# Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point Theory

Authors: Yang Wang; Yansheng Liu; Yujun Cui
Journal: Journal of Function Spaces
 (2017)

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## Abstract

This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem:DT-αaxD0+αux=fx,ux,x∈0,T, u0=uT=0, where α∈1/2,1, ax∈L∞0,T with a0=essinfx∈0,Tax>0, DT-α and D0+α stand for the left and right Riemann-Liouville fractional derivatives of order α, respectively, and f:0,T×R→R is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.


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## Body

## 1. Introduction

Consider the following nonlinear fractional boundary value problem (BVP for short):(1)DT-αaxD0+αux=fx,ux,x∈0,T,u0=uT=0,where α∈1/2,1, ax∈L∞0,T with a0=essinfx∈0,Tax>0, DT-α and D0+α stand for the left and right Riemann-Liouville fractional derivatives of order α, respectively, and f:0,T×R→R is continuous.Fractional calculus has numerous applications in various fields, including signal processing and control, fractal theory, neural network model, mechanics and engineering, and chemical physics. For details, see [1–5] and the references therein. Fractional differential operators have attracted many researchers’ attention because of their applications in modeling physical phenomena exhibiting anomalous diffusion. Recently, the existence of solutions for nonlinear fractional differential equations was obtained by using nonlinear analysis, including some fixed theorems, coincidence degree theory, the monotone iterative method, and the method of upper-lower solutions. It is worth mentioning that the methods above cannot be used to deal with BVP (1) because it is very difficult to find the equivalent integral equation.On the other hand, variational methods have proved to be a very effective approach in dealing with the existence and multiple solutions for fractional boundary value problem including both left and right fractional derivatives; see [6–16].For example, in [7], by using mountain pass theorem, Jiao and Zhou studied the existence of at least one nontrivial solution for the following fractional boundary value problem:(2)DT-αD0+αut=∇Ft,ut,t∈0,T,u0=uT=0,where α∈0,1, F:0,T×RN→R is a suitable given function, and ∇Ft,x is the gradient of F with respect to x.In [8], by means of variational methods and critical point theory due to Bonanno and Marano, the existence of three distinct weak solutions is obtained for the following fractional boundary value problem:(3)DT-αatD0+αut=λFut,ut,vt,t∈0,T,DT-βbtD0+βut=λFvt,ut,vt,t∈0,T,u0=uT=0,v0=vT=0,where λ is a positive real parameter, α,β∈(0,1],a,b∈L∞[0,T] with a0=essinfx∈0,Tax>0 and b0=essinfx∈0,Tax>0, F:0,T×R2→R is a function such that F·,x,y is continuous in 0,T, and Ft,·,· is a C1 function.In [9], by using fountain theorem and dual fountain theorem, the existence of infinitely many solutions is established for the following fractional boundary value problem:(4)ddt12D0+-βu′t+12DT--βu′t+∇Ft,ut=0,t∈0,T,u0=uT=0,where D0+-β and DT--β are the left and right Riemann-Liouville fractional integrals of order 0≤β≤1, respectively.To the best of our knowledge, there are fewer results on multiple solutions for system (1). The purpose of this paper is to fill this gap. Inspired by the above references, we apply variant fountain theorems to study the existence of infinitely many small or high energy solutions for BVP (1). It should be noted that the (P.S.) condition is not needed, which is an important condition assumed in [7, 9]. As is well known, the so-called Ambrosetti-Rabinowitz condition(5)0<θFx,u≤ufx,ufor some θ>2 is often assumed to deduce the boundedness of the (P.S.) sequences. By taking advantage of the variant fountain theorems, the Ambrosetti-Rabinowitz condition is not needed.The rest of this paper is organized as follows. Section2 contains preliminary results. In Section 3, we apply variant fountain theorems to prove the existence of infinitely many small or high energy solutions for BVP (1). In Section 4, two examples are presented to illustrate the main results.


## 2. Preliminaries

To obtain multiple solutions for BVP (1), it is necessary to construct a suitable function space and introduce some definitions, preliminary lemmas.Definition 1 (see [17]).
Letf be a function defined on a,b. The left and right Riemann-Liouville fractional derivatives of order α for function f, denoted by Da+αfx and Db-αfx, are defined by (6)Da+αfx=1Γn-αdndxn∫axx-sn-α-1fsds,Db-αfx=-1nΓn-αdndxn∫xbs-xn-α-1fsds,where x∈a,b, n-1≤α<n, and n∈N.Lemma 2 (see [18]).
Iffa=fb=0, f′∈L∞a,b,R, g∈L1a,b,R, or ga=gb=0, g′∈L∞a,b,R, f∈L1a,b,R, then(7)∫abDa+αfxgxdx=∫abDb-αgxfxdx.
Denote(8)uLp=∫0Tuxpdx1/p,u∞=maxx∈0,Tux.Definition 3 (Let0<α≤1).
The fractional derivative spaceE0α is defined by the closure of C0∞0,T,R with respect to the norm (9)uα=∫0Tux2dx+∫0TaxD0+αux2dx1/2,∀u∈E0α.Remark 4.
(i) The fractional derivative spaceE0α is the space of functions u∈L20,T,R having an α-order Riemann-Liouville fractional derivative D0+αu∈L20,T,R, ax∈L∞0,T with a0=essinfx∈0,Tax>0, and u0=uT=0. (ii) E0α is a reflexive and separable Banach space.Lemma 5 (see [18], let 1/2<α≤1).
For anyu∈E0α, one has(10)uL2≤TαΓα+1D0+αuL2.u∞≤Tα-1/2Γα2α-1D0+αuL2.Remark 6.
(i)
By (10), we can get (11)uL2≤TαΓα+1a0∫0TaxD0+αux2dx1/2,(12)u∞≤Tα-1/2Γαa02α-1∫0TaxD0+αux2dx1/2.(ii)
According to (11) and (12), for p≥1, we can obtain(13)uLp≤Tα-1/2+1/pΓαa02α-1∫0TaxD0+αux2dx1/2,(iii)
From (11), it is easy to see that the norm of E0α defined in (9) is equivalent to the norm (14)uα=∫0TaxD0+αux2dx1/2,∀u∈E0α.(iv)
E0α is a separable Hilbert space and the corresponding inner product is (15)u,v=∫0TaxD0+αuxDT-αvxdx,u,v∈E0α.Lemma 7 (see [18], let 1/2<α≤1).
If the sequenceuk converges weakly to u in E0α, that is, uk⇀u, then uk→u in C0,T, that is, u-uk∞→0 as k→∞.In the following, we denoteE=E0α,u=uα,up=uLp for convenience.Definition 8.
A functionu∈E is a weak solution of BVP (1) if (16)∫0TaxD0+αuxD0+αvx-∫0Tfx,uxvxdx=0,∀v∈E.The energy functionalI:E→R associated with BVP (1) is defined by (17)Iu=12∫0TaxD0+αux2dx-∫0TFx,uxdx,∀u∈E,where Fx,u=∫0ufx,sds. It is easily seen that I∈C1E,R.For anyvx∈E, one has (18)I′u,v=∫0TaxD0+αuxD0+αvx-∫0Tfx,uxvxdx.Moreover, the critical points of I correspond to the weak solutions of BVP (1).To prove our main results, we need the following two variant fountain theorems in [19].LetX be a Banach space with the norm · and X=⨁j∈NXj¯ with dim⁡Xj<∞ for each j∈N. Set Wk=⨁j=1kXj, Zk=⨁j=k∞Xj¯, Bk=u∈Wk:u≤ρk, Sk=u∈Zk:u=rk, where ρk>rk>0.Consider a family ofC1 functions Iλ:X→R defined by (19)Iλu=Au-λBu,λ∈1,2.Lemma 9.
Assume that the functionalIλ defined above satisfies(A1)
Iλ maps bounded sets into bounded sets uniformly for λ∈1,2, and Iλ-u=Iλu for all λ,u∈1,2×X;(A2)
Bu≥0 for all u∈X,Au→∞ or Bu→∞ as u→∞; or(A3)
Bu≤0 for all u∈X, Bu→-∞ as u→∞;(A4)
There existρk>rk>0 such that (20)bkλ=infu∈Zk,u=rk⁡Iλu>akλ=maxu∈Wk,u=ρkIλu,∀λ∈1,2.Then (21)bkλ≤ckλ=infγ∈Γk⁡maxu∈Bk⁡Iλγu,∀λ∈1,2,where Γk=  γ∈CBk,X:γisodd,γ∂Bk=id. Moreover, for almost every λ∈1,2, there exists a sequence unkλ such that(22)supn⁡unkλ<∞,Iλ′unkλ⟶0,Iλunkλ⟶ckλasn⟶∞.Lemma 10.
Assume that the functionalIλ defined above satisfies(B1)
Iλ maps bounded sets into bounded sets uniformly for λ∈1,2, and Iλ-u=Iλu for all λ,u∈1,2×X;(B2)
Bu≥0 for all u∈X, Bu→∞ as u→∞ on any finite dimensional subspace of X;(B3)
There existρk>rk>0 such that (23)akλ=infu∈Zk,u=ρk⁡Iλu≥0,bkλ=maxu∈Wk,∥u∥=rk⁡Iλu<0,∀λ∈1,2,dkλ=infu∈Zk,∥u∥≤ρkIλu⟶0ask⟶∞uniformlyforλ∈1,2.Then there exist λn→1, un(λn)∈Wn such that(24)Iλn′Wnuλn=0,Iλnuλn⟶ckasn⟶∞,where ck∈dk2,bk1. In particular, if uλn has a convergent subsequence for every k, then I1 has infinitely many nontrivial critical points {uk}∈X∖{0} satisfying I1(uk)→0- as k→∞.SinceE is a separable Hilbert space in terms of Remark 6. Let ej be an orthonormal basis of E and define Xj=span⁡ej, Wk=⨁j=1kXj, Zk=⨁j=k∞Xj¯. In order to apply Lemmas 9 and 10 to prove the existence of infinitely many solutions of BVP (1), we define A, B, and Iλ on fractional derivative space E by (25)Au=12∫0TxD0+αux2dx,Bu=∫0TFx,uxdx,Iλu=Au-λBu=12u2-λ∫0TFx,uxdx,∀u∈E,λ∈1,2.For convenience, we list the following assumptions.(H1)
There exist constantsc1>0, c2>0, and p>2 such that (26)fx,u≤c1u+c2up-1,∀x∈0,T,u∈R.(H2)
There existτ>2 and M>0 such that (27)-τFx,u+ufx,u≥0,u≥M,uniformly  forx∈0,T.(H3)
Fx,u≥0,∀x,u∈0,T×R.(H4)
Fx,-u=Fx,u, ∀x,u∈0,T×R.(H5)
There existη∈1,2 and bx∈L2/2-η0,T with bx≥0 such that (28)fx,u≤bxuη-1,∀x,u∈0,T×R.(H6)
There existr∈1,η and d>0 such that (29)limu→∞⁡Fx,uur>d,uniformly  forx∈0,T.(V)
ax∈L∞0,T, and essinfx∈0,T⁡ax=a0>0.(V′)
ax∈L∞0,T, and essinfx∈0,Tax=a0≥(τ/2-1)-12hTα/Γα+12>0, where (30)h=inf⁡l:-τFx,u+ufx,u≤lu2,x∈0,T,u≤M.


## 3. Main Results

Theorem 11.
Assume that (H1)–(H4), (V′) hold. Then BVP (1) possesses infinitely many high energy solutions uk∈E∖0 satisfying(31)12∫0TaxD0+αukx2dx-∫0TFx,ukxdx⟶+∞,ask⟶∞.Proof.
For anyε>0, it follows from (H1) that there exist c3>0 and c4>0 such that (32)Fx,u≤c3u2+c4up.Combining (32) and (13), it is easily seen that Iλ maps bounded sets into bounded sets uniformly for λ∈1,2. By (H4), Iλ-u=Iλu for all λ,u∈1,2×E. Thus, condition (A1) holds. Assumption (H3) implies that Bu≥0. Condition (A2) holds for the fact that A(u)=1/2u2→∞ as n→∞ and Bu≥0.
In what follows, we verify condition (A4). For this sake, we need to prove that there exist two sequences ρk>rk>0 such that (33)bkλ=infu∈Zk,u=rk⁡Iλu>0,∀λ∈1,2,(34)akλ=maxu∈Wk,u=ρkIλu<0,∀λ∈1,2.
First, we prove that (33) is true.Letαrk=supu∈Zk,u=1ur with r≥2. We claim that αrk→0 as k→∞. Indeed, if not, then there exist ε0 and uj⊂E with uj⊥Wkj-1 such that uj=1 and ujr>ε0, where kj→∞ as j→∞. For any v∈E, we may choose wj∈Wkj-1 such that wj→v as j→∞. (35)uj,v=uj,wj-v≤ujwj-v⟶0,j⟶∞.Thus, uj⇀0 in E. By Lemma 7, we have uj→0 in C0,T. Consequently, ujr→0. This contradicts the fact that ujr>ε0.Foru∈Zk, by (32), we have (36)Iλu≥12u2-2c3u22-2c4upp≥12u2-2c3α22ku2-2c4αppkup.Choose rk=1/α2(k)+αp(k). Then rk→∞ as k→∞. Then, for any u∈Zk with u=rk, we obtain (37)Iλu≥12u2-2c3α22kα2k+αpk2-2c4αppkα2k+αpkp≥12rk2-2c3-2c4>0.Therefore,(38)bkλ=infu∈Zk,u=rk⁡Iλu>0,∀λ∈1,2.Next, we prove that (34) is true.We first prove that, for any finite dimensional subspaceH of E, there exists a constant ε0>0 such that (39)meas⁡x∈0,T:ux≥ε0u≥ε0,∀u∈H∖0.If not, there exists a sequence un∈H∖0 such that (40)meas⁡x∈0,T:unx≥1nu<1n,∀n∈N.Set vn=un/un. Then vn=1, and (41)meas⁡x∈0,T:vnx≥1n<1n,∀n∈N.On account of the boundedness of vn and dim⁡H<∞, passing to a subsequence if necessary, we may assume that vn→v in E for some v∈H. According to the equivalence of any norm in finite dimensional space, we get (42)∫0Tvnx-vx2dx⟶0,n⟶∞.Since v≠0, there exists a constant δ0>0 such that (43)meas⁡x∈0,T:vx>δ0≥δ0.Set Ωn=x∈0,T:vnx<1/n, Ωnc=x∈0,T:vnx≥1/n, and Ω0=x∈0,T:vx≥δ0. Then for n large enough, it follows from (41) and (43) that (44)meas⁡Ωn∩Ω0≥meas⁡Ω0-meas⁡Ωnc≥δ0-1n≥23δ0.Consequently, (45)∫0Tvnx-vx2dx≥∫Ωn∩Ω0vnx-vx2dx≥∫Ωn∩Ω0vx2-2vnxvxdx≥δ0δ0-2nmeas⁡Ωn∩Ω0≥29δ03>0.This contradicts (42). Hence, (39) holds.SinceWk is a finite dimensional subspace of E, from (39), for any k∈N, there exists εk>0 such that (46)meas⁡Nu≥εk,u∈Wk∖0,where Nu={x∈0,T:ux≥εku}. In view of (H2), for any k∈N, there exists a constant Lk>0 such that (47)Fx,u≥u2εk3,u≥Lk.By (46), for any u∈Wk with u≥Lk/εk, we get ux≥Lk for all x∈Nu. Therefore, (48)Iλu=12u2-λ∫0TFx,uxdx,≤12u2-εk2u2meas⁡Nu1εk3≤-12u2.If we take ρk>max⁡rk,Lk/εk, we get(49)akλ=maxu∈Wk,u=ρk⁡Iλu<0,∀λ∈1,2.Until now, all the conditions of Lemma9 hold. Hence, for λ∈1,2, there exists a sequence unkλn=1∞ such that (50)Supn⁡unkλ<∞,Iλ′unkλ⟶0,Iλunkλ⟶ckλ,n⟶∞.Furthermore,(51)ckλ≥bkλ≥12rk2-2c3-2c4≔bk¯⟶∞,k⟶∞.ckλ=infγ∈Γk⁡maxu∈Bk⁡Iλγu≤maxu∈Bk⁡I1u≔ck¯.Thus,(52)bk¯≤ckλ≤ck¯,λ∈1,2.Choose a sequenceλm→1 such that (50) holds. Now we show that unkλmn=1∞ possesses a strong convergent subsequence. In fact, the sequence {unk(λm)}n=1∞ is bounded. Then there exists ukλm∈E such that unkλm⇀ukλm as n→∞. By Lemma 7, unkλm→ukλm in C0,T as n→∞. From Iλm′unkλm→0, we know (53)Iλm′unkλm,unkλm-ukλm⟶0asn⟶∞.Also from unkλm⇀ukλm in E, we get (54)Iλm′ukλm,unkλm-ukλm⟶0asn⟶∞.Moreover, by the continuity off, it follows that (55)∫0Tfx,unkλm-fx,ukλmunkλm-ukλmdx⟶0asn⟶∞.Hence, (56)unkλm-ukλm2=Iλm′unkλm,unkλm-ukλm-Iλm′ukλm,unkλm-ukλm+λm∫0Tfx,unkλm-fx,ukλmunkλm-ukλmdx⟶0.Thus, we conclude that unkλm→ukλm in E as n→∞. By (50) and (52), we can get (57)Iλm′ukλm=0,Iλmukλm∈bk¯,ck¯fork≥k1.Next, we shall prove thatukλmm=1∞ is bounded.By (H2), (V′), (11), and (30), we have(58)τIλmukλm-Iλm′ukλm,ukλm=τ2-1ukλm2+λm∫0Tfx,ukλmukλm-τFx,ukλmdx=τ2-1ukλm2+λm∫ukλm≤Mfx,ukλmukλm-τFx,ukλmdx+λm∫ukλm>Mfx,ukλmukλm-τFx,ukλmdx≥τ2-1ukλm2-λm∫ukλm≤Mfx,ukλmukλm-τFx,ukλmdx≥τ2-1ukλm2-2hukλm22≥τ2-11-2hτ2-1-1TαΓα+121a0ukλm2.Therefore, {uk(λm)}m=1∞ is bounded in E. Similar arguments show that uk(λm)→uk in E as m→∞(k≥k1). Then uk is a critical point of I=I1 with I(uk)∈[bk¯,ck¯]. According to bk¯→∞ as k→∞, we know that BVP (1) has infinitely many nontrivial high energy solutions.Theorem 12.
Assume that (H3)–(H6) and (V) hold. Then BVP (1) possesses infinitely many small energy solutions uk∈E∖{0} satisfying(59)12∫0TaxD0+αukx2dx-∫0TFx,ukxdx⟶0-,ask⟶∞.Proof.
By (H5), we can get(60)Fx,u≤1ηbxuη.Combining (60) and (13), it is easily seen that Iλ maps bounded sets into bounded sets uniformly for λ∈1,2. It follows from (H4) that Iλ-u=Iλ(u) for all (λ,u)∈[1,2]×E. Thus, condition (B1) holds. In the following, we verify condition (B2). By (H3), Bu≥0 is obvious. According to (H6), there exists M>0 such that (61)Fx,u≥dur,∀u>M.Suppose H is a finite dimensional subspace of E. As in the proof of Theorem 11, there exists ε0>0 such that (39) holds. Then (62)meas⁡Du≥ε0,where Du=x∈0,T:ux≥ε0u, ∀u∈H∖0. Hence, for any u∈H with u≥M/ε0, from (61), we have (63)Bu=∫0TFx,uxdx≥∫Duduxrdx≥dε0rurmeas⁡Du≥dε01+rur.This means that Bu→∞ as u→∞ on any finite dimensional subspace. Hence, condition (B2) holds.
Next, we claim condition (B3) holds. By (60) and Ho¨lder inequality, we get (64)Iλu≥12u2-1η∫0Tbxuηdx≥12u2-1ηbx2/2-ηu2η.According to the definition of αr(k) in the proof of Theorem 11, we have u2η≤α2ηkuη, ∀u∈Zk. Hence,(65)Iλu≥12u2-1ηbx2/2-ηα2ηkuη.Take ρk=4/ηbx2/2-ηα2ηk1/2-η. Then ρk→0+ as k→∞. Therefore,(66)akλ=infu∈Zk,u=ρk⁡Iλu≥14ρk2>0.Furthermore, for λ∈1,2 and u∈Zk with u≤ρk, we obtain (67)Iλu≥-1ηbx2/2-ηα2ηkρkη⟶0+,k⟶∞.So,(68)dkλ=infu∈Zk,u≤ρk⁡Iλu⟶0ask⟶∞.By (H5) and (H6), we have(69)Fx,u≥dur-1ηbxuη.If u∈Wk, by the equivalence of any norm in finite dimensional space and (69), we get (70)Iλu=12u2-λ∫0TFx,uxdx≤12u2-d∫0Turdx+1η∫0Tbxuηdx≤12u2-c1ur+c2bx2/2-ηuη.Take rk>0 small enough and rk<ρk. It follows that(71)bkλ=maxu∈Wk,u=rk⁡Iλu<0.Hence, condition (B3) holds.Therefore, by Lemma10, for every k∈N, there exist λn→1, unλn∈Wn such that(72)Iλn′Wnuλn=0,Iλnuλn⟶ck∈dk2,bk1asn⟶∞.Now we show thatuλn has a strong convergent subsequence for every k∈N. In fact by (60), Ho¨lder inequality, and Lemma 5, we get (73)uλn2=2Iλnuλn+2λn∫0TFx,uλndx≤2ck+2λn1ηbx2/2-ηuλnη.This means uλn is bounded in E. Without loss of generality, we may assume uλn⇀u in E. Since ej is the completely orthonormal basis of E,Wn=Le1,e2,…,en, u=∑j=1∞ej,uej. Let Pn:E↦Wn be the orthogonal projection operator. We know that Pnu=∑j=1nej,uej and Pnu→u in E as n→∞. Therefore uλn-pnu⇀0 in E as n→∞. Moreover, it follows from I1′u∈E∗ that (74)I1′u,uλn-Pnu⟶0n⟶∞.Also since I1′∈CE→E∗ and Pnu→u in E, we have (75)I1′Pnu-I1′u,uλn-Pnu⟶0n⟶∞.Therefore, by (74) and (75), we get (76)I1′Pnu,uλn-Pnu⟶0n⟶∞.Note that Pnuλn=uλn and Iλn′Pnuλn,Pnuλn-u→0 since uλn∈Wn and Iλn′Wnuλn=0. By the continuity of f and Lemma 7, it is easily seen that (77)λn∫0Tfx,uλnuλn-Pnudx⟶0,n⟶∞.∫0Tfx,Pnuuλn-Pnudx⟶0,n⟶∞.Therefore, (78)uλn-Pnu2=Iλn′uλn,uλn-Pnu-I1′Pnu,uλn-Pnu+λn∫0Tfx,uλnuλn-Pnudx-∫0Tfx,Pnuuλn-Pnudx⟶0.So, uλn-Pnu→0 in E as n→∞. Thus, uλn→u in E as n→∞. By Lemma 10, we know that I=I1 has infinitely many nontrivial critical points uk. Consequently, BVP (1) has infinitely many small energy solutions.


## 4. Examples

In this section, two examples are given to illustrate our results.Example 1.
Consider the following nonlinear fractional boundary value problem:(79)D1-α3Γα+12+xD0+αux=3u2ln⁡u+1+u3u+1,x∈0,1,u0=u1=0,where α∈1/2,1.
First, chooseτ=3 and M=1. A direct calculation shows that (80)Fx,u=u3ln⁡u+1,-3Fx,u+ufx,u=u4u+1≥0.Hence, assumption (H2) holds.Next, we verify assumption(V′). From (30) we have h=1/2. By direct calculation, one has a0=3/Γα+12≥τ/2-1-12hTα/Γα+12>0. Hence, assumption (V′) holds.Finally, it is easy to show that (H1), (H3), and (H4) hold. So, all the assumptions of Theorem 11 are satisfied. Consequently, BVP (79) has infinitely many high energy solutions.Example 2.
Consider the following nonlinear fractional boundary value problem:(81)D1-αx2+1D0+αux=32exu1/2,x∈0,1,u0=u1=0,where α∈1/2,1.
First, notice thatessinfx∈0,1⁡x2+1=1>0, which means that assumption (V) holds.
Moreover,(82)fx,u=32exu1/2≤2exu1/2.Take η=3/2 and bx=2ex. It is easy to show that assumption (H5) are satisfied.Chooser=5/4. By a direct calculation, one has Fx,u=exu3/2 and limu→∞⁡exu3/2/u5/4→∞. Hence, assumption (H6) holds.Finally, it is easy to show that (H3) and (H4) hold. So, all the assumptions of Theorem 12 are satisfied. Consequently, BVP (81) has infinitely many small energy solutions.



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