# The Existence of the Sign-Changing Solutions for the Kirchhoff-Schrödinger-Poisson System in Bounded Domains

Authors: Cun-bin An; Jiangyan Yao; Wei Han
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solutionu0 is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of u0 as the parameters b↘0 and λ↘0.


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## Body

## 1. Introduction and the Main Results

In this paper, the following Kirchhoff-Schrödinger-Poisson system is considered:(1)−a+b∫Ω∇u2dxΔu+λϕx=gu,x∈Ω−Δϕ=u2,x∈Ω,u=ϕ=0,x∈∂Ω,where Ω⊂ℝ3 is a bounded domain with a smooth boundary ∂Ω, a,b,λ∈ℝ+=0,+∞, and g∈Cℝ,ℝ satisfies some basic assumptions.Forb=0, problem (1) reduces to the following Schrödinger-Poisson system:
(2)−aΔu+λϕxu=gu,x∈Ω,−Δϕ=u2,x∈Ω,ϕ,u=0,x∈∂Ω.Alves and Souto [1] studied the above Schrödinger-Poisson system for a=λ=1. Under some suitable assumptions on the nonlinearity gu, by using the deformation lemma and Brouwer’s topological degree theory, they proved that the above system possessed a least-energy sign-changing solution, which changed sign only once.Forλ=0, the problem (1) reduces to the following problem:
(3)−a+b∫Ω∇u2dxΔu=gu,x∈Ωu=0,x∈∂Ω.The problem (3) has been studied in [2, 3]. Under different assumptions on gu, the authors in [2, 3] obtained the existence and some qualitative properties of the sign-changing solution by using the Non-Nehari manifold method and deformation lemma. We can find that the results in [3] improve and generalize the results in [2]. In fact, the studies about the existence of the positive solutions, sign-changing solutions for a class of elliptic equations, have been studied extensively. For more details about such problems, we refer the reader to [4–19].To our best knowledge, the results of the sign-changing solutions for the Kirchhoff-Schrödinger-Poisson system under a weak assumption thatg∈Cℝ,ℝ have not been studied yet. This paper attempts to fill this gap in the literature. Motivated by the above papers, we study the problem (1). For the purpose of getting the results, we use the variational method and some mathematical skills to obtain the existence of the sign-changing solution and its corresponding properties.In this paper, we assumeg∈Cℝ,ℝ satisfies the following four conditions:g1lims→0gs/s=0g2lims→∞gs/s3=∞g3gs<c01+s4,∀s∈ℝ, where c0 is a constantg4 there exists a θ0∈0,1 such that for any s>0 and τ∈ℝ\0,
(4)gττ3−gsτsτ3sign1−s+aθ0λ11−s2sτ2≥0,where λ1 is the first eigenvalue for the following problem:
(5)−Δu=λu,x∈Ω,u=0,x∈∂Ω.Throughout this paper, we will use the following notations.LetH=H01Ω be the usual Sobolev space equipped with the following norm:
(6)u=∫Ω∇u2dx1/2.The usualLp norm is denoted by up=∫Ωupdx1/p. In this way, we know u=∇u2.For the Poisson system,(7)−Δϕ=u2,x∈Ω,ϕ=0,x∈∂Ω,where there exists a unique ϕu=1/4π∫Ωu2y/x−ydy∈H , such that ϕu satisfies the above system. It is known that ϕu satisfies the following conditions [17–19]:
(i)
∫Ωϕuu2dx=∫Ω∇ϕu2dx≤C∇u24(ii)
ϕu≥0 and ϕu>0 for u≠0(iii)
foru=u++u−∈H,ϕu++u−=ϕu++ϕu− and for t≠0,ϕtu=t2ϕu(iv)
ifun⇀u in H01Ω, then ϕun⇀ϕu in H01Ω and limn→∞∫Ωϕunun2dx=∫Ωϕuu2dx; if un±⇀u± in H01Ω, then liminfn→∞∫Ωϕunun±2dx=∫Ωϕuu±2dx.Consequently,u,ϕ∈H01Ω×H01Ω is a solution of problem (1), that is, ϕ=ϕu and u∈H01Ω are a solution of the following problem:
(8)−a+b∫Ω∇u2dxΔu+λϕuu=gu,x∈Ω,u=0,x∈∂Ω.In this paper,u∈H01Ω is called a solution for problem (1), which implies u,ϕu∈H01Ω×H01Ω is a solution of problem (1).Next, we can define the energy functional corresponding to problem (1) F:H01Ω⟶ℝ by
(9)Fu=a2∫Ω∇u2dx+b4∫Ω∇u2dx2+λ4∫Ωϕuu2dx−∫ΩGudx,where Gs=∫0sgtdt. Obviously, the functional F is well defined and belongs to C1H,ℝ. By a simple computation, we have that for any u,φ∈H01Ω,
(10)F′u,φ=∫Ωa∇u∇φdx+b∫Ω∇u2dx∫Ω∇u∇φdx+∫Ωλϕuuφdx−∫Ωguφdx.It is clear that the critical points ofF are the weak solutions for the problem (1). If u∈H is called a sign-changing solution of problem (1), then u±≠0 and for any φ∈H,F′u,φ=0, where u+x=maxux,0,u−x=minux,0.Foru∈H and u=u++u−, by (9) and (10), we have
(11)Fu=Fu++Fu−+b2∇u+22∇u−22+λ4∫Ωϕu+u−2dx+λ4∫Ωϕu−u+2dx,(12)F′u,u+=F′u+,u++b∇u+22∇u−22+∫Ωλϕu−u+2dx,(13)F′u,u−=F′u−,u−+b∇u+22∇u−22+∫Ωλϕu+u−2dx.To get the main results, we restrictu in the following sets:
(14)M=u∈H:u±≠0,F′u,u+=0,F′u,u−=0,N=u∈H:u≠0,F′u,u=0.To get the energy doubling property, we definem=infu∈MFu and c=infu∈NFu.To prove the convergence property, we give the following definitions. Firstly, we define the energy functional corresponding to (2) Fb0:H01Ω⟶ℝ by
(15)Fb0u=a2∫Ω∇u2dx+λ4∫Ωϕuu2dx−∫ΩGudx.Similarly, we have(16)Fb0′u,φ=∫Ωa∇u∇φdx+∫Ωλϕuuφdx−∫Ωguφdx.The setMb0 is defined by Mb0=u∈H:u±≠0,Fb0′u,u+=Fb0′u,u−=0.The energy functionalFλ0:H01Ω⟶ℝ corresponding to (3) can be defined by
(17)Fλ0u=a2∫Ω∇u2dx+b4∫Ω∇u2dx2−∫ΩGudx.Also, we can compute that(18)Fλ0′u,φ=∫Ωa∇u∇φdx+b∫Ω∇u2dx∫Ω∇u∇φdx−∫Ωguφdx.To seek the sign-changing solution of (3), we define the set
(19)Mλ0=u∈H:u±≠0,Fλ0′u,u+=Fλ0′u,u−=0.The main results of this paper are described as follows.Theorem 1.
Assume thatg1−g4 hold, then problem (1) possesses a least-energy sign-changing solution u0∈M such that Fu0=infMF>0, which changes sign only once.Theorem 2.
Assume thatg1−g4 hold. Then problem (1) possesses a solution u1∈N such that Fu1=infNF. Moreover, m>2c.Theorem 3.
Assume thatg1−g4 hold. Then problem (2) possesses a sign-changing solution v0∈Mb0 such that Fb0v0=infMb0Fb0>0, which changes sign only once. Moreover, for any sequence bn with bn↘0asn⟶∞, there exists a subsequence of ubn, still denoted by ubn, such that ubn⟶ub0inH01Ω, where ub0∈Mb0 is a sign-changing solution of problem (2) with Fb0ub0=infMb0Fb0>0.Theorem 4.
Assume thatg1−g4 hold. Then problem (3) has a sign-changing solution w0∈Mλ0such thatFλ0w0=infMλ0Fλ0>0, which changes sign only once. Moreover, for any sequence λnwithλn↘0asn⟶∞, there exists a subsequence of uλn, still denoted byuλn, such thatuλn⟶uλ0inH01Ω, whereuλ0∈Mλ0is a sign-changing solution of (3) with Fλ0uλ0=infMλ0Fλ0>0.The rest of the paper is organized as follows. In Section2, we will give several estimates. In Section 3, some critical lemmas are proved. In Section 4, we will give the proof of the existence of the least-energy sign-changing solution. In section 5, the energy doubling property is proved. Section 6 is devoted to proving the convergence property.


## 2. Several Estimates

Lemma 5.
If the assumptionsg1−g4 hold, then
(20)Fu≥Fsu++tu−+1−s44F′u,u++1−t44F′u,u−+a1−s221−θ04∇u+22+a1−t221−θ04∇u−22+bs2−t224∇u+22∇u−22+λ4∫Ωs2−t2s2ϕu−u+2+t2−s2t2ϕu+u−2dx,∀u=u++u−∈H,s,t≥0.Proof.
According tog4, we can deduce that
(21)1−t44gττ+Gtτ−Gτ+aθ0λ1τ21−t224=∫t1gττ3−gsτsτ3+aθ0λ11−s2sτ2s3τ4ds≥0,∀t≥0,τ∈ℝ\0.
From (9), (12), (13), and (21), we have
(22)Fu−Fsu++tu−=a2∇u22−s∇u++t∇u−22+b4∇u24−s∇u++t∇u−24+λ4∫Ωϕuu2−ϕsu++tu−su++tu−2dx+∫ΩGsu++tu−−Gudx=1−s44a∇u+22+b∇u22∇u+22+∫Ωλϕuu+2dx−∫Ωgu+u+dx+1−t44a∇u−22+b∇u22∇u−22+∫Ωλϕuu−2dx−∫Ωgu−u−dx+a1−s224∇u+22+a1−t224∇u−22+bs2−t224∇u+22∇u−22+λ4∫Ωs2−t2s2ϕu−u+2+t2−s2t2ϕu+u−2dx+∫Ω1−s44gu+u+−Gu++Gsu+dx+∫Ω1−t44gu−u−−Gu−+Gtu−dx≥1−s44F′u,u++1−t44F′u,u−+bs2−t224∇u+22∇u−22+a1−s221−θ04∇u+22+a1−t221−θ04∇u−22+λ4∫Ωs2−t2s2ϕu−u+2+t2−s2t2ϕu+u−2dx+∫Ω1−s44gu+u++Gsu+−Gu++a1−s22θ0λ14u+2dx+∫Ω1−t44gu−u−+Gtu−−Gu−+a1−t22θ0λ14u−2dx≥1−s44F′u,u++1−t44F′u,u−+bs2−t224∇u+22∇u−22+a1−s221−θ04∇u+22+a1−t221−θ04∇u−22+λ4∫Ωs2−t2s2ϕu−u+2+t2−s2t2ϕu+u−2dx.
The above inequality implies that (20) holds.Corollary 6.
If the assumptionsg1−g4 hold and u=u++u−∈M, then
(23)Fu≥Fsu++tu−+a1−s221−θ04∇u+22+a1−t221−θ04∇u−22+bs2−t224∇u+22∇u−22+λ4∫Ωs2−t2s2ϕu−u+2+t2−s2t2ϕu+u−2dx,∀s,t≥0.
Fromu∈M, we have <F′u,u+>=<F′u,u−>=0. Therefore, we can immediately get the above conclusion by (20).
(24)Fu++u−=maxs,t≥0Fsu++tu−.Lemma 7.
Assume thatg4 holds, then
(25)14gττ−Gτ+aθ0λ14τ2≥0,∀τ∈ℝ.
We can get (25) by taking t=0 in (21).Lemma 8.
If the assumptionsg1−g4 hold, then for any u∈H, we have
(26)Fu≥Ftu+1−t44<F′u,u>+a1−θ01−t224∥∇u∥22,∀t≥0.
We can get the conclusion by a similar deduction as Lemma5Corollary 9.
If the assumptionsg1−g4 hold and u∈N, then
(27)Fu≥Ftu+a1−θ01−t224∥∇u∥22,Fu=maxt≥0Ftu,∀t≥0.


## 3. Some Critical Preliminaries

Lemma 10.
If the assumptionsg1−g4 hold and u∈H with u±≠0, then there exists a unique pair su,tu of positive numbers such that suu++tuu−∈M.Proof.
From the definition of the setM, suu++tuu−∈M implies that <F′suu++tuu−,suu+>=<F′suu++tuu−,tuu−>=0. Thus, we assume
(28)f1s,t=<F′su++tu−,su+>=as2∥∇u+∥22+bs4∥∇u+∥24+bs2t2∥∇u+∥22∥∇u−∥22+s4∫Ωλϕu+u+2dx+s2t2∫Ωλϕu−u+2dx−∫Ωgsu+su+dx,(29)f2s,t=<F′su++tu−,tu−>=at2∥∇u−∥22+bt4∥∇u−∥24+bs2t2∥∇u+∥22∥∇u−∥22+t4∫Ωλϕu−u−2dx+s2t2∫Ωλϕu+u−2dx−∫Ωgtu−tu−dx.
If there is a unique pairs,t of positive numbers such that g1s,t=g2s,t=0, then Lemma 10 holds. Next, we will give the detailed proof.
Byg1 and g2, we have that f1s,s>0,f2s,s>0 for s>0 small enough and f1t,t<0,f2t,t<0 for t>0 large enough. Thus, there exists 0<α<β such that
(30)f1α,α>0,f2α,α>0,f1β,β<0,f2β,β<0.
According to (28) and (29), it is clear that f1s,t is increasing on t for fixed s and f2s,t is increasing on s for fixed t. Thus, combining (30) we have
(31)f1α,t>0,f1β,t<0,∀t∈α,β,(32)f2s,α>0,f2s,β<0,∀s∈α,β.
Miranda’s theorem [20] implies that there exists some point su,tu such that f1su,tu=f2su,tu=0, where su,tu∈α,β. Therefore, there exists a positive pair of numbers su,tu such that suu++tuu−∈M.
Next, we will prove thatsu,tu is unique for (28) and (29). Let s1,t1 and s2,t2 be such that siu++tiu−∈M. From Corollary 6, we have
(33)Fs1u++t1u−≥Fs2s1s1u++t2t1t1u−,Fs2u++t2u−≥Fs1s2s2u++t1t2t2u−.
The above two inequalities implies thatFs1u++t1u−=Fs2u++t2u−, that is, s1,t1=s2,t2. The uniqueness is proved. Therefore, Lemma 10 holds.Corollary 11.
If the assumptionsg1−g4 hold and u∈H01Ω\0, then there exists a unique tu>0 such that tuu∈N.Lemma 12.
If the assumptionsg1−g4 hold, then
(34)infu∈MFu=m=infu∈H,u±≠0maxs,t≥0Fsu++tu−.Proof.
Firstly, by Corollary6, one has
(35)infu∈H,u±≠0maxs,t≥0Fsu++tu−≤infu∈Mmaxs,t≥0Fsu++tu−=infu∈MFu=m.
Secondly, for anyu∈H with u±≠0, it follows from Lemma 10 that
(36)maxs,t≥0Fsu++tu−≥Fsuu++tuu−≥infv∈MFv=m.
Combining (35) and (36), we can get Lemma 12.Lemma 13.
If the assumptionsg1−g4 hold, then m>0 can be achieved.Proof.
For allu∈M, we have <F′u,u>=0. According to g1,g3, and the Sobolev embedding theorem, we can get
(37)a∥∇u∥22≤a∥∇u∥22+b∥∇u∥24+λ∫Ωϕuu2dx=∫Ωguudx≤aλ12∥u∥22+c∥u∥55≤a2∥∇u∥22+c5∥∇u∥25,where c and c5 are positive constants. Thus we have ∥∇u∥22≥a/2c52/3>0. Therefore, by (9), (10), (25), and (37), we have
(38)Fu=Fu−14<F′u,u>=a4∥∇u∥22+∫Ω14guu−Gudx≥a4∥∇u∥22−aθ0λ14∥u∥22≥a1−θ04∥∇u∥22≥a1−θ04a2c52/3>0.
Sinceθ0∈0,1, thus, for any u∈M,Fu>0 and m>0. Let un⊂M be such that Fun⟶m. For large n∈ℕ, one has
(39)m+1≥Fun−14<F′un,un>≥a1−θ04∥∇un∥22.
Thus,un is bounded in H01Ω for θ0∈0,1, then there exists u0∈H such that un±⇀u0± in H. From un∈M, we have <F′un,un±>=0, that is
(40)a∥∇un±∥22+b∥∇un∥22∥∇un±∥22+∫Ωλϕunun±2dx=∫Ωgun±un±dx.
By a similar deduction as (37), we have ∥∇un∥22≥μ for all n∈ℕ. From g1 and g3, for any ε>0, there exists cε>0 such that
(41)gss≤εs2+cεs5,∀s∈ℝ.
Thus,μ≤∥∇un±∥22≤1/a∫Ωgun±un±dx≤ε/a∥un±∥22+cε/a∥un±∥55. Since un is bounded in H, there is c1>0 such that ∫Ωun±2dx≤c1, which implies
(42)μ≤εc1a+cεa∥un±∥55.
Chooseε=aμ/2c1, then μ/2≤cε/a∥un±∥55, that is ∥un±∥55≥aμ/2cε. By the compactness of the embedding H01Ω°LqΩ for q∈2,6, we have
(43)∫Ωu0±5dx≥aμ2cε.
Thusu0±≠0. By g1,g3, and the compactness lemma of Strauss [21], we have
(44)limn→∞∫Ωgun±un±dx=∫Ωgu0±u0±dx,(45)limn→∞∫ΩGun±dx=∫ΩGu0±dx.
Also, according to the properties of the solution for the Poisson system, we have(46)liminfn→∞∫Ωϕunun±2dx=∫Ωϕu0u0±2dx.
By the weak semicontinuity of norm, we have(47)a∥∇u0±∥22+b∥∇u0∥22∥∇u0±∥22≤liminfn→∞a∥∇un±∥22+b∥∇un∥22∥∇un±∥22.
From (40), (44), (46), and (47), we have
(48)a∥∇u0±∥22+b∥∇u0∥22∥∇u0±∥22+∫Ωϕu0u0±2dx≤liminfn→∞∫Ωgun±un±dx=∫Ωgu0±u0±dx,that is,
(49)<F′u0,u0±>≤0.
Since∥∇u∥22−θ0λ1∥u∥22≥1−θ0∥∇u∥22, for all u∈H, by (9), (10), (20), and (25), the weak semicontinuity of norm, Fatou’s lemma, and Lemma 12, we can get
(50)m=limn→∞Fun−14<F′un,un>=limn→∞a4∥∇un∥22+∫Ω14gunun−Gundx≥a4liminfn→∞∥∇un∥22−θ0λ1∥un∥22+liminfn→∞∫Ω14gunun−Gun+aθ0λ14un2dx≥a4∥∇u0∥22−θ0λ1∥u0∥22+∫Ω14gu0u0−Gu0+aθ0λ14u02=a4∥∇u0∥22+∫Ω14gu0u0−Gu0dx=Fu0−14<F′u0,u0>≥sups,t≥0Fsu0++tu0−+1−s44<F′u0,u0+>+1−t44<F′u0,u0−>−14<F′u0,u0>=sups,t≥0Fsu0++tu0−−s44<F′u0,u0+>−t44<F′u0,u0−>≥maxs,t≥0Fsu0++tu0−≥m.
Thus,limn→∞∥∇un∥22−θ0λ1∥un∥22=∥∇u0∥22−θ0λ1∥u0∥22. Consequently, un⟶u0 in H01Ω and Fu0=m,u0∈M.Corollary 14.
Assume thatg1−g4 hold. Then
(51)infu∈NFu=c=infu∈H,u≠0maxt≥0Ftuand c>0.Lemma 15 (See for example [3]).
Assume thatg1−g4 hold. Then there exists a constant c∗∈0,c and a sequence un⊂E satisfying
(52)Fun⟶c∗,∥F′un∥1+∥un∥⟶0.Lemma 16.
If the assumptionsg1−g4 hold and u0∈MwithFu0=m,thenu0 is a critical point ofF.Proof.
ForF′u0≠0, there exist σ>0 and ρ>0 such that
(53)u∈H01Ω,u−u0≤3σ⟹∥F′u∥≥ρ.
From (23), we have that
(54)Fsu0++tu0−≤Fu0−a1−θ01−s224∥∇u0+∥22−a1−θ01−t224∥∇u0−∥22=m−a1−θ01−s224∥∇u0+∥22−a1−θ01−t224∥∇u0−∥22∀s,t≥0.
LetD=1/2,3/2×1/2,3/2,gs,t=su0++tu0−. It follows from (54) that
(55)χ=maxs,t∈∂DIsu0++tu0−<m.
Forε=minm−χ/3,1,ρσ/8,S=Bu0,σ, [22]. Lemma 7 yields a deformation η∈C0,1×H,H such that
(i)
η1,u=u, if u∉F−1m−2ε,m+2ε⋂S2σ(ii)
η1,Fm+ε⋂Bu0,σ⊂Fm−ε(iii)
Fη1,u≤Fu,∀u∈H01Ω.
From Corollary6, we have Fsu0++tu0−≤Fu0=m for s,t≥0. For s,t≥0,s−12+t−12<σ2/∥u0∥2, we know su0++tu0−∈Fm+ε⋂Bu0,σ, then it follows from (ii) that Fsu0++tu0−≤m−ε.
According to (iii) and (54), we have that
(56)Fη1,su0++tu0−≤Fsu0++tu0−≤m−a1−θ01−s224∥∇u0+∥22−a1−θ01−t224∥∇u0−∥22≤m−a1−θ04min∥∇u0+∥22,∥∇u0−∥221−s22+1−t22≤m−a1−θ0σ24∥u0∥2min∥∇u0+∥22,∥∇u0−∥22,∀s,t≥0,s−12+t−12≥σ2/∥u0∥2.
Thus,maxs,t∈D¯Fη1,gs,t<m.
Next, we prove thatη1,gD⋂M≠∅, which contradicts to the definition of m. Let us define hs,t=η1,gs,t and
(57)Ψ0s,t=F′gs,tu0+,F′gs,tu0−=F′su0++tu0−u0+,F′su0++tu0−u0−,Ψ1s,t=1sF′hs,th+s,t,1tF′hs,th−s,t.
Lemma10 and the degree theory yields degΨ0s,t,D,0=1. By (55), we can deduce that g=h on ∂Ω. Consequently, degΨ1,D,0=degΨ0,D,0=1. Therefore, we have Ψ1s0,t0=0 for some s0,t0∈D, so that η1,gs0,t0=hs0,t0∈M, which is a contradiction. Thus, (53) does not hold. In other words, u0 is a critical point of F, that is, u0 is a sign-changing solution for problem (1).


## 4. The Existence Result of the Sign-Changing Solutions

In this section, we mainly give the proof of Theorem17.Proof of Theorem 17.
By Lemma13 and Lemma 16, there is a u0∈M such that Fu0=m and F′u0=0. Therefore, u0 is exactly a sign-changing solution of problem (1). Now, we prove that u0 changes sign only once.
We assume by contradiction thatu0=u1+u2+u3, where ui≠0,u1≥0,u2≤0 and suppui⋂suppuj=∅,i≠ji,j=1,2,3.
Letv=u1+u2, then v+=u1,v−=u2, and v±≠0. Note that <F′u0,v+>=0 and <F′u0,v−>=0, we have
(58)<F′v,v+>=−b∥∇u3∥22∥∇v+∥22−λ∫Ωϕu3v+2dx,(59)<F′v,v−>=−b∥∇u3∥22∥∇v−∥22−λ∫Ωϕu3v−2dx.
From (9)–(13), (23), (25), (58), and (59), we have
(60)m=Fu0−14<F′u0,u0>=Fv+Fu3+b2∥∇v∥22∥∇u3∥22+λ4∫Ωϕvu32+ϕu3v2dx−14<F′v,v>+<F′u3,u3>+2b∥∇v∥22∥∇u3∥22+λ∫Ωϕvu32+ϕu3v2dx=Fv+Fu3−14<F′v,v>−14<F′u3,u3>≥sups,t≥0Fsv++tv−+1−s44<F′v,v+>+1−t44<F′v,v−>+Fu3−14<F′v,v>−14<F′u3,u3>≥sups,t≥0Fsv++tv−+s44b∥∇u3∥22∥∇v+∥22+λ∫Ωϕu3v+2dx+t44b∥∇u3∥22∥∇v−∥22+λ∫Ωϕu3v−2dx+a4∥∇u3∥22+∫Ω14gu3u3−Gu3dx≥maxs,t≥0Fsv++tv−+a1−θ04∥∇u3∥22≥m+a1−θ04∥∇u3∥22.
Sinceθ0∈0,1, we have u3=0. Therefore, u0 changes sign only once.


## 5. Energy Doubling Property

Under the above preparations, we give the proof of Theorem18.Proof of Theorem 18.
By Lemma15, we know that there exists a sequence un⊂E satisfying (52), that is,
(61)Fun⟶c∗,<F′un,un>=0.
According to (9), (10), (25), and (61), we have for large n∈ℕ(62)c∗+1≥Fun−14<F′un,un>≥a1−θ04∥∇un∥22,which shows that un is bounded in H01Ω for θ0∈0,1. By a standard argument, we can prove that there exists a u1∈H01Ω\0 such that F′u1=0. This suggests that u1∈N is a nontrivial solution for problem (1) and Fu1≥c. On the other hand, by using (9), (10), and (25), the weak semicontinuity of norm and Fatou’s lemma, we have
(63)c≥c∗=limn→∞Fun−14<F′un,un>=limn→∞a4∥∇un∥22+∫Ω14gunun−Gundx≥a4liminfn→∞∥∇un∥22+θ0λ1∥un∥22+liminfn→∞∫Ω14gunun−Gun+aθ0λ14un2dx≥a4∥∇u1∥22+θ0λ1∥u1∥22+∫Ω14gu1u1−Gu1+aθ0λ14u12dx=a4∥∇u1∥22+∫Ω14gu1u1−Gu1dx=Fu1−14<F′u1,u1>=Fu1,which implies Fu1≤c. Thus, Fu1=c=infu∈NI>0. In view of Theorem 1, there exists u0∈M such that Iu0=m. Therefore, from (11), (24), and Corollary 14, we have
(64)m=Fu0=sups,t≥0Fsu0++tu0−=sups,t≥0Fsu0++Ftu0−+bs2t22∥∇u0+∥22∥∇u0−∥22+λs2t24∫Ωϕu0+u0−2+ϕu0−u0+2>sups≥0Fsu0++supt≥0Ftu0−≥2c,which implies the energy of the least-energy sign-changing solution is strictly larger than twice that of the ground state solutions of the Nehari type.


## 6. The Convergence Property

In this part, we give the convergence property forb↘0. Firstly, we have to give some estimates which will be used in the following process. Choose ω0∈C0∞Ω such that ω0±≠0, then there exists a constant c′ such that
(65)∫Ωϕsω0++tω0+sω0++tω0−2dx≤c′∥s∇ω0++t∇ω0−∥24≤2c′s4∥∇ω0+∥24+t4∥∇ω0−∥24.Thus, byg1 and g3, there exist β1>0 and β2≥max1+λc′∥∇ω0+∥24/∥ω0+∥44,1+λc′∥∇ω0−∥24/∥ω0−∥44 such that
(66)Gt≥β2t4−β1,∀t∈ℝ.Proof of Theorem 19.
It is clear thatb=0 is allowed in Section 3. Therefore, we can deduce that there exists a ub0∈Mb0 such that F′b0ub0=0 and Fb0ub0=mb0=infu∈Mb0Fb0u, that is, problem (2) has the least-energy sign-changing solution, which changes sign only once.
Forb>0, let ub∈Mb=M be a sign-changing solution of (1) obtained in Theorem 1, which changes sign only once and satisfies Fbub=Fub=mb=infu∈MbFbu.
Then, for anyb∈0,1, it follows from Lemma 12 and (65)-(66), we have
(67)Fbub=mb≤maxs,t≥0Fbsω0++tω0−=maxs,t≥0as22∥∇ω0+∥22+bs44∥∇ω0+∥24−∫ΩGsω0+dx+at22∥∇ω0−∥22+bt44∥∇ω0−∥24−∫ΩGtω0−dx+bs2t22∥∇ω0+∥22∥∇ω0−∥22+λ4∫Ωϕsω0++tω0−sω0++tω0−2dx≤maxs,t≥0as22∥∇ω0+∥22+bs44∥∇ω0+∥24−∫ΩGsω0+dx+at22∥∇ω0−∥22+bt44∥∇ω0−∥24−∫ΩGtω0−dx+bs44∥∇ω0+∥24+bt44∥∇ω0−∥24+λc′2s4∥∇ω0+∥24+t4∥∇ω0−∥24≤maxs,t≥0as22∥∇ω0+∥22+1+λc′2s4∥∇ω0+∥24−∫Ωβ2sω0+4dx+at22∥∇ω0−∥22+1+λc′2t4∥∇ω0−∥24−∫Ωβ2tω0−4dx+2β1Ω≤maxs,t≥0as22∥∇ω0+∥22−1+λc′2s4∥∇ω0+∥24+at22∥∇ω0−∥22−1+λc′2t4∥∇ω0−∥24+2β1Ω=Λ0∈0,∞.
For any sequencebn with bn↘0 as n⟶∞, by (9), (10), (25), and (67), we have for large n∈ℕ(68)Λ0+1≥Fbnubn−14<F′bnubn,ubn≥a1−θ04∥∇ubn∥22.
Sinceθ0∈0,1, ubn is bounded in H01Ω. Therefore, there exists a subsequence of bn, still denoted by bn and ub0∈H, such that ubn⟶ub0 in H01Ω. By a standard argument, we can prove ubn±⟶ub0±≠0 in H01Ω. Since
(69)<F′b0ub0,φ>=a∫Ω∇ub0∇φdx+∫Ωλϕub0ub0φdx−∫Ωgub0φdx=limn→∞a+bn∥∇ubn∥22∫Ω∇ubn∇φdx+∫Ωλϕubnubnφdx−∫Ωgubnφdx=limn→∞<F′bnubn,φ>=0.∀φ∈C0∞Ω.
Thus,F′b0ub0=0, ub0∈Mb0, and Fb0ub0≥mb0. Next, we give the proof of Fb0ub0=mb0. Choose bn∈0,1, from g2, there exists a K0 such that
(70)Fbnsv0++tv0−=as22∥∇v0+∥22+bns44∥∇v0+∥24−∫ΩGsv0+dx+at22∥∇v0−∥22+bnt44∥∇v0−∥24−∫ΩGtv0−dx+bns2t22∥∇v0+∥22∥∇v0−∥22+λ4∫Ωϕsv0++tv0−sv0++tv0−2dx≤as22∥∇v0+∥22+1+λc′2s4∥∇v0+∥24−∫ΩGsv0+dx+at22∥∇v0−∥22+1+λc′2t4∥∇v0−∥24−∫ΩGtv0−dx<0,∀s,t≥K0.
From Lemma10, there exists sn,tn such that snv0++tnv0+∈Mbn, (5.4) implies 0<sn,tn<K0. Since F′b0v0=0, then from (9), (10), (15), (16), and (20), we have
(71)mb0=Fb0v0=Fbnv0−bn4∥∇v0∥24≥Fbnsnv0++tnv0−+1−sn44<F′bnv0,v0+>+1−tn44<F′bnv0,v0−>−bn4∥∇v0∥24≥mbn−1+K044∣<F′bnv0,v0+>∣−1+K044∣<F′bnv0,v0−>∣−bn4∥∇v0∥24=mbn−1+K044bn∥∇v0∥22∥∇v0+∥22−1+K044bn∥∇v0∥22∥∇v0−∥22−bn4∥∇v0∥24,which implies
(72)limsupn→∞mbn≤mb0.
According to (9), (15), and (72), we have
(73)mb0≤Fb0ub0=limsupn→∞Fbnubn=limsupn→∞mbn≤mb0.
This showsFb0ub0=mb0, and the convergence property of b↘0 is proved.Proof of Theorem 20.
Since the proof is similar as the proof of Theorem19, we omit the details.



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*Source: 8254898-2020-05-19.xml*