# The Multiplicity of Nontrivial Solutions for a Newpx-Kirchhoff-Type Elliptic Problem

Authors: Chang-Mu Chu; Yu-Xia Xiao
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving apx-Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new px-Kirchhoff problem has at least two nontrivial weak solutions.


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## Body

## 1. Introduction and Main Result

In this paper, we consider the following nonlocalpx-Kirchhoff problem:
(1)−a−b∫Ω1px∇upxdxΔpxu=λcxuqx−2u+dxurx−2u,inΩ,u=0,on∂Ω,whereΩ⊂ℝN is a smooth bounded domain with boundary ∂Ω, a≥b>0 are constants, p∈CΩ¯ with 1<p−≔infx∈Ω¯px≤px≤p+≔supx∈Ω¯px<N, ∆pxu≔div∇upx−2∇u is called px-Laplacian operator. λ is a real parameter, cx,dx>0 for x∈Ω¯, 1<qx<px<rx<p∗x=Npx/N−px.The study of variational problems with nonstandardpx-growth conditions has been a new and interesting topic. These problems arise from the image processing model and stationary thermorheological viscous flows; we can refer to [1, 2]. Problem (1) is related to the stationary problem
(2)ρh∂2u∂t2+δ∂u∂t+f1∂u∂t=p0+Eh2L∫0L∂u∂x2dx∂2u∂x2+f2x,u,whereρ, h, δ, p0, L are constants which represent some physical meanings; E is Young’s modulus; ux,t is the lateral displacement; and f1, f2 are the external forces. When f1=f2, it extends the classical D’Alembert wave equation for free vibrations of elastic strings. Since problem (2) is no longer a pointwise identity, it is often called a nonlocal problem. In recent years, nonlocal elliptic problems have attracted wide attention, and some important and interesting results have been established (see [3–5]).In the past few decades, many people studied the followingpx-Kirchhoff problem:
(3)−M∫Ω1px∇upxdxΔpxu=fx,u,inΩ,u=0,on∂Ω,where Ω be a bounded open subset of ℝN with a C1-boundary ∂Ω, Mt∈C10,+∞ is a bounded below function, f:Ω×ℝ⟶ℝ is a Carathéodory function. For example, Dai in [4] using the three-critical-point theorem obtained the existence of solutions for problem (3). Moreover, when fx,u=λaxuqx−2u+bxurx−2u, Zhou and Ge in [6] studied the existence of the solution for the nonlocal problem (3) by using the fibering map approach for the corresponding Nehari manifold. In recent years, some people are starting to pay attention to the case Mt=a−bta≥b>0 in problem (3). Obviously, Mt=a−bta≥b>0 is not bounded below. Therefore, this is a new class of nonlocal problems. In fact, for the case px≡p, some results are given in [7–9]. However, few literatures have considered this new nonlocal px-Kirchhoff problem. Recently, the authors in [10] have considered the following px-Kirchhoff-type problem:
(4)−a−b∫Ω1px∇upxdxΔpxu=λupx−2u+gx,u,inΩ,u=0,on∂Ω,where a≥b>0 are constants, Ω∈ℝN is a bounded smooth domain, p∈CΩ¯ with 1<px<N, λ is a real parameter, and g is a continuous function. Under appropriate hypotheses, the author used a mountain pass theorem and fountain theorem to obtain the existence and multiplicity of nontrivial solutions for problem (4).Inspired by the above facts and aforementioned papers, the main purpose of this paper is to study the existence of two nontrivial solutions for problem (1). Before stating our main results, we make some assumptions on the functions c, d, p, and r.(H1) cx∈LαxΩ, dx∈LβxΩ, with 1<αx∈CΩ¯, 1<βx∈CΩ¯(H2) qx<αx−1p∗x/αx and rx<βx−1p∗x/βx(H3) 1<p−<p+2<2p−2Now, our main result is as follows:Theorem 1.
Assume that the functionpx, qx, rx∈CΩ¯, cx, dx>0 satisfies 1<q+<p−≤px≤p+<2p−<r−<rx<p∗x. If (H1)–(H3) hold, then there exists λ∗>0 such that problem (1) has at least two nontrivial weak solutions for any λ∈0,λ∗.


## 2. Preliminaries

In order to discuss problem (1), we need some theories on spaces LpxΩ and W1,pxΩ which we will call generalized Lebesgue-Sobolev spaces. For more details on the basic properties of these spaces, we refer the readers to Fan and Zhao [11]. Set C+Ω¯=px;px∈CΩ¯,px>1,∀x∈Ω¯; ζΩ denoted the set of all measurable real functions defined on Ω.For anyp∈C+Ω¯, the variable exponent Lebesgue space LpxΩ is defined as
(5)LpxΩ=u∈ζΩ:∫Ωupxdx<+∞,with the norm(6)uLpxΩ=upx=infλ>0:∫Ωuxλpxdx≤1.The variable exponent Sobolev spaceW1,pxΩ is defined as
(7)W1,pxΩ=u∈LpxΩ:∇u∈LpxΩ,with the norm
(8)uW1,pxΩ=upx+∇upx.DefineW01,pxΩ as the closure of C0∞Ω in W1,pxΩ. From [11], we know that the spaces LpxΩ and W1,pxΩ are all separable and reflexive Banach spaces.Moreover, there is a constantC>0, such that
(9)upx≤C∇upx,for all u∈W01,pxΩ. Therefore, u=∇upx and uW1,pxΩ are equivalent norms on W01,pxΩ. In addition, we can recall the following properties of the variable exponent spaces.Lemma 2 (see [11]).
For anyu∈LpxΩ, v∈Lp′xΩ with 1/px+1/p′x=1, the inequality holds as follows:
(10)∫Ωuvdx≤1p−+1p′−upxvp′x.
Moreover, ifp1, p2, p3∈C+Ω¯ and 1/p1x+1/p2x+1/p3x=1, then for any u∈Lp1xΩ, v∈Lp2xΩ, and w∈Lp3xΩ, the following inequality holds:
(11)∫Ωuvwdx≤1p1−+1p2−+1p3−up1xvp2xwp3x.
Let us now recall the modular functionρpxu:LpxΩ⟶ℝ which plays an important role in the variable order Lebesgue spaces and which is defined by
(12)ρpxu=∫Ωupxdx.Lemma 3 (see [10]).
For anyun,u∈LpxΩ, the following properties hold:
(i)
upx<1=1;>1⇔ρpxu<1=1;>1(ii)
upx<1⇒upxp+≤ρpxu≤upxp−,upx>1⇒upxp−≤ρpxu≤upxp+(iii)
limn⟶∞unpx=0⇔limn⟶∞ρpxun=0, limn⟶∞un−u=0⇔limn⟶∞ρpxun−u=0Lemma 4 (see [10]).
Forp, q∈C+Ω¯ such that 1≤qx≤p∗x for all x∈Ω¯, there is a continuous embedding
(13)W1,pxΩ↪LqxΩ.
If we replace≤ with <, the embedding is compact.Lemma 5 (see [12]).
Assume thath∈L+∞Ω, p∈C+Ω¯. If uhx∈LpxΩ, then we have
(14)minuhxpxh−,uhxpxh+≤uhxpx≤maxuhxpxh−,uhxpxh+.Lemma 6 (see [10, 13]).
LetAu=∫Ω1/px∇upxdx, for all u∈W01,pxΩ, then Au∈C1W01,pxΩ,R,A′u,v=∫Ω∇upx−2∇u∇vdx, and for all u, v∈W01,pxΩ, and the following properties hold:
(i)
A is convex and sequentially weakly lower semicontinuous(ii)
A′:W01,pxΩ⟶W01,pxΩ∗ is a mapping of type S+; that is, if un⇀u in W01,pxΩ and lim¯n⟶∞A′un,un−u≤0, then un⟶u in W01,pxΩ(iii)
A′:W01,pxΩ⟶W01,pxΩ∗ is a strictly monotone operator and homeomorphismDefinition 7.
We say thatu∈W01,pxΩ is a weak solution of problem (1), if for any φ∈W01,pxΩ, it satisfies the following:
(15)a−b∫Ω1px∇upxdx∫Ω∇upx−2∇u∇φdx−λ∫Ωcxuqx−2uφdx−λ∫Ωdxurx−2uφdx=0.
The energy functionalJλu:W01,px0⟶ℝ associated to problem (1) is defined as
(16)Jλu=a∫Ω1px∇upxdx−b2∫Ω1px∇upxdx2−λ∫Ωcxqxuqxdx−λ∫Ωdxrxurxdx.
Obviously,Jλu is a C1W01,pxΩ,ℝ functional and
(17)Jλ′u,φ=a−b∫Ω1px∇upxdx∫Ω∇upx−2∇u∇φdx−λ∫Ωcxuqx−2uφdx−λ∫Ωdxurx−2uφdx,
for allu,φ∈W01,pxΩ. It is well known that the weak solution of (1) corresponds to the critical point of the functional Jλ on W01,pxΩ.Definition 8.
LetW01,pxΩ,⋅ be a Banach space and Jλ∈C1W01,pxΩ.
We say thatJλ satisfies the PSc for c∈ℝ, if any sequence un⊂W01,pxΩ satisfying Jλun⟶c and Jλ′un⟶0 in W01,pxΩ∗ as n⟶∞ has a convergent subsequence.


## 3. Proof of Main Result

In this section, the existence of nontrivial solutions for problem (1) is obtained by using a mountain pass lemma combined with Ekeland’s variational principle.Lemma 9.
Assume that the conditions of Theorem1 hold. Then, the function Jλ satisfies the PSc condition with c<a2/2b for λ>0 small enough.Proof.
Firstly, we prove thatun is bounded in W01,pxΩ. Let un⊂W01,pxΩ be a PSc sequence such that c<a2/2b. Arguing by contradiction, we assume that un⟶+∞asn⟶+∞.
According to (H2), we have 1<qxα′x<p∗x for all x∈Ω¯, where 1/αx+1/α′x=1, by Lemmas 3–5, we have
(18)p+c+1≥p+Jλun−Jλ′un,un≥p+aAun−b2Aun2−λ∫Ωcxqxunqxdx−λ∫Ωdxrxunrxdx−a−bAun∫Ω∇unpxdx−λ∫Ωcxunqxdx−λ∫Ωdxunrxdx≥p+ap+∫Ω∇unpxdx−bp+2p−2∫Ω∇unpxdx2−λp+q−∫Ωcxunqxdx−λp+r−∫Ωdxunrxdx−a∫Ω∇unpxdx+bp+∫Ω∇unpxdx2+λ∫Ωcxunqxdx+λ∫Ωdxunrxdx≥b1p+−p+2p−2∫Ω∇unpxdx2+λ1−p+q−∫Ωcxunqxdx+λ1−p+r−∫Ωdxunrxdx≥b1p+−p+2p−2un2p−−λp+q−−1caxunqxα′x≥b1p+−p+2p−2un2p−−2Cλp+q−−1caxunq+.
Dividing the above inequality byunq+, passing to the limit as n⟶+∞, note that 2p−2>p+2 and 2p−>q+, we obtain a contradiction with λ>0 being sufficiently small. It follows that un is bounded in W01,pxΩ.
Secondly, we will prove thatun has a convergent subsequence in W01,pxΩ.
Sinceun is bounded, there exists a subsequence, still denoted by un and u∈W01,pxΩ such that
(19)un⟶u,weaklyinW01,pxΩ,un⟶u,stronglyinLsxΩ,1≤sx<p∗x,unx⟶ux,a.e.inΩ.
In view of (19) and Lemma 2, we have
(20)∫Ωcxunqx−2unun−udx≤∫Ωcxunqx−1un−udx≤3cxαxunqx−1px/qx−1un−uα1x⟶0,as n⟶∞, where α1x∈C+Ω¯ and 1/α1x+1/αx+qx−1/px=1. Similarly, we also have ∫Ωdxunrx−2unun−udx⟶0 as n⟶∞.
Thus,(21)limn⟶∞∫Ωcxunqx−2unun−udx=0,limn⟶∞∫Ωdxunrx−2unun−udx=0.
According toJλ′un⟶0, we have Jλ′un,un−u⟶0. Therefore,
(22)Jλ′un,un−u=a−b∫Ω1px∇unpxdx∫Ω∇unpx−2∇un∇un−∇udx−λ∫Ωcxunqx−2unun−udx−λ∫Ωdxunrx−2unun−udx⟶0,
asn⟶∞. So, we can deduce from (21) that
(23)a−b∫Ω1px∇unpxdx∫Ω∇unpx−2∇un∇un−∇udx⟶0.
Sinceun is bounded in W01,pxΩ, passing to a subsequence, if necessary, we assume that
(24)∫Ω1px∇unpxdx⟶σ≥0asn⟶+∞.
Ifσ=a/b, then a−b∫Ω1/px∇unpxdx⟶0. For any v∈W01,pxΩ, by (19) and the Hölder inequality, it implies that
(25)limn⟶∞∫Ωcxunqx−2un−uqx−2u+dxunrx−2un−urx−2uvdx=0.
Since(26)Jλ′un,v=a−b∫Ω1px∇unpxdx∫Ω∇unpx−2∇un∇vdx−λ∫Ωcxunqx−2un+dxunrx−2unvdx,and Jλ′un,v⟶0asn⟶∞, we have ∫Ωcxunqx−2un+dxunrx−2unvdx⟶0 as n⟶∞ for fix λ>0. Therefore, ∫Ωcxuqx−2u+dxurx−2uvdx=0.
By the fundamental lemma of the variational method, we obtain thatcxuxqx−2ux+dxuxrx−2ux=0fora.e.x∈Ω.
It follows fromcx>0 and dx>0 that u=0. So
(27)λ∫Ωcxqxunqxdx+λ∫Ωdxrxunrxdx⟶λ∫Ωcxqxuqxdx+λ∫Ωdxrxurxdx=0.
Hence, forσ=a/b, we see that
(28)Jλun=a∫Ω1px∇unpxdx−b2∫Ω1px∇unpxdx2−λ∫Ωcxqxunqxdx−λ∫Ωdxrxunrxdx⟶a22b,
which contradicts the fact thatJλun⟶c<a2/2b. Then, a−b∫Ω1/px∇unpxdx⟶0 is not true. Hence,
(29)a−b∫Ω1px∇unpxdx⟶a−bσ≠0.
By (23), we obtain that
(30)∫Ω∇unpx−2∇un∇un−∇udx⟶0asn⟶+∞.
Invoking theS+ condition (see Lemma 6), we can deduce that un⟶u strongly in W01,pxΩ as n⟶∞. So, Jλ satisfies the PSc condition. The proof is complete.☐Lemma 10.
Assume that the conditions of Theorem1 hold; then, there exist ρ>0, β>0, λ∗>0 such that Jλu≥β>0 for any λ∈0,λ∗ and u∈W01,pxΩ with u=ρ.Proof.
Letρ∈0,1 and u∈W01,pxΩ with u=ρ. By Hölder inequality and Lemmas 3–5, we can deduce that
(31)Jλu=a∫Ω1px∇upxdx−b2∫Ω1px∇upxdx2−λ∫Ωcxqxuqxdx−λ∫Ωdxrxurxdx≥ap+∫Ω∇upxdx−b2p−2∫Ω∇upxdx2−λq−∫Ωcxuqxdx−λr−∫Ωdxurxdx≥ap+ρpx∇u−b2p−2ρpx∇u2−λq−cαxuqxα′x−λr−dβxurxβ′x≥ap+ρpx∇u−b2p−2ρpx∇u2−λq−cαxuα′xqxq++uα′xqxq−−λr−dβxuβ′xrxr++uβ′xrxr−≥ap+up+−b2p−2u2p−−2Cλq−cαxuq−−2Cλr−dβxur−≥ap+−b2p−2u2p−−p+−λ2Cq−cαxuq−−p+−λ2Cr−dβxur−−p+up+.
Sinceρ∈0,1, we can choose ρ sufficiently small such that a/p+−b/2p−2ρ2p−−p+>0. Set
(32)λ∗=a/p+−b/2p−2ρ2p−−p+2Ccαx/q−ρq−−p++Cdβx/r−ρr−−p+,β=λ∗ρp+.
We can deduce that for anyλ∈0,λ∗, there exists β>0 such that for any u∈W01,pxΩ with u=ρ, we have Jλu≥β>0.□Lemma 11.
Assume that the conditions of Theorem1 hold; then, there exists e∈W01,pxΩ with e>ρ>0 such that Jλe<0.Proof.
Choosingψ∈C0∞Ω satisfies ∫Ωdxψrxdx>0 and t>1. We have
(33)Jλtψ=a∫Ωtpxpx∇ψpxdx−b2∫Ωtpxpx∇ψpxdx2−λ∫Ωtqxcxqxψqxdx−λ∫Ωtrxdxrxψrxdx≤atp+p−∫Ω∇ψpxdx−bt2p−2p+2∫Ω∇ψpxdx2−λtq−q+∫Ωcxψqxdx−λtr−r+∫Ωdxψrxdx.
Sincer−>2p−>p+, we obtain Jλtψ⟶−∞ as t⟶+∞. Then, for t>1 large enough, we can take e=tψ so that e>ρ and Jλe<0.□Lemma 12.
Assume that the conditions of Theorem1 hold; there exists ξ∈W01,pxΩ such that ξ≥0, ξ≠0, and Jλtξ<0 for all t>0 small enough.Proof.
Lettingξ∈W01,pxΩ with ξ=1, for all t>0 small enough, we get the estimate
(34)Jλtξ=a∫Ωtpxpx∇ξpxdx−b2∫Ωtpxpx∇ξpxdx2−λ∫Ωtqxcxqxξqxdx−λ∫Ωtrxdxrxξrxdx≤atp−p−∫Ω∇ξpxdx−bt2p+2p+2∫Ω∇ξpxdx2−λtq+q+∫Ωcxξqxdx−λtr+r+∫Ωdxξrxdx≤atp−p−−bt2p+2p+2−λtq+q+∫Ωcxξqxdx.
Then, for anyt<δ1/p−−q+, with 0<δ<min1,bp−/2ap+2+λp−/aq+∫Ωcxξqxdx, we conclude that Jλtξ<0.□Proof of Theorem 1.
By Lemma10, we have inf∂Bρ0Jλu>0, where ∂Bρ0=u∈W01,pxΩ;u=ρ. And from Lemma 12, there exists ξ∈W01,pxΩ such that Jλtξ<0 for t>0 small enough. As in the proof of Lemma 10, for any u∈B0,ρ and λ∈0,λ∗, it follows that
(35)Jλu≥ap+up+−b2p−2u2p−−2Cλ∗q−cαxuq−−2Cλ∗r−dβxur−.
Note that forJλ0=0, we have
(36)−∞<c1≔infBρ0Jλu<0.
By applying Ekeland’s principle inB0,ρ¯ (see [14–16]), we obtain that there exists a PSc1 sequence un⊂B0,ρ¯ of Jλ. It follows from c1<0 and Lemma 9 that Jλ satisfies the PSc1 condition. Therefore, one has a subsequence still denoted by un and u0∈W01,pxΩ such that un⟶u0 in W01,pxΩ and Jλu0=c1, Jλ′u0=0, which implies that u0 is a solution of problem (1).
From Lemmas10 and 11, we see that the functional Jλ has the mountain pass geometry. Define
(37)Γ=γ∈C10,1,W01,pxΩ∣γ0=0,γ1=e,c2=infγ∈Γmaxt∈0,1Jλγt.
Noting thatmaxt>0at−b/2t2=a2/2b, for all u∈W01,pxΩ\0, we have
(38)Jλu≤a∫Ω1px∇upxdx−b2∫Ω1px∇upxdx2≤a22b.
So, one hasc2<a2/2b. Letting un be a PSc2 sequence of Jλ, by Lemma 9, we obtain Jλ which satisfies the PSc2 condition.
By the mountain pass theorem (see [10, 17]), we obtain that problem (1) has the second solution u~ with Jλu~=c2. Noting that Jλu0=c1<0=Jλ0<c2=Jλu~, we know that u0 and u~ are two nontrivial solutions of problem (1).



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