# Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

Authors: Dan Liu; Xuejun Zhang; Mingliang Song
Journal: Journal of Mathematics
 (2021)

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## Abstract

We deal with the following Sturm–Liouville boundary value problem:−Ptx′t′+Btxt=λ∇xVt,x,a.e.t∈0,1x0cosα−P0x′0sinα=0x1cosβ−P1x′1sinβ=0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.


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## Body

## 1. Introduction and Main Results

In this paper, we consider the Sturm–Liouville boundary value problem for second-order Hamiltonian systems:(1)−Ptx′t′+Btxt=λ∇xVt,x,a.e.t∈0,1,x0cosα−P0x′0sinα=0,x1cosβ−P1x′1sinβ=0,where λ>0,α∈0,π,β∈0,π,Bt∈L∞0,1,ℒsRn=Bt=bjkn×n|bjkt=bkjt,t∈0,1,bjkt∈L∞0,1, Pt∈C10,1,ℒsRn with Pt being positive definite for t∈0,1, and ∇xVt,x denotes the gradient of Vt,x for x∈Rn. We suppose that V:0,1×Rn⟶R satisfies the following:V0Vt,x is measurable in t for every x∈Rn and continuously differentiable in x for a.e. t∈0,1. Moreover, if n≥2, then there exist C0>0 and γ>0 such that(2)∇xVt,x≤C01+xγ,∀x∈Rn,a.e.t∈0,1.Ifn=1, then ∇xVt,x=ft,x:0,1×R⟶R is a L1-Carathéodory function.Noticing that whenn=1,α=0,β=π/2 or n=1,α=β=π/2, problem (1) becomes the mixed boundary value problem for Sturm–Liouville equations:(3)−Ptx′t′+Btxt=λft,x,a.e.t∈0,1,x0=x′1=0,or the Neumann boundary value problem for Sturm–Liouville equations:(4)−Ptx′t′+Btxt=λft,x,a.e.t∈0,1,x′0=x′1=0,which show that problems (3) and (4) are two special cases of (1).Recently, the existence of two nontrivial solutions and infinitely many solutions for problems (3) and (4) have been extensively studied and some useful results have been obtained (see [1–12]). Their tools are based on some abstract multiple theorems established by Bonanno and Candito [13], Bonanno and D’Aguì [6], and Ricceri [14]. However, linear term Bt≥0 is necessary in the conditions of theorems discussing the existence of two nontrivial solutions for problem (3) in [1, 2, 9, 10]. Bt≥0 and λ∈λ1,λ2⊂0,+∞ are necessary conditions in the existence theorems of infinitely many solutions for problem (3) in [3, 4]. For problem (4), Bt≡M≥0 and λ=1 are necessary in the existence theorems of two nontrivial solutions in [11, 12]. Bt∈C0,1,R+\0 and λ∈λ1,λ2⊂0,+∞ are also necessary in the existence of infinitely many solutions for problem (4) in [5]. In this paper, we are interested in the assumption without Bt≥0 and λ∈λ1,λ2⊂0,+∞ assumptions.Moreover, with the aid of variational methods, the multiplicity of periodic solutions for Hamiltonian systems has also been extensively investigated in many papers and books (see [15–23] and the references therein). In particular, in [24], using the linking theorem of Schechter [19, 20], Bonanno et al. have discussed the existence of two nontrivial solutions for second-order Hamiltonian systems with subquadratic potentials at zero. For infinitely many solutions of subquadratic second-order Hamiltonian systems, Zou and Li [22] obtained two existence theorems via the minimax technique of Fei [25], and Zhang and Liu [23] obtained an existence theorem via the variant fountain theorem of Zou [26]. After that, by the symmetric mountain pass theorem of Kajikiya [27], Yi and Tang [21] obtained an existence theorem, which unifies and improves upon theorems of Zou and Li and Zhang and Liu [22, 23].Inspired by the ideas of Ye and Tang and Bonanno et al. [21, 24], in this paper, we shall study the existence of two nontrivial solutions and infinitely many solutions for problem (1), where Vt,x is subquadratic at zero. Based on the index theory of Dong [15, 16], the linking theorem of Schechter [19, 20], and the symmetric mountain pass theorem of Kajikiya [27], we will prove the existence of two nontrivial solutions and infinitely many solutions. Applying the results to problems (3) and (4), we obtain some new theorems. Meanwhile, some examples of problems (3) and (4) are given to illustrate the validity of our result and point out that linear terms Bt<0 and λ∈0,+∞ are allowed, which show that our results are also new even in the cases of (3) and (4).Next we use the indexiα,βPB,να,βPB∈N×N defined in [15] (see Section 2) for all Bt∈L∞0,1,ℒsRn to give our main results.Theorem 1.
Assume thatV:0,1×Rn⟶R satisfies V0 andV1iα,βPB=να,βPB=0.V2 There exist η>0,0<ϑ<2 and a function gt∈L10,1,R+ such that(5)Vt,x≤gtxϑ,x≤η,x∈Rn.V3 There exists a constant κ1>κ0=δ0ηϑ−2gL1 such that(6)liminfc⟶0∫01Vt,cφdtc2φL22>κ1λ0,(7)liminfc⟶+∞∫01Vt,cφdtc2φL22>κ1λ0,where φ is an eigenfunction of the operator ΛB corresponding to the first eigenvalue λ0 and δ0>0 is a constant such that(8)x∞2≤δ0ax,x+∫01Btx,xdt,x∈Z=DΛ1/2.
Then, problem (1) has two nontrivial solutions for almost all λ∈1/2κ1,1/2κ0.Remark 1.
In Theorem1, the operator ΛB is defined as ΛBxt=−Ptx′′t+Btxt∀x∈DΛ. DΛ and ax,x are defined by (13) and (15) in Section 2, respectively. The existence of δ0 is proved in (33) of Section 3.Theorem 2.
Assume thatV:0,1×Rn⟶R satisfies V0 andV4Vt,x is even in x for a.e. t∈0,1.V5Vt,0=0 for a.e. t∈0,1 and(9)limx|⟶0|Vt,xx2=+∞uniformly fora.e.t∈0,1.
Then, problem (1) has infinitely many small-energy solutions for each λ>0.Remark 2.
In Theorem2, we do not need any restrictions on iα,βPB and να,βPB, which means that iα,βPB≠0,να,βPB=0 or iα,βPB≠0,να,βPB≠0 or iα,βPB=0,να,βPB≠0 or iα,βPB=να,βPB=0 are allowed.
The paper is arranged as follows. In Section2, we recall some useful conclusions of index theory for linear second-order Hamiltonian systems from [15, 16] and verify that problem (1) possesses a variational construction in Z. In Section 3, using the linking theorem of Schechter [19, 20] and the symmetric mountain pass theorem of Kajikiya [27], we prove Theorems 1 and 2. In Section 4, we investigate their applications to Sturm–Liouville equations with the mixed boundary value conditions and the Neumann boundary value conditions. Meanwhile, some examples are given to show that our results are also new even in the cases of problems (3) and (4).


## 2. Preliminaries and Variational Setting

For the reader’s convenience, we first recall some useful conclusions of index theory for linear second-order Hamiltonian systems given in [15, 16], respectively.Index theory in [15, 16] deals with a classification of L∞0,1,ℒsRn associated with the following Lagrangian systems:(10)−Ptx′′+Btx=0,(11)x0cosα−P0x′0sinα=0,(12)x1cosβ−P1x′1sinβ=0,where Bt∈L∞0,1,ℒsRn, Pt∈C10,1,ℒsRn with Pt being positive definite for t∈0,1, and α∈0,π,β∈0,π.LetX=L20,1,Rn. Set Λxt=−Ptx′′t with(13)DΛ=x∈H20,1,Rn:xsatisfies11and12.By Section 2.3 in [15], we know that Λ is self-adjoint and(14)σΛ=σdΛ=λ∈R:λbelongs to the point spectrum ofΛis bounded from below. We define a bilinear form as follows:(15)ax,y=∫01Ptx′t,y′tdt−x1,y1γβ+x0,y0γα,for all x,y∈Z, where ⋅,⋅ is the usual inner product in Rn, γs=cots as s∈0,π,γs=0 as s=0 or s=π, and(16)Z=x∈H10,1,Rn:x1=0,α=0,β∈0,π;x∈H10,1,Rn:x0=0,α∈0,π,β=π;x∈H10,1,Rn:x′0=x′1=0,α=β=π2;x∈H10,1,Rn:x0=x′1=0,α=0,β=π2;x∈H10,1,Rn:x1=x0=0,α=0,β=π;H10,1,Rn,α,β∈0,π.Similar to the proof of Proposition 1.17 in [28], we know that Z is a Hilbert space and Z=DΛ1/2. By Lemma 2.3.1 of [15], we find that Z=DΛ1/2 can be equipped with the equivalent norm(17)xZ=∫01x′t2+xt2dt1/2,for each x∈Z. Clearly, the embeddings Z⟶L2=X and Z⟶L∞ are compact. For any Bt∈L∞0,1,ℒsRn, we define(18)ψα,βP,Bx,y=ax,y+∫01Btxt,ytdt,for all x,y∈Z.Proposition 1 (see [15], Proposition 2.1.1]).
For anyBt∈L∞0,1,ℒsRn, the space Z has a ψα,βP,B-orthogonal decomposition(19)Z=Z+B⊕Z0B⊕Z−B,such that ψα,βP,B is positive definite, null, and negative definite on Z+B,Z0B, and Z−B, respectively. Moreover, Z0B and Z−B are finitely dimensional.Definition 1 (see [15], Definition 2.3.2]).
For anyBt∈L∞0,1,ℒsRn, we define(20)να,βPB=dimkerΛ+B,iα,βPB=∑λ>0να,βPB+λIn.Proposition 2 (see [16], Definition 7.2.1 and Proposition 7.2.2 (1)]).
For anyB∈L∞0,1,ℒsRn, we have that Z0B is the solution subspace of systems (10)–(12), and(21)να,βPB=dimZ0B,iα,βPB=dimZ−B.
We callνα,βPB and iα,βPB the nullity and index of B with respect to the bilinear form ψα,βP,B⋅,⋅.Proposition 3 (see [29], Proposition 2.6 (5)]).
For anyB∈L∞0,1,ℒsRn, if x∈Z satisfying x=x1+x2 with x1∈Z−B,x2∈Z+B, then −ψα,βP,Bx1,x11/2+ψα,βP,Bx2,x21/2 is an equivalent norm on Z.Proposition 4 (see [15], Proposition 2.3.3 (3)]).
For anyB1,B2∈L∞0,1,ℒsRn satisfying B1t>B2t for all t∈0,1, we have να,βPB1+iα,βPB1≤iα,βPB2.Remark 3.
(see [16], Remark 7.1.2]) Let Pt≡In. Then,(22)ν0,π/2IncIn=nasc=−12+kπ2,ν0,π/2IncIn=0asc≠−12+k2π2,fork∈N,i0,π/2IncIn=0asc≥−π24,i0,π/2IncIn=k+1nasc∈−12+k+1π2,−12+k2π2,;νπ/2,π/2IncIn=nasc=−k2π2,νπ/2,π/2IncIn=0asc≠−k2π2,fork∈N,iπ/2,π/2IncIn=0asc≥0,iπ/2,π/2IncIn=k+1nasc∈−k+12π2,−k2π2.
Next, let us consider the functionalI defined by(23)Ix=ax,x2+12∫01Btx,xdt−λ∫01Vt,xdt,∀x∈Z.
From assumptionV0, using Theorem 1.4 [17], it is easy to check that I is continuously differentiable and weakly lower semicontinuous on Z, and(24)I′xy=ax,y+∫01Btx,ydt−λ∫01∇xVt,x,ydt,for all x,y∈Z. If I′x=0, then we have(25)∫01Ptx′,y′dt−x1,y1γβ+x0,y0γα+∫01Btx,ydt−λ∫01∇xVt,x,ydt=0,for all y∈H10,1,Rn. Choosing y∈H10,1,Rn satisfying y0=y1=0, one has(26)∫01Ptx′,y′dt+∫01Btx,ydt−λ∫01∇xVt,x,ydt=0.
Putet=−Ptx′+∫0tBsxsds−λ∫0t∇xVs,xsds−c such that ∫01etdt=0. Let yt=∫0tesds. Then, we have ∫01et2dt=0 and et=0 for a.e. t∈0,1, which imply that xt satisfies x∈H20,1,Rn and(27)−Ptx′t+Btxt−λ∇xVt,x=0.
Form (25), it follows that(28)0=∫01Ptx′,y′dt−x1,y1γβ+x0,y0γα+∫01Ptx′t′,ydt=P1x′1,y1−P0x′0,y0−x1,y1γβ+x0,y0γα,for all y∈H10,1,Rn. This shows that xt satisfies (11)–(12), which means that the critical points of I correspond to the classical solutions of problem (1).


## 3. Proofs of the Theorems

In order to prove Theorem1, we recall some results of linking given by Schechter [19, 20].LetE be a reflexive Banach space with norm ⋅. The set Φ=Γt:Γt∈CE×0,1,E is to have the following properties:(i)
Γ0=I, the identity map.(ii)
For eacht∈0,1,Γt is a homeomorphism of E onto itself and Γ−1t∈CE×0,1,E.(iii)
Γ1E is a single point uΓ∈E and ΓtA converges uniformly to uΓ as t⟶1 for each bounded set A⊂E.(iv)
For eacht0∈0,1 and each bounded set A⊂E, one has(29)sup0≤t≤t0,u∈AΓtu+Γ−1tu<+∞.Definition 2 (see [19], Definition 3.2]).
A subsetA⊂E links B⊂E [hm] if A∩B=∅, and for each Γt∈Φ, there is a t∈0,1 such that ΓtA∩B≠∅.Theorem 3 (see [20], Theorem 4.1]).
Letℱ,J∈C1E,R be bounded on bounded sets and let(30)Gμu=μℱu−Ju,μ∈ℐ,where ℐ is an open interval contained in 0,+∞. Assume that Gμ satisfiesH1ℱu≥0 for all u∈E and ℱu+Ju⟶+∞ as u⟶+∞.H2 There are sets A,B⊂E such that A links B [hm] and(31)a0=supAGμ≤b0=infBGμ,for eachμ∈ℐ.H3aμ=infΓ∈Φsup0≤s≤1,u∈AGμΓsu is finite for each μ∈ℐ.
Then, for almost allμ∈ℐ, there exists a bounded sequence ukμ∈E such that(32)Gμ′uk⟶0,Gμuk⟶aμask⟶+∞.Lemma 1 (see [19], Example 1 of Section 3.5]).
LetB be an open set in E and let A consist of two points e1,e2 with e1∈B and e2∉B¯. Then, A links ∂B [hm]. ∂B links A [hm] as well if ∂B is bounded.Proof of Theorem 1.
Fromiα,βPB=να,βPB=0 of V1, by Propositions 1 and 2 and Definition 1, we know that Z=Z+B and λ0=minσdΛB>0. By Proposition 3, we can see that ψα,βP,B⋅,⋅1/2=⋅ is an equivalent norm on Z. Since the embedding Z⟶L∞ is compact, there is δ0>0 such that(33)x∞2≤δ0ax,x+∫01Btx,xdt=δ0ψα,βP,Bx,x=δ0x2,x∈Z.
Setφ is an eigenfunction of the operator ΛB corresponding to the first eigenvalue λ0, μ=1/λ, ℱx=1/2ψα,βP,Bx,x=1/2x2,Jx=∫01Vt,xdt, and(34)Gμx=μℱx−Jx,∀x∈Z.
Clearly,ℱ∈C1E,R is bounded on bounded sets, J∈C1Z,R, and Gμ′x=0 is equivalent to I′x=0, which shows that the critical points of Gμ correspond to the solutions of problem (1). By V0, J is also bounded on bounded sets.
Fixμ∈2κ0,2κ1, put r=η/δ0, and set(35)Br=x∈Z:x≤r,∂Br=x∈Z:x=r.
Letε>0 be such that 2κ0<2κ0+ε<μ<2κ1. Then, for every x∈∂Br, we have(36)Gμx≥μ2x2−∫01gtxtϑdt≥μη22δ0−ηϑgL1=μη22δ0−κ0η2δ0>εη22δ0,via V2 and κ0=δ0ηϑ−2gL1, which implies that(37)bμ=infx∈∂BrGμx>0.
For everyx∈Br, by V2 and (33), one has(38)Gμx≥μ2x2−∫01gtxtϑdt≥μ2x2−δ0ϑxϑgL1,dμ=infx∈BrGμx>−∞.
Setliminfc⟶0∫01Vt,cφdt/c2φL22=M0. Fix ϵ∈0,M0−κ1λ0; from (6), we know that there exists ξ>0 such that(39)∫01Vt,cφdtc2φL22>κ1λ0+ϵ,for all c<ξ. Hence, for c small enough, we have cφ∈Br and(40)Gμcφ=μλ0c22φL22−∫01Vt,cφdt≤c2φL22κ1λ0−∫01Vt,cφdtc2φL22≤−ϵc2φL22<0,which shows that 0>infx∈BrGμx=dμ>−∞. There exists a minimizing sequence xk⊂Br such that Gμxk⟶dμ as k⟶∞. Noticing that the embedding Z⟶L∞ is compact, one has a renamed subsequence such that xk⇀x∈Z and xk⟶x∈L∞0,1,R. Then, we have(41)μℱxk⟶dμ+∫01Vt,xdt,μℱx≤liminfμℱxk=dμ+∫01Vt,xdt,as k⟶∞. This shows that Gμx≤dμ<0 and x∉∂Br. Hence, x is in the interior Br, and we have Gμ′x=0.
Byℱx=1/2x2, it is clear that H1 holds. Set liminfc⟶+∞∫01Vt,cφdt/c2φL22=M1. Fix ϵ1∈0,M1−κ1λ0; from (7), we know that there exists ξ1>0 such that ∫01Vt,cφdt/c2φL22>κ1λ0+ϵ1 for all c>ξ1. Hence, for c large enough, we have cφ∉Br and(42)Gμcφ≤c2φL22κ1λ0−∫01Vt,cφdtc2φL22≤−ϵ1c2φL22<0.
From (40) and (42), we can see that there are c1,c2 such that c1φ∈Br,c1φ∉∂Br and c2φ∉Br with Gμciφ<0,i=1,2. Set A=c1φ,c2φ and B=∂Br. By Lemma 1, we know that A links B [hm]. Clearly, a0=supAGμ<0<b0=infBGμ, H2 and H3 holds. Using Theorem 3 and Definition 2, for almost all μ∈ℐ, there exists a bounded sequence ykμ∈Z such that(43)Gμ′yk⟶0,Gμyk⟶aμ=infΓ∈Φsup0≤s≤1,u∈AGμΓsu≥bμ>0,ask⟶+∞.
Since the embeddingZ⟶L∞ is compact, there exists a renamed subsequence such that yk⇀y∈Z and yk⟶y∈L∞0,1,R. Since Gμ′yk⟶0, one has(44)μyk2−∫01∇xVt,yk,ykdt⟶0,μyk2⟶∫01∇xVt,y,ydt,as k⟶∞, which shows that yk⟶y∈Z and Gμ′y=0. Consequently, we have Gμyk⟶Gμy and Gμy=aμ≥bμ>0. Noticing that λ=1/μ,Gμx<0,Gμy>0, and Gμ′x=Gμ′y=0, we know that problem (1) has two nontrivial solutions for almost all λ∈1/2κ1,1/2κ0. The proof is completed.
In order to prove Theorem2, we need to recall some propositions of genus and the symmetric mountain pass theorem given by Kajikiya [27].
LetE be a real Banach space and let Σ denote the family of sets A⊂E\0 such that A is closed in E and symmetric with respect to 0 (i.e., x∈A⇒−x∈A ). For A∈Σ, we define a genus ΥA of A by the smallest integer n such that there exists an odd continuous mapping from A to Rn\0. If such a n does not exist, we define ΥA=∞. Moreover, set Υ∅=0. Let Γk denote the subsets of Σ such that ΥA≥k for all A∈Γk.Proposition 5 (see [18], Propositions 7.5 and 7.7]).
LetA,B⊂Σ. Then, the following hold.Υ1 If there exists an odd continuous mapping from A to B, then ΥA≤ΥB.Υ2 If A⊂B, then ΥA≤ΥB.Υ3 If A is compact, then ΥA<+∞ and ΥNεA=ΥA for ε>0 small enough, where NεA=x∈E:x−A≤ε.Υ4 The n-dimensional sphere Sn has a genus of n+1 by the Borsuk–Ulam theorem.Theorem 4 (see [27], Theorem 1]).
LetE be an infinite-dimensional Banach space and let Ψ∈C1E,R satisfy(1)
Ψx is even, bounded from below, Ψ0=0, and Ψ satisfies the (PS) condition, that is, xk⊂E has a convergent subsequence whenever Ψxk is bounded and Ψ′xk⟶0 as k⟶∞.(2)
For eachk∈N, there exists an Ak∈Γk such that supx∈AkΨx<0.
Then,Ψ possesses a sequence of critical points xk such that Ψxk≤0,xk≠0 and limk⟶∞xk=0.Proof of Theorem 2.
Noticing that the embeddingZ⟶L∞ is compact, there is δ1>0 such that(45)xt≤x∞≤δ1xZ,x∈Z.
Consider the truncated functional(46)Ψx=12∫01Ptx′t,x′tdt+∫01x2dt−ϕxZx12γβ+x02γα2+12∫01x2dt−12∫01Btx,xdt+λ∫01Vt,xdt,for all x∈Z, where ϕ:R+⟶0,1 is a nonincreasing C1 function with ϕt=1 for 0≤t≤1/2δ1 and ϕt=0 for t≥1/δ1. Clearly, Ψ∈C1Z,R and Ψ0=0. Using V4, we have Ψ−x=Ψx for all x∈Z.
SincePt∈C10,1,ℒsRn is positive definite for t∈0,1, there are 0<p1≤p2 such that(47)minp1,1xZ2≤p1∫01x′t2dt+∫01x2dt≤∫01Ptx′t,x′tdt+∫01x2dt≤p2∫01x′t2dt+∫01x2dt≤maxp2,1xZ2,for all x∈Z, which implies that ∫01Ptx′t,x′tdt+∫01x2dt1/2 is also an equivalent norm on Z. Thus, for xZ≥1/δ1, we have(48)Ψx=12∫01Ptx′t,x′tdt+∫01x2dt⟶+∞,asxZ⟶+∞,which shows that Ψ is bounded from below and satisfies the (PS) condition.
It is known that the operatorΛB is self-adjoint and σΛB=σdΛB is bounded from below. This shows that the operator ΛB has a sequence of eigenvalues(49)−∞<λ0≤λ1≤⋯≤λn0≤0<λn0+1≤λn0+2≤⋯≤λn0+n≤⋯⟶+∞,and the system of eigenfunctions en:n∈N corresponding to λn:n∈N forming an orthogonal basis in L2=X. Given any k∈N, let Zk=⊕j=1kXj, where Xj=spanej. By the equivalence of the norms on the finite-dimensional space Zk, we know that there exists a constant ζk>0 such that(50)ζkxL2≥xZ,∀x∈Zk.
Noticing thatBt∈L∞0,1,ℒsRn, we know that there is d1>0 such that(51)∫01Btx,xdt≤d1xL22,∀x∈Z.
By Lemma 2.3.1 of [15], we can see that there is d2>0 such that(52)x02+x12≤∫01x′2dt+d2∫01x2dt,∀x∈Z.
Hence, fixα∈0,π,β∈0,π, and there is d3=maxd2,1γα+γβ+1>0 such that(53)x02+x12γα+γβ≤d3∫01x′2dt+∫01x2dt=d3xZ2,∀x∈Z.
Fixλ>0; using V5, there exists d4>0 such that(54)Vt,x≥κ3λζk2x2,for all x≤d4 and a.e. t∈0,1, where κ3=maxp2,d1+d3. Therefore, for x∈Zk with xZ=ρk:=min1/2δ1,d4/δ1, using (45)–(51) and (53) and (54), we can obtain xt≤d4 and(55)Ψx=Ix=12∫01Ptx′t,x′tdt−x12γβ+x02γα2+12∫01Btx,xdt−λ∫01Vt,xdt≤p22x′L22+d32xZ2+d12xL22−κ3ζk2∫01x2dt≤κ32xZ2−κ3xZ2=−κ32ρk2,which implies that(56)x∈Zk:xZ=ρk⊂x∈Z:Ψx≤−κ32ρk2.
PutAk=x∈Z:Ψx≤−κ3/2ρk2; by Proposition 5, we have(57)ΥAk≥Υx∈Zk:xZ=ρk=k,which shows that Ak∈Γk and(58)supx∈AkΨx≤−κ32ρk2<0.
By Theorem4, we obtain that Ψ possesses a sequence of critical points xk such that Ψxk≤0,xk≠0 and limk⟶∞xk=0. Since limk⟶∞xk=0, for 1/2δ1>0, there is N>0 such that xkZ≤1/2δ1∀k>N. Thus, for k>N, the critical points of Ψ are just critical points of I via Ψx=Ix as xZ≤1/2δ1, which shows that problem (1) has infinitely many small-energy solutions for each λ>0. The proof is completed.


## 4. Applications to Sturm–Liouville Equations and Examples

In this section, we consider the applications of Theorems1 and 2 to Sturm–Liouville equations with the mixed boundary value conditions and the Neumann boundary value conditions. Meanwhile, some examples of problems (3) and (4) are given to illustrate the validity of our result, and the result is also new even in the cases of (3) and (4).As first special case, we consider the mixed boundary value problem (3):(59)−Ptx′t′+Btxt=λft,x,a.e.t∈0,1,x0=x′1=0,where λ>0,Bt∈L∞0,1,R, and Pt∈C10,1,R with Pt being positive definite for t∈0,1. In problem (1), taking n=1,α=0,β=π/2, the following corollary is immediately obtained from Theorem 1.Corollary 1.
Assume thatft,x:0,1×R⟶R is a L1-Carathéodory function and satisfies V1, V2, and V3 with ax,x=∫01Ptx′,x′dt and Vt,x replaced with Ft,x=∫0xft,sds. Then, for problem (3), the conclusion of Theorem 1 is still valid.
An example of Corollary1 is given below.Example 1.
Consider the following problem:(60)−x″t−π28xt=λft,x,x0=x′1=0,where(61)Ft,x=∫0xft,sds=ltxϑ,x≤1,ltx3−3−ϑ2x2+3−ϑ2,x>1,where 1≤ϑ<2 and lt∈C0,1,R+ with lt≡0. Clearly, ft,x is a L1-Carathéodory function. By Remark 3, we have ν0,π/2I1−π2/8=i0,π/2I1−π2/8=0, which implies that V1 holds. Taking that η=1,gt=maxt∈0,1lt=M2, then V2 holds. After a simple calculation, we have δ0=1,λ0=π2/8, and the eigenfunction of Λ−π2/8 corresponding to the first eigenvalue λ0 is φ=sinπ/2t,t∈0,1. Let κ0=δ0ηϑ−2gL1=M0. Noticing that(62)liminfc⟶0∫01Ft,cφdtc2φL22=liminfc⟶0∫01ltφϑdtc2−ϑφL22=+∞,liminfc⟶+∞∫01Ft,cφdtc2φL22=liminfc⟶+∞∫01ltc3φ3−3−ϑ/2c2φ2+3−ϑ/2dtc2φL22=+∞,there exists a constant κ1>κ0=M0 such that (6) and (7) of V3 hold. Hence, Ft,x satisfies the hypotheses of Corollary 1, and for almost all λ∈1/2κ1,1/2κ0, problem (60) has two nontrivial solutions.Remark 4.
Obviously,Ft,x in Example 1 does not satisfy the condition ∫01/2Ft,xdt>0∀x∈0,d of Theorem 3.1 in [1] and the condition Ft,x≤μt1+xss<2,x∈R of Theorem 3.1 in [2]. Moreover, noticing that Bt≥0 must be assumed in the condition of Theorem 3.1 in [1, 2, 9, 10], we easily see that Theorem 3.1 in [1, 2, 9, 10] cannot be applied to Example 1. This shows that Corollary 1 is also new even in the cases of (3).
Takingn=1,α=0,β=π/2, from Theorem 2, we immediately obtain the following corollary.Corollary 2.
Assume thatft,x:0,1×R⟶R is a L1-Carathéodory function and satisfies V4 and V5 with Vt,x replaced with Ft,x=∫0xft,sds. Then, for problem (3), the conclusion of Theorem 2 is still valid.Example 2.
Consider(63)−x″t+btxt=λft,x,x0=x′1=0,where(64)Ft,x=∫0xft,sds=l1tx3/2+ln1+x+x3,λ>0,bt∈L∞0,1,R and l1t∈C0,1,R+ with mint∈0,1lt>0. Clearly, ft,x is a L1-Carathéodory function, Ft,0=0, and V4 holds. Since(65)limx⟶0Ft,xx2=limx⟶0l1tx3/2+ln1+x+x3x2=+∞,V5 holds. By Corollary 2, problem (63) has infinitely many small-energy solutions for each λ>0.Remark 5.
By Remarks2 and 3, we know that bt<0 is also allowed. Noticing that bt≥0 and λ∈λ1,λ2⊂0,+∞ must be assumed in the condition of Theorem 3.1 in [3, 4], we easily see that Theorem 3.1 in [3, 4] cannot be applied to Example 2. This shows that Corollary 2 is also new even in the cases of (3).
As second special case, consider the Neumann boundary value problem (4)(66)−Ptx′t′+Btxt=λft,x,a.e.t∈0,1,x′0=x′1=0,where λ>0,Bt∈L∞0,1,R and Pt∈C10,1,R with Pt being positive definite for t∈0,1. In problem (1), taking n=1,α=β=π/2, the following corollary is immediately obtained from Theorem 1.Corollary 3.
Assume thatft,x:0,1×R⟶R is a L1-Carathéodory function and satisfies V1, V2, and V3 with ax,x=∫01Ptx′,x′dt and Vt,x replaced with Ft,x=∫0xft,sds. Then, for problem (4), the conclusion of Theorem 1 is still valid.
Now, we give an example of Corollary3.Example 3.
Consider the following problem:(67)−x″t+b1txt=λft,x,a.e.t∈0,1,x′0=x′1=0,where Ft,x=∫0xft,sds as defined in Example 1 and(68)b1t=1,t∈0,12,4π2+1,t∈12,1.
Clearly,ft,x is a L1-Carathéodory function. By Remark 2, we have iπ/2,π/2I11/2=0. From b1t>1/2 and Proposition 4, we know that iπ/2,π/2I11/2≥νπ/2,π/2I1b1t+iπ/2,π/2I1b1t≥0, which implies that V1 holds. Taking that η=1,gt=maxt∈0,1lt=M2, then V2 holds. After a simple calculation, we have δ0=1,λ0=4π2+1 and the eigenfunction of Λb1t corresponding to the first eigenvalue λ0 is φ0=cos2πt,t∈0,1/2,−1,t∈1/2,1.. Let κ0=δ0ηϑ−2gL1=M0. Noticing that(69)liminfc⟶0∫01ltφ0ϑdtc2−ϑφ0L22=+∞,liminfc⟶+∞∫01ltc3φ03−3−ϑ/2c2φ02+3−ϑ/2dtc2φ0L22=+∞,then there exists a constant κ1>κ0=M0 such that (6) and (7) of V3 hold. Hence, Ft,x satisfies the hypotheses of Corollary 3, and for almost all λ∈1/2κ1,1/2κ0, problem (67) has two nontrivial solutions.Remark 6.
Noticing thatBt≡M≥0 and λ=1 must be assumed in the condition of Theorem 1 in [11, 12], we easily see that Theorem 1 in [11, 12] cannot be applied to Example 3. This shows that Corollary 3 is also new even in the cases of (4).
Takingn=1,α=β=π/2, from Theorem 2, we have the following corollary.Corollary 4.
Assume thatft,x:0,1×R⟶R is a L1-Carathéodory function and satisfies V4 and V5 with Vt,x replaced with Ft,x=∫0xft,sds. Then, for problem (4), the conclusion of Theorem 2 is still valid.Example 4.
Finally, consider(70)−x″t+btxt=λft,x,x′0=x′1=0,where Ft,x=∫0xft,sds as defined in Example 2, λ>0, and bt∈L∞0,1,R. Clearly, ft,x is a L1-Carathéodory function, Ft,0=0, and V4 and V5 hold. By Corollary 4, problem (70) has infinitely many small-energy solutions for each λ>0.Remark 7.
By Remarks2 and 3, we know that bt<0 is also allowed. Noticing that bt∈C0,1,R+\0 and λ∈λ1,λ2⊂0,+∞ must be assumed in the condition of Theorem 3.1 in [5], we easily see that Theorem 3.1 in [5] cannot be applied to Example 4. This implies that Corollary 4 is also new even in the cases of (4).



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