# Nontrivial Solutions for a Class ofp-Kirchhoff Dirichlet Problem

Authors: Xian Hu; Yong-Yi Lan
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This paper is devoted to the followingp-Kirchhoff type of problems −a+b∫Ω∇updxΔpu=fx,u,x∈Ωu=0,x∈∂Ω with the Dirichlet boundary value. We show that the p-Kirchhoff type of problems has at least a nontrivial weak solution. The main tools are variational method, critical point theory, and mountain-pass theorem.


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## Body

## 1. Introduction and Main Results

Consider the followingp-Kirchhoff type of problems with the Dirichlet boundary value:(1)−a+b∫Ω∇updxΔpu=fx,u,x∈Ω,u=0,x∈∂Ω,where Ω is a smooth bounded domain in ℝN, a>0andb≥0 are real constants, Δp denotes the p-Laplacian operator by △pu=div∇up−2∇u1<p<N, and fx,u is continuous on Ω¯×ℝ.We look for the weak solutions of (1) which are the same as the critical points of the functional I:W01,pΩ⟶ℝ defined by(2)Iu=ap∫Ω∇updx+b2p∫Ω∇updx2−∫ΩFx,udx,where Fx,u=∫oufx,tdt and W01,pΩ is the Sobolev space endowed with the norm(3)u=∇up=∫Ω∇updx1/p.I is of W1Ω and u,v∈W01,pΩ with derivatives given by(4)I′u,v=a+b∫Ω∇updx∫Ω∇up−2∇u⋅∇vdx−∫Ωfx,uvdx.Problem (1) began to attract the attention of researchers after the work of Lions [1], where a functional analysis approach was proposed to attack it. Since then, much attention has been paid to the existence of nontrivial solutions, sign-changing solutions, ground state solutions, multiplicity of solutions, and concentration of solutions.Many researchers studied the existence of weak solution of (1) in N-dimensional whole spaces (see [2–7] and references therein). For example, Wu [2] showed that problem has a nontrivial solution and a sequence of high-energy solutions by using the mountain-pass theorem and the symmetric mountain-pass theorem. Similar consideration can be found in Nie and Wu [3], where radial potentials were considered. Jia and Li [6] studied multiplicity and concentration behaviour of positive solutions for Schro¨dinger-Kirchhoff type equations involving the p-Laplacian. Recently, Li and Niu [7] have obtained the existence of nontrivial solutions for the p-Kirchhoff type equations with critical exponents by the Ekeland variational principle and mountain-pass lemma.Kirchhoff-type problem setting on a bounded domain also attracts a lot of attention, see [8–22] and the references therein. For example, we refer to some recent works [8, 9] in which some interesting results on the problems with Dirichlet or Neumann boundary conditions have been obtained. Chung [10] studied Kirchhoff type problems with Robin boundary conditions and indefinite weights. Chen et al. [11] treated problem (1) when fx,t=λaxuq−2u+bxur−2u1<q<p=2<r<2∗, using the Nehari manifold and fibering maps, and they established the existence of multiple positive solutions for (1). In [14–19], the authors studied the existence of solutions of the p-Kirchhoff problem in the following form −Mupp−1Δpu=fx,u,x∈Ω or p-Kirchhoff problem such as (1). In [16, 17], the authors obtained the existence and multiplicity of solutions for the p-Kirchhoff equation by using the genus theory. In [18, 19], using variational methods, the researchers studied fractional p-Kirchhoff equations and received the multiplicity and centration of solutions. In [20, 21], the authors used the fountain theorem and concentration-compactness principle to consider multiplicity of solutions for p-Kirchhoff equations. By applying a variant of the mountain-pass theorem and the Ekeland variational principle, Cheng et al. [22] obtained the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearity. In [23, 24], the authors studied the equations with nonlinearities which are superlinear in one direction and linear in the other. They use the Ambrosetti-–Rabinowitz (AR) condition to express the superlinear growth at +∞. The AR condition has been used in a technical but crucial way not only in establishing the mountain-pass geometry of the functional but also in obtaining the boundedness of Palais–Smale (PS) sequences. Since the publication of [25], the AR condition has been used extensively throughout the literature (see [26, 27]). However, this condition is very restrictive, eliminating many nonlinearities. There are always many functions that satisfy the AR condition. For example,(5)fx,t=2tln1+t.Under the motivation of Pei and Zhang [28], the aim of this paper is to consider existence of nontrivial solutions for problem (1) with 1<p<N. We shall assume that the nonlinearity term fx,t does not satisfy the AR condition and it is asymmetric as t∈ℝ approaches +∞ and −∞.Nevertheless, to ensure the global compactness, one needs to impose the subcritical growth condition on the nonlinearityfx,t: there exists a position constant C0 such that(6)fx,t≤C01+tq−1,∀t∈ℝ,x∈Ω,where p<q<p∗=pN/N−p. However, under the motivation of Lan and Tang [29], we consider a class of elliptic partial differential equations with a more general growth condition. We assume that the following condition holds:(7)F:limt⟶∞fx,ttp∗−1=0uniformly for almost everyx∈Ω.It is weaker than subcritical growth condition.We next state the other hypotheses onfx,t. Suppose fx,t∈CΩ¯×ℝ, and this satisfies:A1limt⟶0fx,t/t2p−2t=bμ uniformly a.e. x∈Ω, where μ∈0,+∞A2limt⟶0fx,t/t2p−2t=bv uniformly a.e. x∈Ω, where v∈0,+∞A3limt⟶+∞fx,t/t2p−1=+∞ uniformly a.e. x∈ΩA4fx,t/t2p−2t is increasing when t≥0, for a.e. x∈ΩTheorem 1.
Assumefx,u satisfies (F) and A1−A3. If μ<λ1<v<∞, then problem (1) has at least one nontrivial solution when v is not any of the eigenvalues of −Δp on W01,pΩ.Theorem 2.
Leta≤b/2 and assume fx,u satisfies (F) and A1−A3. If μ<λ1=v and limt⟶−∞fx,tt−pFx,t=−∞ uniformly for a.e. x∈Ω, then problem (1) has at least one nontrivial solution.Theorem 3.
Assumefx,u satisfies (F) and A1−A4. If μ<λ1 and v=+∞, then problem (1) has at least one nontrivial solution.Here,λ1 is the first eigenvalue of −Δp,W01,pΩ and there is a corresponding eigenfunction ϕ1>0 in Ω. For any 1≤s<+∞,LsΩ is the usual Lebesgue space endowed with the norm(8)uss:=∫Ωusdx.The paper is organized as follows. In Section1, we introduced the main purpose of this paper and get some conclusions. The proofs of main results will be given in Section 2. We denote various positive constants as C or Cii=0,1,2,3,… for convenience.


## 2. Proofs of the Main Results

Proof of Theorem 1.
Step 1 (thePSc condition). Let un⊂W01,pΩ be a PSc sequence, then we have(9)Iun⟶c,I′un⟶0.First, we prove thatun is bounded. Assume for contradiction that, up to a sequence, un⟶+∞ as n⟶∞. Since I′un⟶0, for every w∈Wo1,pΩ, we can write <I′un,w≥I′unw=o1w, namely,(10)a+b∫Ω∇unpdx∫Ω∇unp−2∇un⋅∇wdx−∫Ωfx,unwdx=o1w.Definezn=un/un. Obviously, zn∈W01,pΩ with zn=1. Then, there exists a subsequence (denoted also by zn) such that(11)zn⇀z0inW01,pΩ,zn⟶z0inLrΩ,wherep≤r<p∗,znx⟶z0xa. e.x∈Ω,znx≤qxa. e.x∈Ω,wherez0∈W01,pΩ and q∈W01,pΩ. Dividing both sides of (10) by un2p−1, we obtain(12)aunp+b∫Ω∇znp−2∇zn⋅∇wdx−∫Ωfx,unun2p−1wdx=o1wun2p−1,for allw∈W01,pΩ.Passing to the limit, we get(13)limn⟶∞∫Ωfx,unun2p−1wdx=b∫Ω∇z0p−2∇z0⋅∇wdx.Let us now prove thatz0x≤0 for a.e. x∈Ω. To verify this, we chose w=z0+=maxz0,0 in (13), and we have(14)limn⟶∞∫Ω+fx,unun2p−1z0dx=b∫Ω+∇z0pdx<+∞,whereΩ+=x∈Ωz0x>0. Moreover, using limn⟶∞unx=+∞a. e.x∈Ω+ and by A3, we have(15)limn⟶∞fx,unun2p−1z0=limn⟶∞fx,unun2p−1zn2p−1xz0x=+∞,a.e.x∈Ω+.Therefore, ifΩ+>0, by Fatou’s lemma, we will obtain that(16)limn⟶∞∫Ω+fx,unxun2p−1z0xdx=+∞,which contradicts (10). Thus, Ω+=0 and the claim (z0x≤0) is proved. Clearly, z0x≡0. By contradiction z0=0, we know that zn⇀0inW01,pΩandzn⟶0inLrΩ, where p≤r≤p∗. Therefore, we obtain zn⟶0, which is a contradiction. When z0<0, we have un=znun⟶−∞asn⟶∞. If this is the case, by I′un,zn=I′unzn=o1, we first obtain(17)a+b∫Ω∇unpdx∫Ω∇unp−2∇un⋅∇zndx−∫Ωfx,unzndx=o1.Dividing both sides of (17) by un2p−1, we obtain(18)aunp+b∫Ω∇znpdx∫Ω∇znpdx−∫Ωfx,unun2p−1zndx=o1.Passing to the limit, we get(19)limn⟶∞∫Ωfx,unun2p−1zndx=b∫Ω∇z0pdx2,Then,(20)limn⟶∞∫Ωfx,unun2p−1zndx=bz02p.Using Lebesgue’s dominated convergence theorem andA2 for the left hand side of (20), we have(21)limn⟶∞∫Ωfx,unun2p−1zndx=∫Ωlimn⟶∞fx,unun2p−1zndx=∫Ωlimn⟶∞fx,unun2p−1zn2pdx=bv∫Ωz02pdx.Then, (20) can be written as follows:(22)bv∫Ωz02pdx=bz02p.This means thatv is an eigenvalue, which contradicts our assumption. So, un is bounded.Now, we prove thatun has a convergent subsequence. By the continuity of embedding, we have unp∗p∗≤C1<∞ for all n. Going if necessary to a subsequence, we have(23)un⇀uinW01,pΩ,un⟶uinLqΩ,wherep≤q<p∗.By (F), for everyε>0, we can find a constant Cε>0 such that(24)fx,tt≤Cε+12C1εtp∗,fort∈ℝ,a.e.x∈Ω.Letδ=ε/2Cε,E⊆Ω means E<δ, we have(25)∫Efx,unundx≤∫Efx,undx≤∫ECεdx+12C1ε∫Eunp∗dx≤ε2+ε2=ε.Hence,∫Ωx,unundx,n∈N is equiabsolutely continuous. Moreover, fx,unun,n∈N is equi-integrable and bounded in L1Ω, and fx,unun⟶fx,uu in measure. From the Vitali convergence theorem, it follows that(26)∫Ωfx,unundx⟶∫Ωfx,uudx.ByF, for every ε>0, we can find a constant Cε>0 such that(27)fx,t≤Cε+12c1c2εtp∗−1,fort∈ℝ,a.e.x∈Ω,wherec1≥∫Ωunp∗dxp∗−1/p∗ for all n and c2=∫Ωup∗dx1/p∗ are positive constants with the assumption of u≡0. Form Ho¨lder’s inequality, for every E⊆Ω, we have(28)∫ECεudx≤CεmeasEp∗−1/p∗∫Eup∗dx1/p∗=CεmeasEp∗−1/p∗c2,∫Eup∗−1dxudx≤∫Eup∗dxp∗−1/p∗∫Eup∗dx1/p∗≤c1c2.Letδ=ε/2c2Cεp∗/p∗−1>0,E⊆ΩmeasE<δ, we have(29)∫Efx,unudx≤∫Efx,unudx≤∫ECεudx+12c1c2ε∫Eunp∗−1udx≤ε2+ε2=ε.Hence,∫Ωfx,unudx,n∈N is equiabsolutely continuous. Moreover, fx,unudx,n∈N is equi-integrable and bounded in L1Ω, and fx,unu⟶fx,uu in measure. From the Vitali convergence theorem, it follows that(30)∫Ωfx,unudx⟶∫Ωfx,uudx.Since(31)I′un,un=a+bunp∫Ω∇unp−2∇un⋅∇undx−∫Ωfx,unundx⟶0,it follows that(32)I′un,u=a+bunp∫Ω∇unp−2∇un⋅∇udx−∫Ωfx,unudx⟶0.Thus, we haveun⟶uinW01,pΩ, which means that I satisfies the PS condition.Step 2 (mountain-pass geometric structure).I has mountain-pass geometry, i.e., there exist e∈W01,pΩ and ρ,α>0 such that(33)Iu≥α>0,whenu=ρ,(34)Ie<0,whene>ρ.Letϕ1>0 be a λ1-eigenfunction with ϕ1=1, and assume A1−A3 and F hold. If v∈λ1,+∞,for anyε>0, there exist C2andC3 so small such that for all x,t∈Ω×ℝ, and I has mountain-pass geometry. By A1 and F, we have(35)Fx,s≤b2pμ+εs2p+C2sp∗.Chooseε>0 such that μ+ε<λ1. By (35), using Poincaré inequality and Sobolev inequality, we get(36)Iu=ap∫Ω∇updx+b2p∫Ω∇updx2−∫ΩFx,udx=apup+b2pu2p−∫ΩFx,udx≥apup+b2pu2p−bμ+ε2pu2p2p−C2up∗p∗≥apup+b2p1−μ+ελ1u2p−C2Kup∗.Then, (33) is proved if we choose u=ρ>0 small enough.On the other hand, byA2, we have(37)Fx,s≥b2pv−εs2p−C3.Takeε>0 such that ν−ε>λ1, and let e=tϕ1 such that e=tϕ1>ρ. Using (37), we have(38)Ie=Itϕ1=atpp∫Ω∇ϕ1pdx+bt2p2p∫Ω∇ϕ1pdx2−∫ΩFx,tϕ1dx≤atpp+bt2p2p−bv−ε2pt2pϕ12p2p+C3Ω≤atpp+b2p1−v−ελ1t2p+C3Ω⟶−∞ast⟶+∞.Thus, it showse>ρ and Ie<0. Therefore, I has mountain-pass geometry.Step 3 (a critical value ofI). For e in Step 2, we define(39)τ=γ:C0,1⟶W01,pΩγ0=0,γ1=e,c=infγ∈τmax0≤t≤1Iγt.It turns out that the mountain-pass theorem holds. Then,c is a critical value of I.Proof of Theorem 2.
Step 1 (theCc condition). Let un⊂W01,pΩ be a Cc sequence, then we have(40)Iun⟶c,I′un1+un⟶0.Similar to the proof of Theorem1, we have z0x≤0,x∈Ω,z0x≡0 and(41)bv∫Ωz02pdx=bz02p.By maximum principle,z0 is an eigenfunction of λ1. When z0<0, we have un=znun⟶−∞asn⟶∞. By our assumption,(42)limn⟶∞fx,unxunx−pFx,unx=−∞,uniformly inx∈Ω, which implies that(43)∫Ωfx,unxunx−pFx,unxdx=−∞asn⟶∞.On the other hand,(44)pIun−I′un,un=a∫Ω∇unpdx+b2∫Ω∇unpdx2−p∫ΩFx,unxdx−a∫Ω∇unpdx−b∫Ω∇unpdx2+∫Ωfx,unxunxdx=−b2unp+∫Ωfx,unxunx−pFx,unxdx⟶pcasn⟶∞,which implies that(45)∫Ωfx,unxunx−pFx,unxdx⟶pc+b2unp⟶+∞asn⟶∞.It contradicts (43). Hence, un is bounded. According to Step 1, the proof of Theorem 1, we have un⟶uinW01,pΩ, which means that I satisfies Cc.Step 2 (mountain-pass geometric structure). Letϕ1>0 be a λ1-eigenfunction with ϕ1=1, and assume A1−A3 and F hold. If v=λ1, similar to the proof of Theorem 1, (33) holds. We only need to show (34) holds. Let e=−tϕ1 be such that e=−tϕ1>ρ. By A2, we have(46)Ie=I−tϕ1=atpp∫Ω∇ϕ1pdx+bt2p2p∫Ω∇ϕ1pdx2−∫ΩFx,−tϕ1dx≤atpp+bt2p2p−1p∫Ωfx,−tϕ1−tϕ1dx−∫ΩFx,−tϕ1−1pfx,−tϕ1−tϕ1dx≤atpp+bt2p2p−bpλ1−tϕ12p−∫ΩFx,−tϕ1−1pfx,−tϕ1−tϕ1dx≤atpp−bt2p2p−∫ΩFx,−tϕ1−1pfx,−tϕ1−tϕ1dx.By (43), we get(47)∫ΩFx,−tϕ1−1pfx,−tϕ1−tϕ1dx⟶∞.By our assumptiona≤b/2, we have(48)Ie=I−tϕ1⟶−∞ast⟶+∞.
Thus, (34) is proved. Therefore, I has mountain-pass geometry.Proof of Theorem 3.
Step 1 (theCc condition). Let un⊂W01,pΩ be a Cc sequence. We can prove that un is bounded in W01,pΩ. Assume un⟶+∞asn⟶∞. Similar to the proof of Theorem 1, we have z0≤0, and when z0<0,un=znun⟶−∞asn⟶∞. Let(49)sn=4pc/b2pun,wn=snun=4pc/b2punun.Sincewn is bounded in W01,pΩ, we have a subsequence denoted also by wn) such that wn⇀w0inW01,pΩ. From the proof of Theorem 1, we know Ω+=0, so Ω−=x∈Ω:w0x<0, and v=+∞. By A3, we have(50)fx,unun2p−2un≥Muniformly for allx∈Ω−,whereM is a large enough constant. Since I′un,un⟶0, then we get(51)aunp+bun2p=∫Ωfx,unxundx.Therefore, we have(52)a4pcb+4pc=limn⟶∞awnp+bwn2p=limn⟶∞∫Ωfx,wnwn2p−2wnwn2pdx≥limn⟶∞∫Ω−fx,wnwn2p−2wnwn2pdx≥M∫Ω−w02pdx.So,w0≡0 for a.e. x∈Ω. However, if w0≡0, then ∫ΩFx,wndx⟶0. Hence,(53)Iwn=apwnp+b2pwn2p+o1≥2c+o1.On the other hand, byI′un,un⟶0 as n⟶∞, for a suitable subsequence, we may assume that(54)−1n<I′un,un=a+bunpunp−∫Ωfx,unxunxdx<1nfor alln.From (54), we have(55)bun2p<−aunp+∫Ωfx,unxunxdx+1n.Then, for anyt≥0,(56)Itun=atppunp+bt2p2pun2p−∫ΩFx,tunxdx≤atppunp+t2p2p−aunp+1n+∫Ωfx,tunxunxdx−Fx,tunxdx=t2p2np+atpp1−tp2unp+∫Ωt2p2pfx,tunxunx−Fx,tunxdx.We set(57)ht=t2p2pfx,unxunx−Fx,tunx.Then,(58)h′t=t2p−1fx,unun−fx,tunun=t2p−1unfx,un−fx,tunt2p−1=t2p−1unfx,1un12p−1−fx,tunt2p−1.ByA4, hence(59)h′t≥0,0<t≤1,h′t≤0,t≥1.So,ht≤h1 for all t≥0.Similarly, we set(60)gt=atpp1−tp2.Then,(61)g′t=atp−1−t2p−1.Hence,(62)g′t≥0,0<t≤1,g′t≤0,t≥1.So,gt≤g1 for all t≥0.Therefore,(63)Itun=t2p2np+gtunp+∫Ωhtdx≤t2p2np+g1unp+∫Ωh1dx=t2p2np+a2punp+∫Ω12pfx,unxunx−Fx,unxdx.On the other hand, from (54), we have(64)bun2p>−aunp−1n+∫Ωfx,unxunxdx.Then,(65)Iun=apunp+b2pun2p−∫ΩFx,unxdx≥apunp+12p−aunp−1n+∫Ωfx,unxunxdx−∫ΩFx,unxdx=a2punp−12np+∫Ω12pfx,unxunx−Fx,unxdx.That is,(66)a2punp+∫Ω12pfx,unxunx−Fx,unxdx≤12pn+Iun.Thus, we have(67)Itun≤t2p2pn+12pn+Iun=1+t2p2np+Iun,for allt≥0. By un⟶∞asn⟶∞, we have sn⟶0 (defined by (49)) as n⟶∞. Thus, we get(68)Iwn=Isnun≤1+sn2p2np+Iun⟶c+o1asn⟶∞.Obviously, it contradicts (53). So, un is bounded in W01,pΩ. According to the proof of Theorem 1, we have un⟶uinW01,pΩ which means that I satisfies Cc.Step 2 (mountain-pass geometric structure). Letϕ1>0 be a λ1-eigenfunction with ϕ1=1, and assume A1−A4 and F hold. If v=+∞, for any ε>0, there exists C4 so small such that for all x,t∈Ω×ℝ, and I has mountain-pass geometry. Similar to the proof of Theorem 1, (33) holds. We only need to show that (34) holds. By A2, we have(69)Fx,s≥b2pλ1+εs2p−C4.Lete=tϕ1 such that e=tϕ1>ρ. Using (69), we have(70)Ie=Itϕ1=atpp∫Ω∇ϕ1pdx+bt2p2p∫Ω∇ϕ1pdx2−∫ΩFx,tϕ1dx≤atpp+bt2p2p−bλ1+ε2pt2pϕ12p2p+C4Ω≤atpp+b2p1−λ1+ελ1t2p+C4Ω⟶−∞ast⟶+∞.Thus, it showse>ρ and Ie<0. Therefore, I has mountain-pass geometry.



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