therefore, the proof of Theorem6 is finished.Remark 4.
When (g0), (g2), and (g4) are established, the author of [23] has studied the multiplicity of solutions to equation (1) and assumed that(35)g1′∃R>0such thatℱx,t≔tfx,t−2p+Fx,t≥0for allx∈Ω,t≥R.
However,(36)tfx,t−2p+Fx,t≥0≥−μtp−,∀t>R.Therefore, condition g1 is weaker than condition g1′, which shows that we generalize the original results and our results are better. In other words, our conclusion is more universal.Proof of Theorem7.
By (g4) and Lemma3, we will deduce that Η satisfies (M1) and (N4). Next step, all we have to do is check (N1)–(N3).
The first step is to prove (N1). Due to Lemma2 and (27), for k≥k0andu∈Zk∩T1, with T1=u|u∈Zk,0<u<1,u=ρ¯k=1, we deduce that(37)Ηu≥ap+∫ΩΔupxdx+b2p+2∫ΩΔupxdx2−ε∫Ωup+dx−c7∫Ωuαxdx≥ap+up++b2p+2u2p+−εc¯up+−c7ξkuα−,uαx≤1,c7ξkuα+,uαx>1,≥a3p++b2p+2u2p+−c7ξkα−uα−,uαx≤1,c7ξkα+uα+,uαx>1,≥L2−c7ξkα−,uαx≤1,c7ξkα+,uαx>1,since ξk⟶0 belongs to k⟶+∞; thus, it is clear that Ηu≥L2≥0. That is to say, (N1) was established.
Then, we verify that (N2) is true.∀u∈Yk and u are small enough, by (g5), we have(38)Ηu≤ap−up−+b2p−2u2p−−c12∫Ωuγxdx≤2L3up−−c12∫Ωuγxdx,due todimYk<+∞p−>γ+thusmk≔maxu∈Yk,u=λ¯kΗu<0.
Now, it only remains to show (N3). SinceYk∩Zk≠ϕ and λ¯k<ρ¯k, then(39)nk≔infu∈Zk,u≤ρ¯kΗu≤mk≔maxu∈Yk,u=λ¯kΗu<0.From (37), ∀u∈Zk∩T1, we gain(40)Ηu≥L2−c7ξkα−,uαx≤1,c7ξkα+,uαx>1,≥−c7ξkα−,uαx≤1,c7ξkα+,uαx>1,noting that ξk⟶0 as k⟶+∞; hence, nk⟶0, i.e., (N3) is satisfied. Consequently, the Theorem 7 is proved.



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