# The Existence Result for a Class of p-Kirchhoff-Type Problem with a Multilinear Growth Nonlinearity

Authors: Jiangyan Yao; Wei Han
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In this paper, we firstly discuss the existence of the least energy sign-changing solutions for a class of p-Kirchhoff-type problems with a(2p-1)-linear growth nonlinearity. The quantitative deformation lemma and Non-Nehari manifold method are used in the paper to prove the main results. Remarkably, we use a new method to verify that Mb≠∅. The main results of our paper are the existence of the least energy sign-changing solution and its corresponding energy doubling property. Moreover, we also give the convergence property of the least energy sign-changing solution as the parameter b↘0.


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## Body

## 1. Introduction and the Main Results

In this paper, we are devoted to investigating the existence of the least energy sign-changing solutions for the following p-Kirchhoff-type problem with a (2p-1)-linear growth nonlinearity:(1)-a+b∫Ω∇updxΔpu=λup-2u+u2p-2u,x∈Ω,u=0,x∈∂Ω,where Ω is a bounded domain in RN(N=1,2,3), a,b>0, λ<aλ1, λ1 is the first eigenvalue of the following problem:(2)-Δpϕ=λϕp-2ϕ,x∈Ω,ϕ=0,x∈∂Ω.In fact, the related problems have been studied extensively, especially on the existence of the positive solutions, multiple solutions, ground state solutions, and least energy sign-changing solutions. In [1], Li and Sun studied the existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities; the mountain pass theorem was used in the paper. Guo, Ma, and Zhang [2] studied a class of autonomous Kirchhoff-type equation. By a simple transformation, they found that the solutions of autonomous Kirchhoff-type equation or system could be obtained by using the known solutions of the corresponding local equation or system, which is very interesting. In [3], Ying Li and Lin Li considered the existence and multiplicity of solutions to a class of p(x)-Laplacian-like equations. They introduced a revised Ambrosetti-Rabinowitz condition and obtained that the problem had a nontrivial solution and infinitely many solutions, respectively. Meanwhile, in [4], Luca Vilasi proved an eigenvalue theorem for a stationary p(x)-Kirchhoff problem by using variational techniques, and the author also provided an estimate for the range of such eigenvalues. For more details, we refer the reader to [5–30].In [31, 32], the authors studied the following Kirchhoff-type problems in bounded domains:(3)-a+b∫Ω∇u2Δu=fu,x∈Ω,u=0,x∈∂Ω,under different assumptions on f(u), the authors mainly use the quantitative deformation lemma and the degree theory to get the existence of the least energy sign-changing solution and its corresponding convergence property as the parameter b↘0. From the assumptions on f(u), we can easily find that both in [31, 32] f(u) satisfies 3-superlinear growth at infinity and superlinear growth at zero.Later, some scholars made some expanding work; we can find some details in [33]. In [33], we know that the nonlinearity f satisfies (2p-1)-superlinear growth condition at infinity.Motivated by the above works, a natural question is that if there exists a ground state sign-changing solution for problem (1). However, up to now, no paper has appeared in the literature which discusses the existence and convergence property of the solution for the p-Kirchhoff-type problem with a (2p-1)-linear growth nonlinearity. This paper attempts to fill this gap in the literature.Throughout this paper, we will make full use of the following notations. LetW=W01,p(Ω) be the usual Sobolev space equipped with the following norm:(4)u=∫Ω∇updx1/p.·s denotes the usual Lebesgue space Ls(Ω) norm. S is the best Sobolev constant for the embedding of W01,p(Ω) in L2p(Ω); that is,(5)u2p≤S-1/pu.From the above definition, we give the energy functional corresponding to problem (1) by(6)Ibu=ap∫Ω∇updx+b2p∫Ω∇updx2-λp∫Ωupdx-12p∫Ωu2pdx.Clearly, Ib is well defined on W and is of C1 class. For each u,υ∈W, by a simple calculation, we have(7)Ib′u,υ=a+b∫Ω∇updx∫Ω∇up-2∇u·∇υdx-∫Ωλup-2uυdx-∫Ωu2p-2uυdx.Obviously, the critical points of Ib are corresponding to the weak solutions of problem (1). If u∈W is a sign-changing solution of problem (1), then(i)u is a solution of problem (1), that is, u is a critical point of Ib;(ii)u±≠0, where u+=max{u(x),0},u-=min{u(x),0}.Foru=u++u-, from (6) and (7), we have(8)Ibu=Ibu++Ibu-+bp∇u+pp∇u-pp;(9)Ib′u,u+=Ib′u+,u++b∇u+pp∇u-pp;(10)Ib′u,u-=Ib′u-,u-+b∇u+pp∇u-pp.Whenb=0, problem (1) reduces to the following problem:(11)-aΔpu=λup-2u+u2p-2u,x∈Ω,u=0,x∈∂Ω.The corresponding energy functionalI0:W→R is defined by(12)I0u=ap∫Ω∇updx-λp∫Ωupdx-12p∫Ωu2pdx. Also, we can compute that(13)I0′u,υ=∫Ωa∇up-2∇u·∇υdx-∫Ωλup-2uυdx-∫Ωu2p-2uυdx.Forb>0, problem (1) is called a nonlocal problem since the appearance of the nonlocal term (∫Ω‍∇updx)Δpu. The differences posed by the nonlocal term make the method in solving problem (11) cannot be applied to solve problem (1), which makes the study of our paper very interesting and meaningful.In our paper, we restrictu in the following sets to find the ground state sign-changing solutions of (1) and (11),(14)Mb=u∈W:u±≠0,Ib′u,u+=Ib′u,u-=0,M0=u∈W:u±≠0,I0′u,u+=I0′u,u-=0,and we define mb=infu∈Mb⁡Ib(u) and m0=infu∈M0⁡I0(u).To get the ground state solutions, we define the following sets:(15)Nb=u∈W:u≠0,Ib′u,u=0,N0=u∈W:u≠0,I0′u,u=0and consider the following minimization problem:(16)cb=infu∈Nb⁡Ibu,c0=infu∈N0⁡I0u.SinceMb⊂Nb, we can immediately get mb≥cb. The main results of the paper are described as follows.Theorem 1.
Forb∈(0,1/2S2) and λ<aλ1, problem (1) has at least one ground state sign-changing solution, which precisely has two nodal domains. Moreover, mb>2cb.Theorem 2.
For eachλ<aλ1, for any sequence {bn} small enough with bn↘0 as n→∞, there exists a subsequence still denoted by {bn}, such that ubn convergent to u0 strongly in W01,p(Ω), where u0 is a ground state sign-changing solution of problem (11), which changes sign only once.Our paper is organized as follows. In Section2, some preliminary lemmas are given to prove the main results. In Sections 3 and 4, we are devoted to proving the main results of the paper.


## 2. Some Critical Preliminaries

The following several lemmas are crucial to prove our main results.Lemma 3.
Ifb>0, λ<aλ1, u∈W satisfies u±≠0 and(17)b∇u+p2p+b∇u+pp∇u-pp<∫Ωu+2pdx,b∇u-p2p+b∇u+pp∇u-pp<∫Ωu-2pdx,then there exists a unique pair (su,tu) of positive numbers such that
(i)suu++tuu-∈Mb;
(ii)Ib(suu++tuu-)=maxs,t≥0⁡Ib(su++tu-).Proof.
(i) Ifsu++tu-∈Mb, then from (7), (9), and (10), we have(18)Ib′su++tu-,su+=asp∇u+pp+bs2p∇u+p2p+bsptp∇u+pp∇u-pp-sp∫Ωλu+pdx-s2p∫Ωu+2pdx=0and(19)Ib′su++tu-,tu-=atp∇u-pp+bt2p∇u-p2p+bsptp∇u+pp∇u-pp-tp∫Ωλu-pdx-t2p∫Ωu-2pdx=0.Let S=sp and T=tp, the above equations correspond to the following system:(20)S∫Ωu+2pdx-bS∇u+p2p-bT∇u+pp∇u-pp=a∇u+pp-∫Ωλu+pdx,T∫Ωu-2pdx-bT∇u-p2p-bS∇u+pp∇u-pp=a∇u-pp-∫Ωλu-pdx.Obviously, if we can prove that system (20) has the unique solution (S,T), then (s=S1/p,t=T1/p) is the unique solution for (18) and (19).
Let(21)D=∫Ωu+2pdx-b∇u+p2p-b∇u+pp∇u-pp-b∇u+pp∇u-pp∫Ωu-2pdx-b∇u-p2p>0.For λ<aλ1, we have a∇u±pp>λ∫Ω‍u±p. Since ∫Ω‍u-2p-b∇u-p2p>0, then(22)DS=a∇u+pp-∫Ωλu+pdx-b∇u+pp∇u-ppa∇u-pp-∫Ωλu-pdx∫Ωu-2pdx-b∇u-p2p>0. Similarly, we have(23)DT=∫Ωu+2pdx-b∇u+p2pa∇u+pp-∫Ωλu+pdx-b∇u+pp∇u-ppa∇u-pp-∫Ωλu-pdx>0.From (21)-(23), we have S=DS/D>0,T=DT/D>0, and (S,T) is the unique solution for system (20). Accordingly, (s=S1/p,t=T1/p) is the unique positive solution for (18) and (19). Thus, (i) is proved.
(ii) Next, we give the proof of (ii).
From (6), we have(24)Ibsu++tu-=aspp∇u+pp+bs2p2p∇u+p2p-spp∫Ωλu+pdx-s2p2p∫Ωu+2pdx+atpp∇u-pp+bt2p2p∇u-p2p-tpp∫Ωλu-pdx-t2p2p∫Ωu-2pdx+bsptpp∇u+pp∇u-pp.By a simple computation, we have(25)∂2Ib∂s2=p-1sp-2a∇u+pp-λ∫Ωu+pdx+btp∇u+pp∇u-pp+2p-1s2p-2b∇u+p2p-∫Ωu+2pdx,and(26)∂2Ib∂t2=p-1tp-2a∇u-pp-λ∫Ωu-pdx+bsp∇u+pp∇u-pp+2p-1t2p-2b∇u-p2p-∫Ωu-2pdx.
Fromsuu++tuu-∈Mb, we have(27)∂2Ib∂s2su,tu=psu2p-2b∇u+p2p-∫Ωu+2pdx<0,∂2Ib∂t2su,tu=ptu2p-2b∇u-p2p-∫Ωu-2pdx<0,and(28)∂2Ib∂t∂ssu,tu=pbsup-1tup-1∇u+pp∇u-pp.We consider the Hessian matrix of Ib(su++tu-); then from (17), we have (29)Hsu,tu=∂2Ib∂s2∂2Ib∂s∂t∂2Ib∂s∂t∂2Ib∂t2=p2su2p-2tu2p-2∫Ωu+2pdx-b∇u+p2p∫Ωu-2pdx-b∇u-p2p-p2b2su2p-2tu2p-2∇u+p2p∇u-p2p>0.The above deduction implies that (su,tu) is a maximal point of Ib(su++tu-) for s,t≥0. Since we cannot get the maximal point of Ib on the boundary of R+, (su,tu) is the unique maximal point; that is, Ib(suu++tuu-)=maxs,t≥0Ib(su++tu-).Lemma 4.
Assume thatλ<aλ1 and u∈Mb, then (17) holds.Proof.
Foru=u++u-∈Mb, we have(30)Ib′u,u+=a∇u+pp+b∇upp∇u+pp-∫Ωλu+pdx-∫Ωu+2pdx=0,Ib′u,u-=a∇u-pp+b∇upp∇u-pp-∫Ωλu-pdx-∫Ωu-2pdx=0.Since λ<aλ1, we have a∇u±pp>∫Ω‍λu±pdx. Thus, we have(31)b∇u+p2p+b∇u+pp∇u-pp<∫Ωu+2pdxb∇u-p2p+b∇u+pp∇u-pp<∫Ωu-2pdx.Lemma 5.
Assume thatb>0,λ<aλ1, u∈W with u±≠0 and Ib′(u),u±≤0, there exists a unique pair (su,tu)∈(0,1]×(0,1] such that suu++tuu-∈Mb.Proof.
Ifu∈W with u±≠0 and Ib′u,u±≤0, we have(32)a∇u+pp+b∇upp∇u+pp≤∫Ωλu+pdx+∫Ωu+2pdxa∇u-pp+b∇upp∇u-pp≤∫Ωλu-pdx+∫Ωu-2pdx.Since a∇u±pp>λ∫Ω‍u±pdx, then(33)b∇upp∇u+pp<∫Ωu+2pdxb∇upp∇u-pp<∫Ωu-2pdx.From Lemma 3, there is a unique pair (su,tu) of positive numbers such that suu++tuu-∈Mb, which implies that (sup,tup) is the solution of system (20). Then, we have(34)Dsup=a∇u+pp-∫Ωλu+pdx∫Ωu-2pdx-b∇u-p2p+b∇u+pp∇u-ppa∇u-pp-∫Ωλu-pdx≤∫Ωu+2pdx-b∇upp∇u+pp∫Ωu-2pdx-b∇u-p2p+b∇u+pp∇u-pp∫Ωu-2pdx-b∇upp∇u-pp=∫Ωu+2pdx-b∇u+p2p∫Ωu-2pdx-b∇u-p2p-b2∇u+p2p∇u-p2p=D.Therefore, we have sup=Dsup/D≤1. Similarly, we have tup=Dtup/D≤1. Thus, there exists a unique pair (su,tu)∈(0,1]×(0,1] such that suu++tuu-∈Mb.Lemma 6.
Ifλ<aλ1 for any u∈W with b∇up2p<∫Ω‍u2pdx, there exists a unique su¯>0 such that su¯u∈Nb. Moreover, Ib(su¯u)>Ib(su) for all s≥0 and s≠su¯.Proof.
Ifλ<aλ1 and u∈W satisfies b∇up2p<∫Ω‍u2pdx, su∈Nb implies that(35)Ib′su,su=asp∇upp+bs2p∇up2p-sp∫Ωλupdx-s2p∫Ωu2pdx=0.Thus, there exists a unique su¯=(a∇upp-∫Ω‍λupdx/∫Ω‍u2pdx-b∇up2p)1/p>0 satisfying (35). From (6), we have(36)Ibsu=aspp∇upp+bs2p2p∇up2p-spp∫Ωλupdx-s2p2p∫Ωu2pdx.By a simple deduction, we have(37)∂2Ib∂s2su¯=p-1sp-2a∇upp-∫Ωλupdx+2p-1s2p-2b∇up2p-∫Ωu2pdxsu¯=-psu¯p-2a∇upp-∫Ωλupdx<0.Thus, Ib(su) attains its maximal point at s=su¯. In other words, we have Ib(su¯u)>Ib(su) for all s≥0 and s≠su¯.Lemma 7.
Assumeλ<aλ1; we have that
(i) if0<b<1/S2, cb>0 is attained by some vb∈Nb and vb is a constant sign critical point of Ib, where S is given by (5);
(ii) if0<b<1/2S2, mb>0 is attained by some ub∈Mb and ub is a sign-changing critical point of Ib.Proof.
(i) Firstly, we will show that for all0<b<1/S2, there exists u∈W such that b∇up2p<∫Ω‍u2pdx, which implies Nb≠∅. From (5), we know that there exists e1∈W such that e12p=S-1/pe1. For 0<b<1/S2, we have(38)b∇e1p2p<1S2∇e1p2p=∫Ωe12pdx.Thus, we have Nb≠∅.
For eachu∈Nb, it follows from λ<aλ1 and (5) that(39)a∇upp+b∇up2p=λ∫Ωupdx+∫Ωu2pdx≤λλ1∇upp+1S2∇up2p.
Then,∇upp≥a-λ/λ1/1/S2-b. Thus, we have(40)Ibu=Ibu-12pIb′u,u=12pa∇upp-∫Ωλupdx≥12pa-λλ1∇upp≥a-λ/λ12S22p1-bS2,that is, cb=infu∈Nb⁡Ib(u)≥(a-λ/λ1)2S2/2p(1-bS2)>0 and Ib is coercive and bounded below on Nb for 0<b<1/S2 and λ<aλ1.
Let{vn}⊂Nb be a minimizing sequence for Ib. From Ib(vn)=Ib(vn) and vn∈Nb, we assume that vn(x)≥0 in Ω for all n∈N. Since Ib is coercive and bounded below on Nb, the sequence {vn} is bounded in W, so that, up to subsequences, vn⇀vb in W and vb≥0. Next, we will prove that vn→vb strongly in W. We suppose by contradiction that vb<liminfn→∞⁡vn. Therefore, we have(41)a∇vbpp+b∇vbp2p<λ∫Ωvbpdx+∫Ωvb2pdx.If vb=0, the above inequality makes a contradiction. Thus, we have vb≠0 in Ω. From the fact that a∇vbpp>λ∫Ω‍vbpdx, we have b∇vbp2p<∫Ω‍vb2pdx. By Lemma 6, there exists a unique sv>0 such that svvb∈Nb and Ib(svvn)≤Ib(vn) for all vn∈Nb. Thus, we have(42)cb≤Ibsvvb=ap∇svvbpp+b2p∇svvbp2p-λp∫Ωsvvbpdx-12p∫Ωsvvb2pdx<liminfn→∞ap∇svvnpp+b2p∇svvnp2p-λp∫Ωsvvnpdx-12p∫Ωsvvn2pdx=liminfn→∞⁡Ibsvvn≤liminfn→∞⁡Ibvn=cb,which leads to a contradiction. Therefore, we have vb=liminfn→∞vn, vn→vb strongly in W and Ib(vb)=cb. Then, by a standard argument, which is similar to the discussion in [34], we can deduce that vb is a constant sign critical point of Ib.
(ii) From a similar deduction as (i), we know that for0<b<1/2S2, there exists u1∈W such that(43)b∇u1p2p<12S2∇u1p2p=12∫Ωu12pdx.Obviously, if u∈W such that u1 satisfies (43), then u1∈W also satisfies (43). Therefore, we assume that u1(x)≥0 a.e. in W. We let suppu1⊂Bρ(x0) and define u2(x)=-u1(-x) for all x∈Bρ(-x0), where Bρ(x0)={x∈Ω:x-x0<ρ} and ρ>0. Then, from (43), we have(44)∫Ωu12pdx∇u1p2p=∫Ωu22pdx∇u2p2p>2b.Let u=u1+u2; we can obtain that u∈W and u+=u1,u-=u2 and(45)b∇u+p2p+b∇u+pp∇u-pp<12∫Ωu12pdx+12u12ppu22pp=∫Ωu12pdx,that is,(46)b∇u+p2p+b∇u+pp∇u-pp<∫Ωu+2pdx.Similarly, we also have(47)b∇u-p2p+b∇u+pp∇u-pp<∫Ωu-2pdx.By Lemma 3, we know that Mb≠∅ for 0<b<1/2S2.
Assume that{un}⊂Mb is a minimizing sequence for Ib, such that Ib(un)→mb. Since Ib is coercive on Nb, the sequence {un} is bounded in W; going if necessary to a subsequence, still denoted by {un}, we can assume that there exists a ub∈W such that for n sufficiently large,(48)un±⇀ub±weaklyinW,unx→ubxalmosteverywhereonΩ,un±→ub±stronglyinLsΩforp≤s<p∗.From {un}⊂Mb, we have Ib′(un),un±=0; that is,(49)a∇un±pp+b∇unpp∇un±pp=λ∫Ωun±pdx+∫Ωun±2pdx.Therefore,(50)a∇un±pp≤λ∫Ωun±pdx+∫Ωun±2pdx.In the same way, we have (a-λ/λ1)∇un±pp≤1/S2∇un±p2p and ∇un±pp≥S2(a-λ/λ1)>0. Passing to the limit, we have(51)0<S2a-λλ12≤liminfn→∞⁡a-λλ1∇un±pp≤∫Ωub±2pdx,which implies that ub±≠0 and(52)a∇ub+pp+b∇ubpp∇ub+pp≤λ∫Ωub+pdx+∫Ωub+2pdx,a∇ub-pp+b∇ubpp∇ub-pp≤λ∫Ωub-pdx+∫Ωub-2pdx.From a∇ub±pp>λ∫Ω‍ub±pdx, Lemmas 3 and 5, there exists a unique pair (su,tu)∈(0,1]×(0,1] such that(53)suub++tuub-∈Mb.From the definition of mb, we have(54)mb≤Ibsuub++tuub-=Ibsuub++tuub--12pIb′suub++tuub-,suub++tuub-=12pa∇suub++tuub-pp-λ∫Ωsuub++tuub-pdx=12psupa∇ub+pp-λ∫Ωub+pdx+tupa∇ub-pp-λ∫Ωub-pdx≤12pa∇ub+pp-λ∫Ωub+pdx+a∇ub-pp-λ∫Ωub-pdx=12pa∇ubpp-λ∫Ωubpdx≤liminfn→∞⁡Ibun-12pIb′un,un=mb.Thus, su=tu=1, ub∈Mb, and Ib(ub)=mb, ub is the required minimizer.
Next, we will prove thatub is indeed a sign-changing solution; that is, Ib′(ub)=0. We mainly use the quantitative deformation lemma [35] to prove the results.
IfIb′(ub)≠0, there exists δ>0 and α>0, such that(55)u∈W,Ib′u≥α,∀u-ub≤3δ.Let D=(1/2,3/2)×(1/2,3/2), ψ(s,t)=sub++tub-, and (s,t)∈D. It follows from Lemma 3 that(56)mb¯=max∂DIb∘ψ<mb.Let ε=min{mb-mb¯/3,αδ/8} and Sδ={u∈W:u-ub≤δ}; there exists a deformation η∈C([0,1]×W,W) such that
(i)η(1,u)=u if u∉Ib-1([mb-2ε,mb+2ε])∩S2δ;
(ii)η(1,Ibmb+ε∩Sδ)⊂Ibmb-ε;
(iii)Ib(η(1,u))≤Ib(u),∀u∈W.
From (56), Lemma 3 and (ii), we can easily get(57)maxs,t∈D¯Ibη1,ψs,t<mb.We prove that η(1,ψ(D))∩Mb≠∅, which contradicts the definition of mb. We define g(s,t)=η(1,ψ(s,t)) and(58)Φ0s,t=Ib′ψs,tub+,Ib′ψs,tub-=Ib′sub++tub-ub+,Ib′sub++tub-ub-,Φ1s,t=1sIb′gs,tg+s,t,1tIb′gs,tg-s,t.Lemma 3 and the degree theory yield deg(Φ0,D,0)=1. From (56), we know that g=ψ on ∂D. Consequently, we have deg(Φ0,D,0)=deg(Φ1,D,0)=1. Therefore, Φ1(s0,t0)=0 for some (s0,t0)∈D; that is, η(1,ψ(s0,t0))=g(s0,t0)∈Mb, which is a contradiction. From this point, ub is a sign-changing critical point of Ib and Ib′(ub)=0.


## 3. The Existence of the Sign-Changing Solutions

In this part, we are devoted to proving Theorem1.Proof of Theorem1.
In view of Lemma7, we know that for 0<b<1/2S2 and λ<aλ1, there exists a ub∈Mb such that mb=Ib(ub) and Ib′(ub)=0; that is, ub is a ground state sign-changing solution for problem (1). Then by Lemma 4, we have that(59)b∇ub+p2p+b∇ub+pp∇ub-pp<∫Ωub+2pdxb∇ub-p2p+b∇ub+pp∇ub-pp<∫Ωub-2pdx,which implies(60)b∇ub+p2p<∫Ωub+2pdxb∇ub-p2p<∫Ωub-2pdx.Then from Lemma 6, there exist s1,t1>0 such that s1ub+,t1ub-∈Nb. Therefore, we have(61)mb=Ibub≥Ibs1ub++t1ub-=Ibs1ub++Ibt1ub-+bs1pt1pp∇ub+pp∇ub-pp>Ibs1ub++Ibt1ub-≥2cb.Therefore, the energy doubling property is proved.
Next, we prove thatub changes sign only once; that is, ub has exactly two nodal domains. We assume by contradiction that ub=u1+u2+u3 with ui≠0,u1≥0,u2≤0,u3≥0 and supp(ui)∩supp(uj)=∅ for i≠j,(i,j=1,2,3).
SinceIb′(ub)=0, we can get(62)Ib′u1+u2,u1=Ib′ub,u1-b∇u1pp∇u3pp<0,Ib′u1+u2,u2=Ib′ub,u2-b∇u2pp∇u3pp<0.Then, by Lemma 5, there exists a pair (s′,t′)∈(0,1]×(0,1] such that s′u1+t′u2∈Mb and Ib(s′u1+t′u2)≥mb.
Fromλ<aλ1, Ib′(ub),ub=0, and Ib′(s′u1+t′u2),s′u1+t′u2=0, we have(63)mb=Ibub-12pIb′ub,ub=12pa∇ubpp-λ∫Ωubpdx=12pa∇u1pp-λ∫Ωu1pdx+a∇u2pp-λ∫Ωu2pdx+a∇u3pp-λ∫Ωu3pdx>12pa∇u1pp-λ∫Ωu1pdx+a∇u2pp-λ∫Ωu2pdx≥12ps′pa∇u1pp-λ∫Ωu1pdx+t′pa∇u2pp-λ∫Ωu2pdx=Ibs′u1+t′u2-12pIb′s′u1+t′u2,s′u1+t′u2=Ibs′u1+t′u2≥mb,which leads to a contradiction; thus, ub has exactly two nodal domains.


## 4. The Convergence Property ofub as b↘0

In this part, we regardb>0(b∈(0,1/2S2)) as a small parameter in (1) and discuss the convergence property of the least energy sign-changing solution ub, where ub∈Mb and ub changes sign only once.Proof of Theorem2.
We choose a nonzero functionw0∈C0∞(Ω) and γ>0 such that w0±≠0 and(64)a∇w0+pp+γ∇w0pp∇w0+pp≤λ∫Ωw0+pdx+∫Ωw0+2pdxa∇w0-pp+γ∇w0pp∇w0-pp≤λ∫Ωw0-pdx+∫Ωw0-2pdx.Thus, for any b∈[0,γ], we have Ib′(w0),w0±≤0. It follows from Lemma 5 that there is a unique pair (sb,tb)∈(0,1]×(0,1] such that sbw0++tbw0-∈Mb. Thus, we have(65)Ibsbw0++tbw0-=Ibsbw0++tbw0--12pIb′sbw0++tbw0-,sbw0++tbw0-=12pa∇sbw0++tbw0-pp-λ∫Ωsbw0++tbw0-pdx<a2p∇sbw0++tbw0-pp≤a2pw0p=Θ.For any sequence {bn} with bn↘0 as n→∞, there exists ubn∈Mb such that ubn is a ground state sign-changing critical point of Ibn(u) and(66)Θ+1≥Ibnubn-12pIbn′ubn,ubn=12pa∇ubnpp-λ∫Ωubnpdx≥12pa-λλ1∇ubnpp.The above inequality shows that ubn is bounded in W; then there exists a subsequence of {bn}, still denoted by {bn}, such that ubn⇀u0 weakly in W. By the compactness of the embedding W↪Ls(Ω) for p≤s<p∗, using a standard argument, we have that ubn±→u0± in W and u0±≠0. Moreover, we have that for all u∈W,(67)0=limn→∞Ibn′ubn,u=limn→∞∫Ωa∇ubnp-2∇ubn∇udx+bn∫Ω∇ubnpdx∫Ω∇ubnp-2∇ubn∇udx-λ∫Ωubnp-2ubnudx-∫Ωubn2p-2ubnudx=∫Ωa∇u0p-2∇u0∇udx-λ∫Ωu0p-2u0udx-∫Ωu02p-2u0udx=I0′u0,u,which implies that(68)I0′u0=0,u0∈M0,I0u0≥m0.
Secondly, in the Proof of Theorem1, b=0 is allowed; then, there exists a ν0∈M0 such that(69)I0ν0=m0=infu∈M0⁡I0u,and ν0 is a sign-changing solution for (11) which changes sign only once. Similarly, we can pick up ε>0 which is independent of bn such that(70)ε∇ν0+p2p+ε∇ν0+pp∇ν0-pp<∫Ων0+2pdxε∇ν0-p2p+ε∇ν0+pp∇ν0-pp<∫Ων0-2pdx.According to Lemma 3, there exists a unique pair (s0,t0) of positive numbers such that s0ν0++t0ν0-∈Mε.
Letbn∈[0,ε]; we know that(71)Ibn′s0ν0++t0ν0-,s0ν0+=a∇s0ν0+pp+bn∇s0ν0+p2p+bn∇s0ν0+pp∇t0ν0-pp-λ∫Ωs0ν0+pdx-∫Ωs0ν0+2pdx≤a∇s0ν0+pp+ε∇s0ν0+p2p+ε∇s0ν0+pp∇t0ν0-pp-λ∫Ωs0ν0+pdx-∫Ωs0ν0+2pdx=Iε′s0ν0++t0ν0-,s0ν0+=0.In the same way, we can obtain that(72)Ibn′s0ν0++t0ν0-,t0ν0-≤Iε′s0ν0++t0ν0-,t0ν0-=0.It follows from Lemma 5 that for all bn∈[0,ε], there exists a unique pair (sn,tn)∈(0,s0]×(0,t0] such that snν0++tnν0-∈Mbn. Then, for any sequence {bn} with bn↘0 as n→∞, we have as n→∞,(73)bnsn2p∇ν0+p2p→0,bnsnptnp∇ν0+pp∇ν0-pp→0,bntn2p∇ν0-p2p→0.According to Ibn′(snν0++tnν0-),snν0+=Ibn′(snν0++tnν0-),tnν0-=0, we have(74)a∇ν0+pp+∘1=λ∫Ων0+pdx+snp∫Ων0+2pdxa∇ν0-pp+∘1=λ∫Ων0-pdx+tnp∫Ων0-2pdx. From I0′(ν0),ν0±=0, we have(75)a∇ν0+pp=λ∫Ων0+pdx+∫Ων0+2pdxa∇ν0-pp=λ∫Ων0-pdx+∫Ων0-2pdx. Combining (74) with (75), we have that as n→∞, sn→1, tn→1. Lastly, we only need to show I0(u0)=I0(ν0); then by (68), u0 is a ground state sign-changing solution for (11), which changes sign only once. In fact,(76)I0ν0≤I0u0=limn→∞⁡Ibnubn≤limn→∞⁡Ibnsnν0++tnν0-=I0ν0++ν0-=I0ν0.Then, the proof of Theorem 2 is complete.



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*Source: 3713909-2019-07-07.xml*