# Multiple Nontrivial Solutions for a Class of Biharmonic Elliptic Equations with Sobolev Critical Exponent

Authors: Xiaoyong Qian; Jun Wang; Maochun Zhu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic equation with Sobolev critical exponent in a bounded domain. By using the idea of the previous paper, we generalize the results and prove the existence and multiplicity of nontrivial solutions of the biharmonic elliptic equations.


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## Body

## 1. Introduction and Main Results

In the present paper, we are concerned with the existence of multiple solutions to the following biharmonic elliptic equation with perturbation(1)Δ2u=up-2u+f,x∈Ω,u=∇u=0,x∈∂Ω,where Ω is a bounded domain in RN(N≥5), Δ2 is the biharmonic operator, and p=2∗∗=2N/N-4 is the Sobolev critical exponent.The second-order semilinear and quasilinear problems have been object of intensive research in the last years. Brezis and Nirenberg [1] have studied the existence of positive solutions of (1). Particularly, when f=λu, where λ∈R is a constant, they have discovered the following remarkable phenomenon: the qualitative behavior of the set of solutions of (1) is highly sensitive to N, the dimension of the space. Precisely, Brezis and Nirenberg [1] have shown that, in dimension N≥4, there exists a positive solution of (1), if and only if λ∈(0,λ1); while, in dimension N=3 and when Ω=B1 is the unit ball, there exists a positive solution of (1), if and only if λ∈(λ1/4,λ1), where λ1>0 is the first eigenvalue of -Δ in Ω. For more results on this direction we refer the readers to [2–5] and the references therein.During the last decades many works have been orientated to the analysis of biharmonic nonlinear Schrödinger equation (BHNSE)(2)iφt-Δ2φ+gx,φφ=0,φ0,x=φ0x∈H02Ω,Ω⊂RN is an open domain N≥5. For instance, paper [6] proved that some of the properties and characteristics for the second-order semilinear problems can be extended to BHLSE. Paper [7] proved the existence of blow-up solutions. In papers [8–10], the authors proved the existence of global solutions, in particular, looking for standing wave solutions for (2) of the form(3)φt,u=eiλtusuch that u is a solution satisfying the equation(4)Δ2u+λu=g^x,u,u∈H02Ω.If λ=0 and g^(x,u)=up-2u, we know that (4) admits no positive solutions if Ω is star shaped under the Navier or Dirichlet boundary conditions (see [11, Theorem 3.3] and [12, Corollary 1]). If λ>0 and Ω is a ball, paper [13] proved the existence of positive radially symmetric solutions. For more general results on this direction one can refer to [14, 15, 15–21] and the references therein.Motivated by the above results, we study the case thatλ=0, g^(x,u)=up-2u+f(x), and Ω⊂RN is a bounded domain. Precisely, we shall generalize the results of Tarantello [22] to the biharmonic and critical exponent case. Our main tool here is the Nehari manifold method which is similar to the fibering method of Pohozaev’s.In order to state the main results, we shall give some notation and assumptions. LetD=H02(Ω), and up=(∫Ωupdx)1/p be the usual Lp(Ω) norm. Obviously, D is a Hilbert space under the inner product 〈u,v〉=∫ΩΔuΔvdx. Correspondingly, the norm is denoted by ·; i.e., u2=∫ΩΔu2dx. Assume that f∈Lq(Ω)(q=2N/N+4)(f≠0) satisfies(5)fq≤CNSN+4/8,where S is the best Sobolev embedding constant of D↪Lp(Ω)(p=2∗∗=2N/N-4), and(6)CN=8N-4N-4N+4N+4/8,S=infu∈D∖0Δu22up2.Let(7)uεx=NN-4N2-4N-4/8εN-4/2ε2+x2N-4/2be an extremal function for the Sobolev inequality in RN. For a∈Ω, let uε,a(x)=uε(x-a) and ξa∈C0∞(Ω) with ξa≥0 and ξa=1 near a. We point out that the embedding D↪Lp(Ω) is not compact. This leads to the lack of compactness for the proved existence and multiplicity of nontrivial solutions of (1). Motivated by [1, 22], we recover the local compactness by dividing the Nehari manifold into three parts and give some estimates for the least energy of (1)It is easy to see that the energy functional of (1) is denoted by(8)Iu=12∫ΩΔu2dx-1p∫Ωupdx-∫Ωfudx,u∈D.Hence, I is well defined (under (5)) and of the class C2(Ω). Moreover, all the critical points of I are precisely the solutions of (1). We define the Nehari manifold N associated with the functional by (9)N=u∈D∣I′u,u=0. It is clear that all critical points lie in the Nehari manifold, and it is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. For fixed u∈D∖0, we set(10)ϕt=Itu=t22∫ΩΔu2dx-tpp∫Ωupdx-t∫Ωfudx,t≥0.The mapping is called fibering map. Such maps are often used to investigate Nehari manifolds for various semilinear problems. From the relationship between I and ϕ(t), we can divide N into three parts (11)N+=u∈N∣Δu22-p-1upp>0,N0=u∈N∣Δu22-p-1upp=0,N-=u∈N∣Δu22-p-1upp<0.It turns out that under the assumption (5), we infer that N0=0 (see Lemma 5 below). Now the main result in this paper can be stated as follows.Theorem 1.
Assume thatf≠0 satisfies (5). Then(12)infNI=infN+I=c0is achieved at a point u0∈N. Furthermore, u0 is a critical point of I, and u0≥0 when f≥0.In the following we study the second infimum problem(13)infN-I=c1.In this case we have the following results.Theorem 2.
Assume thatf≠0 satisfies (5). Then c1>c0 and the infimum in (13) is achieved at a point u1∈N-, which is a critical point of I.The proofs of Theorems1–2 rely on the Ekeland’s variational principle and careful estimates (see [1]) of minimizing sequence.


## 2. Some Preliminary Results

In this section we prove some preliminary results for the proof of Theorems1–2. The main ideas are coming from [1, 22]. We begin with the following lemma which states the purpose of assumption (5).Lemma 3.
Supposed thatf≠0 satisfies (5). For every u∈D∖0, there exists a unique t1=t1(u)>0 such that t1u∈N-. Particularly, we have(14)t1>Δu22p-1upp1/p-2≔tmaxand I(t1u)=maxt≥tmaxI(tu). Moreover, if ∫Ωfudx>0, then there exists a unique t2=t2(u)>0 such that t2u∈N+. In particular, one has (15)t2<Δu22p-1upp1/p-2and I(t2u)≤I(tu),∀t∈[0,t1].Proof.
Recall that the fibering map is defined by(16)ϕt=Itu=t22∫ΩΔu2dx-tpp∫Ωupdx-t∫Ωfudx.Then (17)ϕ′t=t∫ΩΔu2dx-tp-1∫Ωupdx-∫Ωfudx≔gt-∫Ωfudx.We deduce from g′(t)=0 that (18)t=tmax=Δu22p-1upp1/p-2.If 0<t<tmax, we have ϕ′′(t)=g′(t)>0, and if t>tmax, one sees ϕ′′(t)=g′(t)<0. A direct computation shows that g(t) achieves its maximum at tmax, and (19)gtmax=8N-4N-4N+4N+4/8Δu2N+4/4upN/4=CNΔu2N+4/4upN/4>0.We divide the following two cases to accomplish our results.
Case 1.If ∫Ωfudx≤0, then ϕ′(tmax)=g(tmax)-∫Ωfudx>0. It is easy to see that if t→+∞, we have ϕ′(t)<0. So, there exists unique t1>tmax such that ϕ′(t1)=0 and g(t1)=∫Ωfudx. We infer from the monotonicity of g(t) that, for t1>tmax, (20)ϕ′′t1=g′t1<0,t12g′t1=Δt1u22-p-1t1upp<0.This shows that t1u∈N-.
Case 2.If ∫Ωfudx>0, we infer from assumption (5) that ∫Ωfudx<g(tmax)∀u∈D. Then ϕ′(tmax)=g(tmax)-∫Ωfudx>0. Since ϕ′(0)=-∫Ωfudx<0, there exists a unique t2∈[0,tmax] such that ϕ′(t2)=0 and g(t2)=∫Ωfudx. A direct computation shows that t2u∈N+ and I(t2u)≤I(tu),∀t∈[0,t1].Lemma 4.
Assume thatf≠0 satisfies (5). We infer that the infimum(21)infup=1CNΔu2N+4/4-∫Ωfudx≔μ0is achieved, where μ0>0.The proof of Lemma4 is technical and the idea of the proof is mainly motivated by paper [23]. We shall prove it in the appendix. Next we study the property of the set N0.Lemma 5.
Letf≠0 satisfy (5). Then for every u∈N,u≠0, we can get the conclusion that N0=0.Proof.
We use the contradiction arguments. Assume that, for someu∈N,u≠0, we have u∈N0. That is,(22)Δu22-p-1upp=0.Since u∈N, it follows that Δu22-upp-∫Ωfudx=0. Hence, we get(23)upp-∫Ωfudx=0.By Sobolev inequality, we deduce that (p-2)up≥(S/p-1)1/p-2. For u≠0, we set(24)Au=CNΔu2N+4/4upN/4-∫Ωfudx.For t≥0 and up=1, a direct computation shows that (25)Atu=tCNΔu2N+4/4-∫Ωfudx.We derive from Lemma 4 that, for γ>0,(26)infu≥γAu≥γμ0.Let γ=(S/p-1)1/p-2>0. We infer from (26) that (27)0<γμ0≤Au=CNΔu2N+4/4upN/4-∫Ωfudx=p-2uppΔu22p-1uppp-1/p-2-1=0,which is a contradiction.Lemma 6.
Letf≠0 satisfy (5). For each u∈N∖0, there exist ε>0 and a differentiable function t=t(w)>0,w∈D,w<ε, satisfying the following: (28)t0=1,twu-w∈N,∀w<ε,t′0,w=2∫ΩΔuΔwdx-p∫Ωup-2uwdx-∫ΩfwdxΔu22-p-1upp.Proof.
We defineF:R×D→R by (29)Ft,w=tΔu-w22-tp-1u-wpp-∫Ωfu-wdx.Since F(1,0)=0 and Ft(1,0)=Δu22-(p-1)upp≠0 (Lemma 5), by using the implicit function theorem at the point (1,0) we know that the results of the lemma hold.


## 3. Proof of Theorem1

In this part we shall give the proof of Theorem1.Proof of Theorem1.
We first claim that the functionalI is bounded from below in N. For u∈N, we have I′(u),u=0. That is, Δu22-upp-∫Ωfudx=0. One deduces from (2) and Hölder inequality that(30)Iu=12∫ΩΔu2dx-1p∫Ωupdx-∫Ωfudx=12-1pΔu22-1-1p∫Ωfudx≥2NΔu22-1-1pfqup≥2NΔu22-1-1pCNSN/8Δu2≥-N1-1/p2CN2SN/48.Hence, we know that the infimum c0 is also bounded from below. Second, we can get an upper bound for c0. Let v∈D be the solution for Δ2u=f. For f≠0, one obtains that (31)∫Ωfvdx=Δv22>0. Set t0=t2(v)>0 as defined by Lemma 3. Thus, we have that t0v∈N+ and(32)c0≤It0v=t022Δv22-t0ppvpp-t0∫Ωfvdx=-t022Δv22+p-1pt0pvpp<-2t02NΔv22<0.For any minimizing sequence un⊂N, we can use Ekeland’s variational principle (see [24]) to get following properties:(i)
I(un)<c0+1/n,(ii)
I(w)≥I(un)-1/nΔ(w-un)2,∀w∈N.
Hence forn large enough, we obtain (33)Iun=12-1pΔun22-1-1p∫Ωfundx<c0+1n<-2t02NΔv22.This implies (34)∫Ωfundx≥4t02N+4Δv22>0,andun≠0.Since I(un)<0, we infer from Hölder’s inequality that(35)∃M>0,Δun22≤M.At the same time, we observe that(36)4t02N+4Δv22≤∫Ωfundx.One deduces from (5) and Hölder’s and Sobolev’s inequalities that(37)∃m>0,Δun22≥m>0.So we derive from (35) and (37) that(38)0<m≤Δun22≤M,where m and M only depend on f and Ω.
Next we shall prove thatI′un→∞, as n→∞. Applying Lemma 6 with u=un and w=δI′(un)/I′un(δ>0), we can find some tnδ=tδI′un/I′un such that (39)wδ=tnδun-δI′unI′un∈N.By condition (ii) we have(40)1nΔw-un2≥Iun-Iwδ=1-tnδI′wδ,un+δtnδI′wδ,uI′unI′un+oδ.Dividing by δ and letting δ→0, we get(41)1n1+tn′0Δun2≥-tn′0I′un,un+I′un=I′un,where tn′(0)=t′(0),I′(un)/I′un. So, we conclude that(42)I′un≤Cn1+tn′0,where C is a constant. In order to complete the proof we need to prove that tn′(0) is bounded uniformly on n. By Lemma 6 we can get(43)tn′0≤CΔun22-p-1unpp.Thus, there exists subsequence un (still denote by un) such that(44)Δun22-p-1unpp=o1.We infer from un⊂N that(45)Δun22-unpp=∫Ωfundx,∫Ωfundx=p-2unpp+o1.By the estimate of Δun2 from (38), we have that unp≥γ>0 and(46)0<μ0γ5N/4≤unpN/4Aun≤CNΔun2N+4/4-p-2unpp+N+4/4=p-21p-1N+4/8Δun2N+4/4-p-2unpp+N+4/4=p-2Δun22p-1p-2/p-1-unppp-2/p-1=o1.This is impossible. So, Δun22-(p-1)unpp is away from zero. Thus, we conclude that(47)I′un→0asn→∞.Let u0∈D be the weak limit in D of un. From (47) we can get that u0 is a weak solution for (1). In fact, u0∈N and (48)c0≤Iu0≤limn→+∞Iun=c0.So, we have that un→u0 strongly in D and I(u0)=c0=infu∈NI(u). Moreover, u0∈N+. By using standard method, we can prove that u0 is a global minimum for I in D (See [25]).


## 4. Proof of Theorem2

In this section, we shall give the proof of Theorem2. Since the embedding D↪L2N/N-4(Ω) is not compact, we need to find some way to recover this compactness. Motivated by previous works of [1, 22, 23], we will seek the level in which (PS)c-condition will recover. Then we shall use the Mountain-Pass principle to get the second nontrivial solution of (1). The related problems have been studied in [23], and such an approach has been used. The threshold is found in the following lemma to obtain the compactness.Lemma 7.
Assume that the sequenceun⊂D satisfying(i)
I(un)→c with c<c0+2/NSN/4, where c0 is defined in (12).(ii)
I′un→0 as n→∞.
Thenun has a convergent subsequence.Proof.
It is clear thatΔun22 is uniformly bounded from condition (i) and (ii). For a subsequence of un, we can get a w0∈D such that(49)un⇀w0inD.So, from (ii), we obtain that(50)I′w0,w=0,for∀w∈D.Then w0 is a weak solution of (1), w0≠0, and w0∈N, I(w0)≥c0. Let un=w0+vn. So, vn⇀0 in D. By Brezis-Lieb lemma (see [24]), we conclude that (51)unpp=w0+vnpp=w0pp+vnpp+o1.Thus, for n large enough, we get (52)c0+2NSN/4>Iw0+vn=Iw0+12Δvn22-1pvnpp+o1≥c0+12Δvn22-1pvnpp+o1,which means(53)12Δvn22-1pvnpp<2NSN/4+o1.Moreover, we infer from condition (ii) that(54)o1=I′un,un=I′w0,w0+Δvn22-vnpp+o1=Δvn22-vnpp+o1,and then we obtain(55)Δvn22-vnpp=o1.Next we shall prove that if (53) and (55) hold, then there exists the subsequence of vn (still denoted by vn), which satisfies (56)Δvn2→0,n→+∞.We assume vn is bounded away from 0; that is (57)∃C>0,Δvn2≥C,∀n∈N.So from (55) we can get (58)vnpp-2≥S+o1,vnpp≥SN/4+o1.We infer from (53) and (55) that (59)2NSN/4≤2Nvnpp+o1=12Δvn22-1pvnpp+o1<2NSN/4for n large. This is contradiction. So, we can get un→w0 strongly in D.Note thatu0≠0. Following [23], we set Σ⊂Ω to be a set of positive measures such that u0>0 on Σ. Let us define(60)Uε,ax=ξaxuε,ax,x∈RN,where uε,a(x) and ξa(x) are defined in Section 1. Without loss of generality, we take uε,a(x)=ε(N-4)/2/(ε2+x-a2)(N-4)/2. Then we have the following estimates for Uε,a.Lemma 8.
∀ R > 0 and a.e.a∈Σ, there exists ε0>0 such that(61)Iu0+RUε,a<c0+2NSN/4for every 0<ε<ε0.Proof.
LetB=∫RNΔu1x2dx and A=∫RNu1x2dx. By the definition of uε(x), we can get the Sobolev embedding exponent S=B/A2/p. A direct computation shows that (62)∫ΩΔξax2uε,ax2dx=OεN-4,∫Ω∇ξax2∇uε,ax2dx=OεN-4,∫ΩΔξaxξaxuε,axΔuε,axdx=OεN-4,∫ΩΔξax∇ξax∇uε,axuε,axdx=OεN-4,∫Ω∇ξaxξaxΔuε,ax∇uε,axdx=OεN-4,∫Ωξax2Δuε,ax2dx=B+OεN-4,Uε,axpp=A+OεN.Now we take the C0∞(Ω) function ξa(x) such that(63)ξax≡1,whenx-a≤r0,0≤ξax≤1,whenr0≤x-a≤2r0,ξax≡0,whenx-a≥2r0,∇ξax≤Cr0,Δξax≤Cr02,where r0>0. On the other hand, we see that(64)uε,ax=εN-4/2ε2+x-a2N-4/2,∇uε,a=N-4εN-4/2x-aε2+x-a2N-2/2,Δuε,a=N-4εN/2-N-4N-3εN-4/2x-a2ε2+x-a2N/2.So, by direct computation we infer that(65)∫ΩΔξax2uε,ax2dx≤Cr04∫r0≤x-a≤2r0εN-4ε2+x-a2N-4dx=CωN-1r04∫r0/ε2r0/εε4rN-11+r2N-4dr≤CωN-1ε4r04∫r0/ε2r0/εr7-Ndr=OεN-4,where ωN-1 is the measure of the unit sphere in RN. Moreover, we have that (66)∫Ω∇ξax2∇uε,ax2dx≤Cr02∫r0≤x-a≤2r0εN-4x-a2ε2+x-a2N-2dx=CωN-1r02∫r0/ε2r0/εε2rN+11+r2N-2dr≤CωN-1ε2r02∫r0/ε2r0/εr5-Ndr=OεN-4,∫ΩΔξaxξaxuε,axΔuε,axdx≤Cr02∫r0≤x-a≤2r0εN-2+εN-4x-a2ε2+x-a2N-2dx=CωN-1r02∫r0/ε2r0/εε2rN-11+r2N-2dr+∫r0/ε2r0/εε2rN+11+r2N-2dr≤CωN-1r02∫r0/ε2r0/εε2r3-Ndr+∫r0/ε2r0/εε2r5-Ndr=OεN-2+OεN-4=OεN-4,∫ΩΔξax∇ξax∇uε,axuε,axdx≤Cr03∫r0≤x-a≤2r0εN-4x-aε2+x-a2N-3dx=CωN-1r03∫r0/ε2r0/εε3rN1+r2N-3dr≤CωN-1ε3r03∫r0/ε2r0/εr6-Ndr=OεN-4,∫Ω∇ξaxξaxΔuε,ax∇uε,axdx≤Cr0∫r0≤x-a≤2r0εN-2x-a+εN-4x-a3ε2+x-a2N-1dx=CωN-1r0∫r0/ε2r0/εεrN1+r2N-1dr+∫r0/ε2r0/εεrN+21+r2N-1dr≤CωN-1r03∫r0/ε2r0/εεr2-Ndr+∫r0/ε2r0/εεr4-Ndr=OεN-2+OεN-4=OεN-4,∫Ωξax2Δuε,ax2dx≤∫x-a≤r0Δuε,ax2dx+∫r0≤x-a≤2r0Δuε,ax2dx≤C∫r0/ε2r0/εrN-11+r2N+rN+31+r2Ndr+rN+11+r2Ndr+∫RNy21+y2Ndy≤B+OεN+OεN-4+OεN-2=B+OεN-4.Thus, we infer from [23] that(67)u0+RUε,app=u0pp+RpUε,app+pR∫Ωu0p-2u0Uε,adx+pRp-1∫ΩUε,ap-1u0dx+oεN-4/2. From all of the above, noticing that u0∈N, one has that(68)Iu0+RUε,a=12∫ΩΔu0+RUε,a2dx-1p∫Ωu0+RUε,apdx-∫Ωfu0+RUε,adx=12∫ΩΔu02+R∫ΩΔu0ΔUε,adx+12R2∫ΩΔUε,a2dx-1p∫Ωu0+RUε,apdx-∫Ωfu0+RUε,adx=12∫ΩΔu02-1p∫Ωu0pdx-∫Ωfu0dx+R∫ΩΔu0ΔUε,adx-∫Ωu0p-2u0Uε,adx-∫ΩfUε,adx+R22B-RppA-Rp-1∫ΩUε,ap-1u0dx+oεN-4/2=Iu0+R22B-RppA-Rp-1∫ΩUε,ap-1u0dx+oεN-4/2.By using an estimate obtained by G. Folland [26] and setting u0=0 outside Ω, one gets that (69)∫ΩUε,ap-1u0dx=εN-4/2u0aE+oεN-4/2,where(70)E=∫RNdx1+x2N+4/2,11+x2N+4/2∈L1RN.Consequently, we have (71)Iu0+RUε,a=c0+R22B-RppA-Rp-1u0aEεN-4/2+oεN-4/2.We set(72)hs=B2s2-Apsp-u0aEεN-4/2sp-1,s>0,and assume h(s) achieves its maximum at s1>0, which satisfies(73)s1B-s1p-1A=p-1u0aEεN-4/2sp-2.We define(74)s0=BA1/p-2,which is the maximum point of h1(s)=B/2s2-A/psp. We can conclude that 0<s1<s0, and s1→s0(ε→0). Let s1=s0(1-δ). It is easy to see that δ→0(ε→0). From (73) we can get (75)s01-δB-s0p-11-δp-1A=p-1u0aEεN-4/2s0p-2(1-δp-2and then expanding for δ, we can get(76)p-2Bp-1/p-2A1/p-2δ=p-1BAu0aEεN-4/2+oεN-4/2.So, one sees that (77)Iu0+RUε,a=c0+R22B-RppA-Rp-1u0aEεN-4/2+oεN-4/2≤c0+s122B-s1ppA-s1p-1u0aEεN-4/2+oεN-4/2=c0+s022B-s0ppA-s02Bδ+s0pAδ-s0p-1u0aEεN-4/2+oεN-4/2=c0+2NSN/4-s0p-1u0aEεN-4/2+oεN-4/2.When we take small ε0>0, we arrive at (78)Iu0+RUε,a<c0+2NSN/4,∀0<ε<ε0.This finishes the proof.Now we are ready to give the proof of Theorem2.Proof of Theorem2.
It is clear that the uniqueness oft1(u) satisfies the following condition: (79)t1uu∈N-,It1uu=maxt≥tmax⁡Itu,foreveryu∈D,u=1.At the same time, t1(u) is a continuous function of u. And N- divides D into two components D1 and D2, which are disconnected from each other. Let (80)D1=u=0oru:u<t1uu,D2=u=0oru:u>t1uu.Obviously, D-N-=D1∪D2, and we can check N+⊂D1, u0∈D1. We can choose a constant C0, which satisfies(81)0<t1u≤C0,∀u=1,and claim that(82)w=u0+R0Uε,a∈D2,where R0=(1/BC02-u02)1/2+1. In fact, a direct computation shows that (83)w2=u02+R02Uε,a2+2R0∫ΩΔu0ΔUε,adx=u02+R02B+o1>C02≥t1ww2for ε>0 small enough. Thus, claim (82) holds.
We fixε>0 such that both (61) and (82) hold by the choice of R0 and a∈Σ. We set(84)Γ=γ∈C0,1,D:γ0=u0,γ1=u0+R0Uε,a,and take h(t)=u0+tR0Uε,a, which belongs to Γ. From Lemma 7, we conclude that (85)c=infh∈Γmaxt∈0,1Iht<c0+2NSN/4.Since every h∈Γ intersects N-, we get that(86)c1=infN-I≤c<c0+2NSN/4.Next we use Mountain-Pass lemma to prove Theorem 2. Let un⊂N- be such that (87)Iun→c1,I′un→0.We deduce from Lemma 7 that there exists a subsequence (still denoted by un) of un, and u1∈D such that (88)un→u1inD.So, u1 is a critical point for I, u1∈N- and I(u1)=c1.Remark 9.
We point out that the results of Theorems1–2 can be generalized to polyharmonic problem. Precisely, we can consider the semilinear polyharmonic problem(89)-Δmu=up-2u+f,x∈Ω,u=Du=⋯=Dm-1u=0,x∈∂Ω,where Ω is a smooth bounded domain in RN(N≥2m+1).m∈N+,p=2N/N-2m denotes the critical Sobolev exponent for (-Δ)m, and f∈Lq(Ω)(q=2N/N+2m)(f≠0) is small enough. We can define the energy functional: (90)Iu=12um2-1p∫Ωupdx-∫Ωfudx,u∈H,where(91)H=H0mΩ=v∈HmΩ∣Div=0on∂Ω,∀0≤i<m.H is Hilbert space and endowed with the scalar product(92)
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dand ·m is the corresponding norm. Let (93)uεx=CN,mεN-2m/2ε2+x2N-2m/2be an extremal function for the Sobolev inequality in RN, and the constant CN,m be independent of ε. By dividing the Nehari manifold, we can prove (PS)c condition when c<c0+m/NSm/2N, where c0=I(u0) and u0 is the first solution. By using the same idea of this article, one can obtain that (89) has at least two nontrivial solutions.



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*Source: 8212785-2018-11-21.xml*