# Nontrivial Solutions of the Kirchhoff-Type Fractionalp-Laplacian Dirichlet Problem

Authors: Taiyong Chen; Wenbin Liu; Hua Jin
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we consider the new results for the Kirchhoff-typep-Laplacian Dirichlet problem containing the Riemann-Liouville fractional derivative operators. By using the mountain pass theorem and the genus properties in the critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence and multiplicity of nontrivial weak solutions for the Kirchhoff-type fractional Dirichlet problem withp-Laplacian of the form
(1)a+b∫0T0Dtαutpdtp−1DtTαϕpD0tαut=ft,ut,t∈0,T,u0=uT=0,where a, b>0, and p>1 are constants, 0Dtα and tDTα are the left and right Riemann-Liouville fractional derivatives of order α∈1/p,1, respectively, and ϕp:ℝ⟶ℝ is the p-Laplacian [1] defined by
(2)ϕps=sp−2s,s≠0,ϕp0=0,and f∈C0,T×ℝ,ℝ. It should be pointed out that the weak solutions of the boundary value problem (BVP for short) (1) mean the critical points of the associated energy functional.The fractional derivativeD0tα is nonlocal and reduces to the local first-order differential operator when α=1. Moreover, the p-Laplacian ϕp is nonlinear and reduces to the linear identity operator when p=2. If b=0, BVP (1) reduces to the following fractional p-Laplacian BVP [2]:
(3)DtTαϕpD0tαut=ft,ut,t∈0,T,u0=uT=0.In contrast to BVP (3), if b≠0, another nonlocal term,
(4)∫0TD0tαutpdt,makes BVP (1) rough when one deals with it by the variational methods.Recently, many important results on the fractional differential equations [3–18] and the Kirchhoff equations [19–24] have been obtained. Motivated by the above works, we study the solvability of BVP (1). More precisely, we prove that BVP (1) possesses at least one nontrivial weak solution when ft,x is p2−1-superlinear or p2−1-sublinear in x at infinity and possesses infinitely many nontrivial weak solutions when ft,x is p2−1-sublinear in x at infinity. The main ingredients used here are the mountain pass theorem and the genus properties in the critical point theory.Note that, since the Kirchhoff-typep-Laplacian is a nonlinear operator, it is usually difficult to verify the Palais-Smale condition ((PS)-condition for short). Now, we make the following assumptions on the nonlinearity f.(H11).
There exist two constantsμ>p2,R>0 such that
(5)0<μFt,x≤xft,x,∀t∈0,T,x∈ℝwithx≥R,where Ft,x=∫0xft,sds.(H12).
ft,x=oxp−1 as x⟶0 uniformly for ∀t∈0,T.(H21).
There exists a constant1<r1<p2 and a function d∈L10,T,ℝ+ such that
(6)ft,x≤r1dtxr1−1,∀t,x∈0,T×ℝ.(H22).
There exists an open intervalI⊂0,T and three constants η,δ>0,1<r2<p2 such that
(7)Ft,x≥ηxr2,∀t,x∈I×−δ,δ.(H23).
ft,x=−ft,−x,∀t,x∈0,T×ℝ.We are now to state our main results.Theorem 1.
Let (H11) and (H12) be satisfied. Then, BVP (1) possesses at least one nontrivial weak solution.Theorem 2.
Let (H21) and (H22) be satisfied. Then, BVP (1) possesses at least one nontrivial weak solution.Theorem 3.
Let (H21)–(H23) be satisfied. Then, BVP (1) possesses infinitely many nontrivial weak solutions.


## 2. Preliminaries

### 2.1. Fractional Sobolev Space

In this subsection, we present some basic definitions and notations of the fractional calculus [25, 26]. Moreover, we introduce a fractional Sobolev space and some properties of this space [14].Definition 4 (see [25]).
Forβ>0, the left and right Riemann-Liouville fractional integrals of order β of a function u:a,b⟶ℝ are given by
(8)Dat−βut=1Γβ∫att−sβ−1usds,Dtb−βut=1Γβ∫tbs−tβ−1usds,respectively, provided that the right-hand-side integrals are pointwise defined on a,b, where Γ⋅ is the gamma function.Definition 5 (see [25]).
Forn−1≤β<n (n∈ℕ), the left and right Riemann-Liouville fractional derivatives of order β of a function u:a,b⟶ℝ are given by
(9)Datβut=dndtnDatβ−nut,Dtbβut=−1ndndtnDtbβ−nut.Remark 6.
Whenβ=1, one can obtain from Definitions 4 and 5 that
(10)Dat1ut=u′t,tDb1ut=−u′t,where u0 is the usual first-order derivative of u.Definition 7 (see [14]).
For0<α≤1 and 1<p<∞, the fractional derivative space E0α,p is defined by the closure of C0∞0,T,ℝ with respect to the following norm:
(11)uEα,p=uLpp+0DtαuLpp1/p,where uLp=∫0Tutpdt1/p is the norm of Lp0,T,ℝ.Remark 8.
It is obvious that, foru∈E0α,p, one has
(12)u,0Dtαu∈Lp0,T,ℝ,u0=uT=0.Lemma 9 (see [14]).
Let0<α≤1 and 1<p<∞. The fractional derivative space E0α,p is a reflexive and separable Banach space.Lemma 10 (see [14]).
Let0<α≤1 and 1<p<∞. For u∈E0α,p, one has
(13)uLp≤CpD0tαuLp,where
(14)Cp=TαΓα+1>0,is a constant. Moreover, if α>1/p, then
(15)u∞≤C∞0DtαuLp,where u∞=maxt∈0,Tut is the norm of C0,T,ℝ,
(16)C∞=Tα−1/pΓααq−q+11/q>0,q=pp−1>1.Remark 11.
By (13), we can consider the space E0α,p with norm
(17)uEα,p=0DtαuLp,in what follows.Lemma 12 (see [14]).
Let1/p<α≤1 and 1<p<∞. The imbedding of E0α,p in C0,T,ℝ is compact.


### 2.2. Critical Point Theory

Now, we present some necessary definitions and theorems of the critical point theory [27, 28].LetX be a real Banach space, and I∈C1X,ℝ which means that I is a continuously Fréchet differentiable functional. Moreover, let Bρ0 be an open ball in X and ∂Bρ0 denote its boundary.Definition 13 (see [27]).
LetI∈C1X,ℝ. If any sequence uk⊂X for which Iuk is bounded and I′uk⟶0 as k⟶∞ possesses a convergent subsequence in X, then we say that I satisfies the (PS)-condition.Lemma 14 (see [28]).
LetX be a real Banach space, and I∈C1X,ℝ satisfying the (PS)-condition. Suppose that I0=0 and C1, there are constants ρ,σ>0 such that I∂Bρ0≥σ; C2 there is an e∈X\Bρ0¯ such that Ie≤0.
Then,I possesses a critical value c≥σ. Moreover, c can be characterized as
(18)c=infγ∈Γmaxs∈0,1Iγs,where
(19)Γ=γ∈C0,1,Xγ0=0,γ1=e.Lemma 15 (see [27]).
LetX be a real Banach space, and I∈C1X,ℝ satisfies the (PS)-condition. If I is bounded from below, then c=infXI is a critical value of I.In order to find the infinitely many critical points ofI, we introduce the following genus properties. Let
(20)Σ=A⊂X−0AisclosedinXandsymmetricwithrespectto0,Kc=u∈XIu=c,I′u=0,Ic=u∈XIu≤c.Definition 16 (see [28]).
ForA∈Σ, we say that the genus of A is n denoted by γA=n if there is an odd map G∈CA,ℝn\0 and n is the smallest integer with this property.Lemma 17 (see [28]).
LetI be an even C1 functional on X and satisfy the (PS)-condition. For any n∈ℕ, set
(21)Σn=A∈ΣγA≥n,cn=infA∈∑nsupu∈AIu.(i)
IfΣn≠0 and cn∈ℝ, then cn is a critical value of I(ii)
If there existsl∈ℕ such that cn=cn+1=⋯=cn+l=c∈ℝ, and c≠I0, then γKc≥l+1Remark 18.
From Remark 7.3 in [28], we know that if Kc∈Σ and γKc>1, then Kc contains infinitely many distinct points; that is, I has infinitely many distinct critical points in X.


## 2.1. Fractional Sobolev Space

In this subsection, we present some basic definitions and notations of the fractional calculus [25, 26]. Moreover, we introduce a fractional Sobolev space and some properties of this space [14].Definition 4 (see [25]).
Forβ>0, the left and right Riemann-Liouville fractional integrals of order β of a function u:a,b⟶ℝ are given by
(8)Dat−βut=1Γβ∫att−sβ−1usds,Dtb−βut=1Γβ∫tbs−tβ−1usds,respectively, provided that the right-hand-side integrals are pointwise defined on a,b, where Γ⋅ is the gamma function.Definition 5 (see [25]).
Forn−1≤β<n (n∈ℕ), the left and right Riemann-Liouville fractional derivatives of order β of a function u:a,b⟶ℝ are given by
(9)Datβut=dndtnDatβ−nut,Dtbβut=−1ndndtnDtbβ−nut.Remark 6.
Whenβ=1, one can obtain from Definitions 4 and 5 that
(10)Dat1ut=u′t,tDb1ut=−u′t,where u0 is the usual first-order derivative of u.Definition 7 (see [14]).
For0<α≤1 and 1<p<∞, the fractional derivative space E0α,p is defined by the closure of C0∞0,T,ℝ with respect to the following norm:
(11)uEα,p=uLpp+0DtαuLpp1/p,where uLp=∫0Tutpdt1/p is the norm of Lp0,T,ℝ.Remark 8.
It is obvious that, foru∈E0α,p, one has
(12)u,0Dtαu∈Lp0,T,ℝ,u0=uT=0.Lemma 9 (see [14]).
Let0<α≤1 and 1<p<∞. The fractional derivative space E0α,p is a reflexive and separable Banach space.Lemma 10 (see [14]).
Let0<α≤1 and 1<p<∞. For u∈E0α,p, one has
(13)uLp≤CpD0tαuLp,where
(14)Cp=TαΓα+1>0,is a constant. Moreover, if α>1/p, then
(15)u∞≤C∞0DtαuLp,where u∞=maxt∈0,Tut is the norm of C0,T,ℝ,
(16)C∞=Tα−1/pΓααq−q+11/q>0,q=pp−1>1.Remark 11.
By (13), we can consider the space E0α,p with norm
(17)uEα,p=0DtαuLp,in what follows.Lemma 12 (see [14]).
Let1/p<α≤1 and 1<p<∞. The imbedding of E0α,p in C0,T,ℝ is compact.


## 2.2. Critical Point Theory

Now, we present some necessary definitions and theorems of the critical point theory [27, 28].LetX be a real Banach space, and I∈C1X,ℝ which means that I is a continuously Fréchet differentiable functional. Moreover, let Bρ0 be an open ball in X and ∂Bρ0 denote its boundary.Definition 13 (see [27]).
LetI∈C1X,ℝ. If any sequence uk⊂X for which Iuk is bounded and I′uk⟶0 as k⟶∞ possesses a convergent subsequence in X, then we say that I satisfies the (PS)-condition.Lemma 14 (see [28]).
LetX be a real Banach space, and I∈C1X,ℝ satisfying the (PS)-condition. Suppose that I0=0 and C1, there are constants ρ,σ>0 such that I∂Bρ0≥σ; C2 there is an e∈X\Bρ0¯ such that Ie≤0.
Then,I possesses a critical value c≥σ. Moreover, c can be characterized as
(18)c=infγ∈Γmaxs∈0,1Iγs,where
(19)Γ=γ∈C0,1,Xγ0=0,γ1=e.Lemma 15 (see [27]).
LetX be a real Banach space, and I∈C1X,ℝ satisfies the (PS)-condition. If I is bounded from below, then c=infXI is a critical value of I.In order to find the infinitely many critical points ofI, we introduce the following genus properties. Let
(20)Σ=A⊂X−0AisclosedinXandsymmetricwithrespectto0,Kc=u∈XIu=c,I′u=0,Ic=u∈XIu≤c.Definition 16 (see [28]).
ForA∈Σ, we say that the genus of A is n denoted by γA=n if there is an odd map G∈CA,ℝn\0 and n is the smallest integer with this property.Lemma 17 (see [28]).
LetI be an even C1 functional on X and satisfy the (PS)-condition. For any n∈ℕ, set
(21)Σn=A∈ΣγA≥n,cn=infA∈∑nsupu∈AIu.(i)
IfΣn≠0 and cn∈ℝ, then cn is a critical value of I(ii)
If there existsl∈ℕ such that cn=cn+1=⋯=cn+l=c∈ℝ, and c≠I0, then γKc≥l+1Remark 18.
From Remark 7.3 in [28], we know that if Kc∈Σ and γKc>1, then Kc contains infinitely many distinct points; that is, I has infinitely many distinct critical points in X.


## 3. Proof of Theorem1

In this section, we discuss the existence of nontrivial weak solutions of BVP (1) when the nonlinearity ft,x is p2−1-superlinear in x at infinity.Define the functionalI:E0α,p⟶ℝ by
(22)Iu=1bp2a+b∫0T0Dtαutpdtp−∫0TFt,utdt−apbp2=1bp2a+buEα,pp−∫0TFt,utdt−apbp2.It is easy to verify from (15), (17), and f∈C0,T×ℝ,ℝ that the functional I is well defined on E0α,p and is a continuously Fréchet differentiable functional; that is, I∈C1E0α,p,ℝ. Furthermore, we have
(23)I′u,v=a+buEα,ppp−1∫0Tϕp0Dtαut0Dtαvtdt−∫0Tft,utvtdt,∀u,v∈E0α,p,which yields
(24)I′u,u=a+buEα,ppp−1uEα,pp−∫0Tft,ututdt.In the following, for simplicity, let(25)Mpk=a+bukEα,pp,Mp=a+buEα,pp.Lemma 19.
Assume that (H11) holds. Then, I satisfies the (PS)-condition in E0α,p.Proof.
Letuk⊂E0α,p be a sequence such that
(26)Iuk≤K,I′uk⟶0ask⟶∞,where K>0 is a constant. We first prove that uk is bounded in E0α,p. From the continuity of μFt,x−xft,x and (H11), we obtain that there exists a constant c>0 such that
(27)Ft,x≤1μxft,x+c,∀t,x∈0,T×ℝ.Thus, by (22) and (24), we have
(28)K≥Iuk≥1bp2Mpkp−1μ∫0Tft,uktuktdt−cT−apbp2=1bp2Mpkp−1μMpkp−1ukEα,pp+1μI′uk,uk−cT−apbp2≥Mpkp−11p2−1μukEα,pp+abp2−1μI′ukE0α,p∗ukEα,p−cT−apbp2,which together with I′uk⟶0 as k⟶∞ yields
(29)K≥Mpkp−11p2−1μukEα,pp+abp2−ukEα,p−cT−apbp2ask⟶∞.Then, it follows fromμ>p2 that uk is bounded in E0α,p.SinceE0α,p is a reflexive Banach space (see Lemma 9), going if necessary to a subsequence, we can assume uk⟶u in E0α,p. Hence, from I′uk⟶0 as k⟶∞ and the definition of weak convergence, we have
(30)I′uk−I′u,uk−u=I′uk,uk−u−I′u,uk−u≤I′ukE0α,p∗uk−uEα,p−I′u,uk−u⟶0ask⟶∞.In addition, we obtain from (15), (17), and Lemma 12 that uk is bounded in C0,T,ℝ and uk−u∞⟶0 as k⟶∞. Thus, there exists a constant c1>0 such that
(31)ft,ukt−ft,ut≤c1,∀t∈0,T,which yields
(32)∫0Tft,ukt−ft,utukt−utdt≤c1Tuk−u∞⟶0ask⟶∞.Moreover, by the boundedness ofuk in E0α,p, one has
(33)Mpkp−1−Mpp−1∫0TϕpD0tαut0Dtαukt−D0tαutdt=Mpkp−1−Mpp−1I1′u,uk−u⟶0ask⟶∞,where I1′ is the Fréchet derivative of I1:E0α,p→ℝ defined by
(34)I1u=1p∫0TD0tαutpdt.From (23), we have
(35)I′uk−I′u,uk−u+∫0Tft,ukt−ft,utukt−utdt=Mpkp−1∫0TϕpD0tαuktD0tαukt−D0tαutdt−Mpp−1∫0TϕpD0tαutD0tαukt−D0tαutdt=Mpkp−1∫0TϕpD0tαukt−ϕpD0tαutD0tαukt−D0tαutdt+Mpkp−1−Mpp−1∫0TϕpD0tαutD0tαukt−D0tαutdt,which together with (30)–(33) yields
(36)∫0TϕpD0tαukt−ϕpD0tαutD0tαukt−D0tαutdt⟶0,as k⟶∞.Following (2.10) in [29], there exist two constants c2,c3>0 such that
(37)∫0TϕpD0tαukt−ϕpD0tαutD0tαukt−D0tαutdt≥c2∫0TD0tαukt−D0tαutpdt,p≥2,c3∫0TD0tαukt−D0tαut2D0tαukt+D0tαut2−pdt,1<p<2.When1<p<2, based on the Hölder inequality, we get
(38)∫0TD0tαukt−D0tαutpdt≤∫0TD0tαukt−D0tαut2D0tαukt+D0tαut2−pdtp/2⋅∫0TD0tαukt+D0tαutpdt2−p/2≤c4ukEα,pp+uEα,pp2−p/2∫0TD0tαukt−D0tαut2D0tαukt+D0tαut2−pdtp/2,where c4=2p−12−p/2>0 is a constant, which together with (37) implies
(39)∫0TϕpD0tαukt−ϕpD0tαutD20tαukt−D0tαutdt≥c3c4−2/pukEα,pp+uEα,ppp−2/puk−uEα,p2,1<p<2.Whenp≥2, by (37), we have
(40)∫0TϕpD0tαukt−ϕpD0tαutD0tαukt−D0tαutdt≥c2uk−uEα,pp,p≥2.Then, it follows from (36), (39), and (40) that
(41)uk−uEα,p⟶0ask⟶∞.Hence,I satisfies the (PS)-condition.Proof of Theorem 1.
From (H12), there exist two constants 0<ε<1,δ>0 such that
(42)Ft,x≤1−εap−1pCppxp,∀t∈0,T,x∈ℝwithx≤δ,where Cp>0 is a constant defined in (13). Let ρ=δ/C∞ and σ=εαp−1ρp/p, where C∞>0 is a constant defined in (15). Then, by (15) and (17), we have
(43)u∞≤C∞uEα,p=δ,∀u∈E0α,pwithuEα,p=ρ,which together with (13), (17), (22), and (42) implies
(44)Iu=1bp2a+buEα,ppp−∫0TFt,utdt−apbp2≥ap−1puEα,pp−1−εap−1pCpp∫0Tutpdt≥ap−1puEα,pp−1−εap−1puEα,pp=εap−1puEα,pp=σ,∀u∈E0α,pwithuEα,p=ρ,which means that the condition C1 in Lemma 14 is satisfied.From (H11), a simple argument using the very definition of the derivative shows that there exist two constants c1,c2>0 such that
(45)Ft,x≥c1xµ−c2,∀t,x∈0,T×ℝ.Then, for anyu∈E0α,p\0,ξ∈ℝ+, we can obtain from (22) and μ>p2 that
(46)Iξu=1bp2a+bξuEα,ppp−∫0TFt,ξutdt−apbp2≤1bp2a+bξuEα,ppp−c1∫0Tξutμdt+c2T−apbp2=1bp2a+bξpuEα,ppp−c1ξμuLμμ+c2T−apbp2⟶−∞asξ⟶∞.Thus, takingξ0 large enough and letting e=ξ0u, we have Ie≤0. Hence, the condition C2 in Lemma 14 is also satisfied.Finally, byI0=0, Lemmas 14 and 19, we get a critical point u∗ of I satisfying Iu∗≥σ>0, and so u∗ is a nontrivial solution of BVP (1).


## 4. Proofs of Theorems2 and 3

In this section, we discuss the existence and multiplicity of nontrivial weak solutions of BVP (1) when the nonlinearity ft,x is p2−1-sublinear in x at infinity.Lemma 20.
Suppose that (H21) is satisfied. Then, I is bounded from below in E0α,p.Proof.
From (H21), one has
(47)Ft,u≤dtur1,∀t,u∈0,T×ℝ,which together with (15)–(22) yields
(48)Iu≥1bp2a+buEα,ppp−∫0Tdtutr1dt−apbp2≥1bp2a+buEα,ppp−dL1u∞r1−apbp2≥1bp2a+buEα,ppp−C∞r1dL1uEα,pr1−apbp2.Since1<r1<p2, (48) yields Iu⟶∞ as uEα,p⟶∞. Hence, I is bounded from below.Lemma 21.
Assume that (H21) holds. Then, I satisfies the (PS)-condition in E0α,p.Proof.
Let be a sequence such that(49)Iuk≤K,I′uk⟶0ask⟶∞,where K>0 is a constant. Then, (48) implies that uk is bounded in E0α,p. The remainder of proof is similar to the proof of Lemma 19, so we omit the details.Proof of Theorem 2.
From Lemmas15, 20, and 21, we obtain c=infE0α,pIu which is a critical value of I; that is, there exists a critical point u∗∈E0α,p such that Iu∗=c.Now, we showu∗≠0. Let u0∈W01,2I,ℝ∩E0α,p\0 and u0∞=1, from (22) and (H22), we get
(50)Isu0=1bp2a+bsu0Eα,ppp−∫0TFt,su0tdt−apbp2=1bp2a+bspu0Eα,ppp−∫IF,su0tdt−apbp2≤1bp2a+bspu0Eα,ppP−ηsr2∫Iu0tr2dt−apbp2,0<s≤δ.Since1<r2<p2, (50) implies I(su0) < 0 for s > 0 small enough. Then, Iu∗=c<0; hence, u∗ is a nontrivial critical point of I, and so u∗ is a nontrivial solution of BVP (1).Proof of Theorem 3.
From Lemmas20 and 21, we obtain that I∈C1E0α,p,ℝ is bounded from below and satisfies the (PS)-condition. In addition, (22) and (H23) show that I is even and I0=0.Fixingn∈ℕ, we take n disjoint open intervals Ii such that ∪i=1nIi⊂I.Letui∈W01,2Ii,ℝ∩E0α,p\0 and uiEα,p=1, and
(51)En=spanu1,u2,⋯,un,Sn=u∈EnuEα,p=1.Foru∈En, there exists λi∈ℝ, such that
(52)ut=∑i=1nλiuit,∀t∈0,T.Thus, we get(53)uEα,pp=∫0TD0tαutpdt=∑i=1nλip∫IiD0tαuitpdt=∑i=1nλip∫0TD0tαuitpdt=∑i=1nλipuiEα,pp=∑i=1nλip,∀u∈En.From (15)–(22), (52), and (H22), for u∈Sn, one has
(54)Isu=1bp2a+bsuEα,ppp−∫0TFt,sutdt−apbp2=1bp2a+bspp−∑i=1n∫IiFt,sλiuitdt−apbp2≤1bp2a+bspp−ηsr2∑i=1nλir2∫Iiuitr2dt−apbp2,0<s≤δC∞λ∗,where λ∗=maxi∈1,2,⋯,nλi>0 is a constant. Since 1<r2<p2, it follows from (54) that there exist constants ϵ,σ>0 such that
(55)Iσu<−ε,∀u∈Sn.Let(56)Snσ=σuu∈Sn,Λ=λ1,λ2,⋯,λn∈ℝn∑i=1nλip<σp.Then, we obtain from (55) that
(57)Iu<−ε,∀u∈Snσ,which, together with the fact that I is even and I0=0, yields
(58)Snσ⊂I−ε∈∑.From (53), it is seen that the mapping λ1,λ2,⋯,λn⟶∑i=1nλiuit from ∂Λ to Snσ is odd and homeomorphic. Hence, by some properties of the genus (Propositions 7.5 and 7.7 in [28]), we deduce that
(59)γI−∈≥γSnσ=n.Thus,I−ε∈∑n and so Σn≠0. Let
(60)cn=infA∈∑nsupu∈AIu.It follows fromI is bounded from below that −∞<cn≤−ε<0. That is, for any n∈ℕ, cn is a real negative number. Hence, by Lemma 17, I admits infinitely many nontrivial critical points, and so BVP (1) possesses infinitely many nontrivial negative energy solutions.Obviously the following assumption implies (H21)–(H23).(H24).
ft,x=rgtxr−2x, where 1<r<p2 is a constant, g∈C0,T,ℝ, and there exists an open interval I⊂0,T such that gt>0,∀t∈I.As a direct result, we have the following result.Corollary 22.
Let (H24) be satisfied. Then, BVP (1) possesses infinitely many nontrivial weak solutions.



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