# A Variational Approach to Perturbed Discrete Anisotropic Equations

Authors: Amjad Salari; Giuseppe Caristi; David Barilla; Alfio Puglisi
Journal: Abstract and Applied Analysis
 (2016)

---

## Abstract

We continue the study of discrete anisotropic equations and we will provide new multiplicity results of the solutions for a discrete anisotropic equation. We investigate the existence of infinitely many solutions for a perturbed discrete anisotropic boundary value problem. The approach is based on variational methods and critical point theory.


---

## Body

## 1. Introduction

The aim of this paper is to investigate the existence of infinitely many solutions for the following perturbed discrete anisotropic problem:Pλf,g-ΔαkΔuk-1pk-1-2Δuk-1=λfk,uk+μgk,uk,k∈1,T,u0=uT+1=0,where T≥2 is a fixed positive integer number, [1,T] is the discrete interval {1,…,T}⊂N, Δuk-1=uk-uk-1, α:1,T+1→0,+∞, and p:[0,T]→1,+∞ are some fixed functions, and f,g:[1,T]×R→R are two continuous functions. Let (1)p-=mink∈0,Tpk,p+=maxk∈0,Tpk,α-=mink∈1,T+1αk,α+=maxk∈1,T+1αk.Many problems in applied mathematics lead to the study of discrete boundary value problems and difference equations. Indeed, common among many fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, and economics, is the fact that the mathematical modelling of fundamental questions is usually tended towards considering discrete boundary value problems and nonlinear difference equations. Regarding these issues, a thoroughgoing overview has been given in, as an example, the monograph [1] and the reference therein. On the other hand, in recent years some researchers have studied the existence and multiplicity of solutions for equations involving the discrete p-Laplacian operator by using various fixed point theorems, lower and upper solutions method, critical point theory and variational methods, Morse theory, and the mountain-pass theorem. For background and recent results, we refer the reader to [2–20] and the references therein. For example, Atici and Guseinov in [3] investigated the existence of positive periodic solutions for nonlinear difference equations with periodic coefficients by employing a fixed point theorem in cone. Atici and Cabada in [2], by using the upper and lower solution method, obtained the existence and uniqueness results for a discrete boundary value problem. Henderson and Thompson in [12] gave conditions on the nonlinear term involving pairs of discrete lower and discrete upper solutions which led to the existence of at least three solutions of discrete two-point boundary value problems, and in a special case of the nonlinear term they gave growth conditions on the function and applied their general result to show the existence of three positive solutions. Chu and Jiang in [9] based on fixed point theorem in a cone, due to Krasnoselskii, characterized the eigenvalues and showed the existence of positive solutions to a discrete boundary value problem. Jiang and Zhou in [13] employing a three-critical point theorem, due to Ricceri, established the existence of at least three solutions for perturbed nonlinear difference equations with discrete boundary conditions. Wang and Guan in [20], by using the Five Functionals Fixed Point Theorem, obtained the existence criteria for three positive solutions of a p-Laplacian difference equation. Bonanno and Candito [5], employing critical point theorems in the setting of finite dimensional Banach spaces, investigated the multiplicity of solutions for nonlinear difference equations involving the p-Laplacian. Cabada et al. in [6], based on three critical points’ theorems, investigated different sets of assumptions which guarantee the existence and multiplicity of solutions for difference equations involving the discrete p-Laplacian operator. In [4] Bian et al. by using critical point theory studied a class of discrete p-Laplacian periodic boundary value problems; some results were obtained for the existence of two positive solutions, three solutions, and multiple pairs of solutions of the problem when the parameter lies in some suitable infinite or finite intervals. In [15], by using variational methods and critical point theory, the existence of infinitely many solutions for perturbed nonlinear difference equations with discrete Dirichlet boundary conditions was discussed.There seems to be increasing interest in the existence of solutions to discrete anisotropic equations, because of their applications in many fields such as models in physics [21–24], biology [25, 26], and image processing (see, for example, Weickert’s monograph [27]). We also mention Fragalà et al. [28] and El Hamidi and Vétois [29] as essential references in treating the nonlinear anisotropic problems. Besides, Mihăilescu et al. (see [30, 31]) were the first authors who studied anisotropic elliptic problems with variable exponents. On the other hand, numerous researches have been undertaken on the existence of solutions for discrete anisotropic boundary value problems (BVPs) in recent years. As to the background and latest results, the readers can refer to [32–38] and the references therein. For example, Mihăilescu et al. in [36], by using critical point theory, obtained the existence of a continuous spectrum for a family of discrete boundary value problems. Galewski and Wieteska in [34] investigated the existence of solutions of the system of anisotropic discrete boundary value problems using critical point theory, while in [33], using variational methods, they derived the intervals of the numerical parameter for which the problem Pλf,g has at least 1, exactly 1, or at least 2 positive solutions. They also derived some useful discrete inequalities. Molica Bisci and Repovš in [37] considered advantage of a recent critical point theorem to establish the existence of infinitely many solutions for anisotropic difference equation: (2)-ΔΔuk-1pk-1-2Δuk-1=λfkuk,k∈Z1,T,u0=uT+1=0,where λ is a positive parameter, fk:R→R is a continuous function for every k∈Z[1,T] (with T≥2), and Δu(k-1)≔u(k)-u(k-1) is the forward difference operator, assuming that the map p:Z0,T→R satisfies p-≔minZ0,Tp(k)>1 as well as p+≔maxZ0,Tp(k)>1. Stegliński in [38], based on critical point theory, obtained the existence of infinitely many solutions for the parametric version of the problem Pλf,g, in the case where μ=0.Motivated by the above works, in the present paper, by employing a smooth version of [39, Theorem  2.1], which is more precise version of Ricceri’s Variational Principle [40, Theorem  2.5] under some hypotheses on the behavior of the nonlinear terms at infinity, we prove the existence of definite intervals about λ and μ in which the problem Pλf,g admits a sequence of solutions which is unbounded in the space E which will be introduced later (Theorem 6). Furthermore, some consequences of Theorem 6 are listed. A partial case of main result is formulated as Theorem 5. Replacing the conditions at infinity on the nonlinear terms, by a similar one at zero, we obtain a sequence of pairwise distinct solutions strongly converging at zero; see Theorem 14. Three examples of applications are pointed out (see Examples 8, 13, and 16).


## 2. Preliminaries

Our main tool to ensure the existence of infinitely many solutions for the problemPλf,g is a smooth version of Theorem  2.1 of [39] which is a more precise version of Ricceri’s Variational Principle [40] that we now recall here.Theorem 1.
LetX be a reflexive real Banach space and let Φ,Ψ:X→R be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every r>infXΦ, let one put (3)φr≔infu∈Φ-1-∞,rsupv∈Φ-1-∞,r⁡Ψv-Ψur-Φu,γ≔lim⁡infr→+∞⁡φr,δ≔lim⁡infr→infX⁡Φ+⁡φr.Then, one has the following: (a)
For everyr>infXΦ and every λ∈0,1/φr, the restriction of the functional Iλ=Φ-λΨ to Φ-1-∞,r admits a global minimum, which is a critical point (local minimum) of Iλ in X.(b)
Ifγ<+∞, then, for each λ∈0,1/γ, the following alternative holds:either
(b1)
Iλ possesses a global minimumor(b2)
there is a sequenceun of critical points (local minima) of Iλ such that (4)limn→+∞Φun=+∞.(c)
Ifδ<+∞, then, for each λ∈0,1/δ, the following alternative holds:either
(c1)
there is a global minimum ofΦ which is a local minimum of Iλor(c2)
there is a sequence of pairwise distinct critical points (local minima) ofIλ which weakly converges to a global minimum of Φ.We refer the reader to the paper [41–47] in which Theorem 1 was successfully employed to ensure the existence of infinitely many solutions for boundary value problems.Here and in the sequel we take theT-dimensional Banach space (5)E≔u:0,T+1⟶R:u0=uT+1=0, endowed with the norm (6)u≔∑k=1T+1Δuk-121/2.Remark 2.
We consider that wheneverE is a finite dimensional Banach space in Theorem 1, in order to show the regularity of the derivative of Φ and Ψ, it is merely enough to indicate that Φ′ and Ψ′ are two continuous functionals on E∗.Lemma 3 (see [48, Section 2]).
a . 1 For every u∈E with u>1 we have (7)∑k=1T+1Δuk-1pk-1≥T2-p-/2up--T.
a . 2 For every u∈E with u≤1 we have(8)∑k=1T+1Δuk-1pk-1≥Tp+-2/2up+.
a . 3 For every u∈E and for any m≥2 we have (9)T+12-m/2um≤∑k=1T+1Δuk-1m≤T+1um.
a . 4 If p+≥2 there exists positive constant Cp+ such that for every u∈E(10)∑k=1T+1Δuk-1pk-1≤2p+T+1Cp+up++1.
a . 5 For every u∈E and for any m≥2 we have(11)∑k=1T+1Δuk-1m≤2m∑k=1Tukm.
a . 6 For every u∈E and for any p,q>1 such that 1/p+1/q=1 we have (12)u∞=maxk∈1,Tuk≤T+11/q∑k=1T+1Δuk-1p1/p.Put(13)Fk,t≔∫0tfk,ξdξ∀k,t∈1,T×R.Remark 4.
We recall that a mapf:1,T×R→R is continuous if it is continuous as a map of the topological space 1,T×R into the topological space R. In this paper, the topology on 1,T will be the discrete topology.A special case of our main result is the following theorem.Theorem 5.
Letf:R→R be a continuous function and put F(t)=∫0tfξdξ for all t∈R. Assume that (14)lim⁡infξ→+∞⁡Fξξp=0,lim⁡supξ→+∞⁡Fξξp=+∞.Then, for every continuous function g:R→R whose G(t)=∫0t‍g(ξ)dξ, for every t∈R, is a nonnegative function satisfying the condition(15)g⋆≔Tpplimξ→+∞supt≤ξ⁡Gtξp<+∞and for every μ∈0,μ⋆,λ, where μ⋆,λ≔1/g⋆1-λTppliminfξ→+∞F(ξ)/ξp, the problem (16)-ΔΔuk-1p-2Δuk-1=λfuk+μguk,k∈1,T,u0=uT+1=0 has an unbounded sequence of solutions.


## 3. Main Results

We present our main result as follows.Theorem 6.
Assume that
A 1  lim ⁡ inf ξ → + ∞ ⁡ ∑ k = 1 T s u p t ≤ ξ F k , t / ξ p - < p - α - / 2 α + T p - p + lim ⁡ sup ξ → + ∞ ⁡ ∑ k = 1 T ‍ F ( k , ξ ) / ξ p +.
Then, for eachλ∈]λ1,λ2[, where (17)λ1≔2α+p-lim⁡supξ→+∞⁡∑k=1TFt,ξ/ξp+,λ2≔α-Tp-p+lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,t/ξp-,for every continuous function g:[1,T]×R→R whose G(k,t)=∫0t‍g(k,ξ)dξ for every (k,t)∈[1,T]×R, is a nonnegative function satisfying the condition(18)g∞≔Tp-p+α-limξ→+∞∑k=1Tsupt≤ξ⁡Gk,tξp-<∞and for every μ∈[0,μg,λ[, where (19)μg,λ≔1g∞1-λTp-p+α-lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,tξp-,the problem Pλf,g has an unbounded sequence of solutions.Proof.
Fixλ¯∈]λ1,λ2[ and let g be a function satisfying the condition (18). Since λ¯<λ2, one has μg,λ¯>0. Fix μ¯∈[0,μg,λ¯[ and put ν1≔λ1 and ν2≔λ2/1+μ¯/λ¯λ2g∞. If g∞=0, clearly, ν1=λ1, ν2=λ2, and λ¯∈]ν1,ν2[. If g∞≠0, since μ¯<μg,λ¯, we obtain λ¯/λ2+μ¯g∞<1, and so λ2/1+μ¯/λ¯λ2g∞>λ¯, namely, λ¯<ν2. Hence, since λ¯>λ1=ν1, one has λ¯∈]ν1,ν2[. Now, set Q(k,t)=F(k,t)+μ¯/λ¯G(k,t) for all (k,t)∈[1,T]×R. Take X=E and define on X two functionals Φ and Ψ as follows: (20)Φu=∑k=1T+1αkpk-1Δuk-1pk-1,Ψu=∑k=1TFk,uk+μ¯λ¯∑k=1TGk,uk.Since X is a finite dimensional Banach space, Ψ is a Gâteaux differentiable functional and sequentially weakly upper semicontinuous whose Gâteaux derivative at the point u∈X is the functional Ψ′u∈X∗, given by(21)Ψ′uv=∑k=1Tfk,ukvk+μ¯λ¯∑k=1Tgk,ukvk,for every v∈X, and Ψ′:X→X∗ is a compact operator. Moreover, Φ is a Gâteaux differentiable functional of which Gâteaux derivative at the point u∈X is the functional Φ′u∈X∗, given by (22)Φ′uv=∑k=1T+1αkΔuk-1pk-1-2Δuk-1Δvk-1for every v∈X. Furthermore, Φ is sequentially weakly lower semicontinuous (see [5, Remarks  2.2 and  2.3]). Put Iλ¯≔Φ-λ¯Ψ. We observe that the solutions of the problem Pλf,g are exactly the solutions of the equation Iλ¯′u=0. So, our end is to apply Theorem 1 to Φ and Ψ. Now, we wish to prove that γ<+∞, where γ is defined in Theorem 1. Let ξn be a real sequence such that ξn>0 for all n∈N and ξn→+∞ as n→∞ and (23)limn→∞∑k=1Tsupt≤ξn⁡Qk,tξnp-=lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Qk,tξp-.Put rn=α-/Tp-p+ξnp- for all n∈N. Let n0∈N be such that p+/α-rn>1 for all n>n0. We claim that(24)Φ-1-∞,rn⊆u:u∞≤ξn∀k∈0,T+1.
Indeed, ifv∈X and ϕv<rm, one has (25)∑k=1T+1αkpk-1Δuk-1pk-1<rm.Then, (26)Δuk-1<pk-1αkrm1/pk-1≤p+α-rm1/p-for every k∈[1,T+1]. Consequently, since u∈X, we deduce by easy induction that (27)uk≤Δuk-1+uk-1<p+α-rn1/p-+uk-1≤kp+α-rn1/p-≤Tp+α-rn1/p-=ξnfor every k∈1,T and this gives (24). Hence, taking into account the fact that Φ0=Ψ0=0, for every n large enough, one has (28)φrn=infu∈Φ-1-∞,rnsupv∈Φ-1-∞,rn⁡Ψv-Ψurn-Φu≤supv∈Φ-1-∞,rn⁡Ψvrn≤∑k=1Tsupt≤ξn⁡Qk,tα-ξnp-/Tp-p+=∑k=1Tsupt≤ξn⁡Fk,t+μ¯/λ¯Gk,tα-ξnp-/Tp-p+≤∑k=1Tsupt≤ξn⁡Fk,tα-ξnp-/Tp-p++μ¯λ¯∑k=1Tsupt≤ξn⁡Gk,tα-ξnp-/Tp-p+. Moreover, it follows from Assumption A1 that(29)lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,tξnp-<+∞,which concludes that(30)limn→∞∑k=1Tsupt≤ξn⁡Fk,tξnp-<+∞.Then, in view of (18) and (30), we have (31)limn→∞∑k=1Tsupt≤ξn⁡Fk,tξnp-+limn→∞μ¯λ¯∑k=1Tsupt≤ξn⁡Gk,tξnp-<+∞,which follows(32)limn→∞∑k=1Tsupx≤ξn⁡Fk,t+μ¯/λ¯Gk,tξnp-<+∞.Therefore,(33)γ≤lim⁡infn→+∞⁡φrn≤limn→∞∑k=1Tsupt≤ξn⁡Fk,t+μ¯/λ¯Gk,tα-ξnp-/Tp-p+<+∞.Since(34)∑k=1Tsupt≤ξn⁡Qk,tξnp-≤∑k=1Tsupt≤ξn⁡Fk,tξnp-+μ¯λ¯∑k=1Tsupt≤ξn⁡Gk,tξnp-,taking (18) into account, one has(35)Tp-p+α-lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Qk,tξp-≤Tp-p+α-lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,tξp-+μ¯λ¯g∞.Moreover, since G is nonnegative, we have(36)lim⁡supξ→+∞⁡∑k=1TQk,ξξp+≥lim⁡supξ→+∞⁡∑k=1TFk,ξξp+.Therefore, from (35) and (36) and from Assumption A1 and (33), one has(37)λ¯∈ν1,ν2⊆2α+p-lim⁡supξ→+∞⁡∑k=1TQk,ξ/ξp+,α-Tp-p+lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Qk,t/ξp-⊆0,1γ.For fixed λ¯, inequality (33) assures that condition (b) of Theorem 1 can be used and either Iλ¯ has a global minimum or there exists a sequence {un} of solutions of the problem Pλf,g such that limn→∞u=+∞.
The other step is to verify that the functionalIλ¯ has no global minimum. Since (38)1λ¯<p-2α+lim⁡supξ→+∞⁡∑k=1TFk,ξξp+,we can consider a real sequence {γn} with γn>1 for all n∈N and a positive constant τ such that γn→+∞ as n→∞ and(39)1λ¯<τ<p-2α+∑k=1TFk,γnγnp+for each n∈N large enough. Thus, we consider a sequence {wn} in X defined by setting(40)wnk=γnfork∈1,T0fork=0,T+1.Thus (41)wn2=∑k=1T+1Δwnk-12=2γn2,(42)2α-p+γnp-≤Φwn≤2α+p-γnp+.On the other hand, since G is nonnegative, we observe(43)Ψwn≥∑k=1TFk,γn.So, from (39), (42), and (43), we conclude that (44)Iλ¯wn=Φwn-λ¯Ψwn≤2α+p-γnp+-λ¯∑k=1TFk,γn<2α+1-λ¯τp-γnp+,for every n∈N large enough. Hence, the functional Iλ¯ is unbounded from below, and it follows that Iλ¯ has no global minimum. Therefore, Theorem 1 assures that there is a sequence {un}⊂X of critical points of Iλ¯ such that limn→∞⁡Φun=+∞, which from Lemma 3 follows that limn→∞⁡un=+∞. Hence, we have the conclusion.Remark 7.
Under the conditions(45)lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,tξp-=0,lim⁡supξ→+∞⁡∑k=1TFk,ξξp+=∞,Theorem 6 assures that for every λ>0 and for each μ∈0,1/g∞ the problem Pλf,g admits infinitely many solutions. Moreover, if g∞=0, the result holds for every λ>0 and μ≥0.Now, we give an application of Theorem6 as follows.Example 8.
LetT=10, let pk=2+k/5 for all k∈[0,10], let αk=1+1/k for all k∈[1,11], and let an be a sequence defined by (46)a1=4,an+1=1+1022an2forn≥2and let bn be a sequence such that b1=44 and bn=an4-an-14 for all n≥2. Consider the problem(47)-Δ1+1kΔuk-1k-1/5Δuk-1=λfk,uk+μekuk,k∈1,10,u0=u11=0,where fk,t=ekht for all k,t∈1,10×R with (48)ht=∑n=1∞2bn1-2t-an+12χan-1,ant∀t∈R,where χ[α,β] denotes the characteristic function of the interval [α,β]. According to the above data we have p-=2, p+=4, α-=12/11, α+=2, and Gk,t=ekt2/2 for all (k,t)∈[1,10]×R and it is easy to verify that an+1-1>an and ∫an-1anhtdt=bn for all n∈N. Then, one has Fk,an=ek∫0anh(ξ)dξ=an4, and it is easy to see that(49)lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,tξp-=limn→+∞∑k=1TekFan+1-1an+1-12=12200limn→∞an4∑k=110ekan4=12200∑k=110ek,lim⁡supξ→+∞⁡∑k=1TFk,tξp+=limn→+∞∑k=1TFk,anan4=∑k=110ek.Hence, using Theorem 6, since (50)g∞≔Tp-p+α-limξ→+∞∑k=1Tsupt≤ξ⁡Gk,tξp-=8003limξ→+∞ξ2∑k=1Tekξ2=800∑k=110ek3<+∞,problem (47) for every λ∈2/∑k=110ek,6/∑k=110ek and μ∈0,6-λ∑k=110ek/1600∑k=110ek has an unbounded sequence of solutions in the space (51)E10≔u:0,11⟶R:u0=u11=0.Remark 9.
AssumptionA1 in Theorem 6 could be replaced by the following more general condition:
A 2 there exist two sequences {θn} and {ηn} with ηn>0 for every n∈N and 2α+/p-θnp+<α-/Tp-p+ηnp- for all n∈N and limn→+∞⁡ηn=+∞ such that (52)limn→+∞∑k=1Tsupt≤ηn⁡Fk,t-∑k=1TFk,θnα-/Tp-p+ηnp--2α+/p-θnp+<p-2α+lim⁡supξ→+∞⁡∑k=1TFk,ξξp+.Indeed, clearly, by choosing θn=0 for all n∈N, from A2 we obtain A1. Moreover, if we assume A2 instead of A1 and choose rn=α-ηnp-/Tp-p+ for all n∈N, applying the same argument in the proof of Theorem 6, we obtain (53)φrn≤supv∈Φ-1-∞,rn⁡Ψv-∑k=1TFk,wnkrn-∑k=1T+1αk/pk-1Δuk-1pk-1≤∑k=1Tsupx≤ηn⁡Fk,t-∑k=1TFk,θnα-/Tp-p+ηnp--2α+/p-θnp+,where wnt is the same as (40) but γn is replaced by θn. We have the same conclusion as in Theorem 6 with the interval ]λ1,λ2[ replaced by the interval (54)Λ′=2α+p-lim⁡supξ→+∞⁡∑k=1TFk,ξ/ξp+,1limn→+∞⁡∑k=1Tsupt≤ηn⁡Fk,t-∫k=1TFk,θn/α-/Tp-p+ηnp--2α+/p-θnp+.Here, we point out a simple consequence of Theorem6.Corollary 10.
Assume that
A 3  lim⁡infξ→+∞⁡∑k=1Tsupt≤ξF(k,t)/ξp-<α-/Tp-p+;
A 4  lim⁡supξ→+∞⁡∑k=1TFk,ξ/ξp+>2α+/p-.
Then, for every arbitrary functiong∈C([1,T]×R,R) whose G(k,t)=∫0t‍g(k,ξ)dξ for every (k,t)∈[1,T]×R is a nonnegative function satisfying condition (18) and for every μ∈[0,μg,1[, where (55)μg,1≔1g∞1-Tp-p+α-lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,tξp-,the problem (56)-ΔαkΔuk-1pk-1-2Δuk-1=fk,uk+μgk,uk,k∈1,T,u0=uT+1=0has an unbounded sequence of solutions.Remark 11.
Theorem5 is an immediate consequence of Corollary 10.We here give the following consequence of the main result.Corollary 12.
Letf1∈C1,T×R,R and let F1(k,t)=∫0t‍f1(k,ξ) for all t∈R. Assume that
A 5   lim ⁡ inf ξ → + ∞ ⁡ ∑ k = 1 T sup t ≤ ξ ⁡ F 1 ( k , t ) / ξ p - < + ∞;
A 6   lim ⁡ sup ξ → + ∞ ⁡ ∑ k = 1 T F 1 k , ξ / ξ p + = + ∞.
Then, for every functionfi∈C([1,T]×R,R), denoting Fi(k,t)=∫0t‍fi(k,ξ)dξ for all t∈R for 2≤i≤n, satisfying (57)max⁡supξ∈R⁡Fik,ξ;2≤i≤n≤0,min⁡lim⁡infξ→+∞⁡Fik,ξξp-;2≤i≤n>-∞,for each (58)λ∈0,α-Tp-p+lim⁡infξ→+∞⁡F1t,ξ/ξp-,for every arbitrary function g∈C1,T×R,R whose G(k,t)=∫0tgk,ξdξ for every k,t∈1,T×R, is a nonnegative function satisfying condition (18) and for every μ∈0,μg,λ, where(59)μg,λ≔1g∞1-λTp-p+α-lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡F1k,tξp-,the problem (60)-ΔαkΔuk-1pk-1-2Δuk-1=λ∑i=1nfik,uk+μgk,uk,k∈1,T,u0=uT+1=0has an unbounded sequence of solutions.Proof.
SetFk,ξ=∑i=1n‍Fik,ξ for all ξ∈R. Assumption (A6) along with the condition (61)min⁡lim⁡infξ→+∞⁡Fik,ξξp-;2≤i≤n>-∞ensures that (62)lim⁡supξ→+∞⁡∑k=1TFk,ξξp+=lim⁡supξ→+∞⁡∑i=1n∑k=1TFik,ξξp+=+∞.Moreover, Assumption (A5), together with the condition (63)max⁡supξ∈R⁡Fik,ξ;2≤i≤n≤0,implies that (64)lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡Fk,tξp-≤lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡F1k,tξp-<+∞. Hence, the conclusion follows from Theorem 6.Here we give the following example to illustrate Corollary12.Example 13.
LetT=100, p(k)=k+100/50 for all k∈0,100, and α(k)=ek for all k∈1,101. Let f1,f2:1,100×R→R be defined by (65)f1k,t=hkt45+10sin2⁡ln⁡t+2sin⁡2ln⁡tifk,t∈1,100×0,+∞,0ifk,t∈1,100×-∞,0,f2k,t=-lk2t1+t22,respectively, where h,l:1,100→R are two nonnegative continuous functions. Consider the problem(66)-ΔekΔuk-1k-1/50Δuk-1=λf1+f2k,uk,k∈1,100,u0=u101=0.A direct calculation shows p-=2, p+=4, and (67)F1k,t=hkt51+2sin2⁡ln⁡tifk,t∈1,100×0,+∞,0ifk,t∈1,100×-∞,0,F2k,t=-lkt21+t2.Put (68)am=mifmis even,e-mπifmis odd,bm=emπfor everym∈N. Then (69)limm→+∞⁡∑k=1Tsupt≤am⁡F1k,tam2=0ifmis odd,+∞ifmis even,lim⁡supm→+∞⁡∑k=1TF1k,bmbm4=+∞. So, (70)lim⁡infξ→+∞⁡∑k=1Tsupt≤ξ⁡F1k,tξp-=0<+∞,lim⁡supξ→+∞⁡∑k=1TF1k,tξp+=+∞. Moreover, (71)supξ∈R⁡F2k,ξ=0,lim⁡infξ→+∞⁡F2k,ξξp-=lim⁡infξ→+∞⁡-lkξ2ξ21+ξ2=0>-∞.Hence, all assumptions of Corollary 12 with μ=0 are satisfied. So, for every λ∈(0,+∞), problem (66) has an unbounded sequence of solutions in the space (72)E100≔u:0,101⟶R:u0=u101=0.Arguing as in the proof of Theorem6 but using conclusion (c) of Theorem 1 instead of (b), one establishes the following result.Theorem 14.
Assume that
A 7  lim⁡infξ→0+⁡∑k=1Tsupt≤ξF(k,t)/ξp-<p-α-/2α+p+Tp-lim⁡supξ→0+⁡∑k=1TF(k,ξ)/ξp+.
Then, for eachλ∈]λ3,λ4[, where(73)λ3≔2α+p-lim⁡supξ→0+⁡∑k=1TFk,ξ/ξp+,λ4≔α-Tp-p+lim⁡infξ→0+⁡∑k=1Tsupt≤ξ⁡Fk,t/ξp-,for every arbitrary function g∈C1,T×R,R whose G(k,t)=∫0t‍g(k,ξ)dξ for every (k,t)∈[1,T]×R is a nonnegative function satisfying the condition(74)g0≔Tp-p+α-limξ→0+∑k=1Tsupt≤ξ⁡Gk,tξp-<+∞and for every μ∈0,μg,λ where (75)μg,λ≔1g01-λTp-p+α-lim⁡infξ→0+⁡∑k=1Tsupt≤ξ⁡Fk,tξp-,the problem Pλf,g has a sequence of pairwise distinct solutions which strongly converges to 0 in E.Proof.
Fixλ¯∈]λ3,λ4[ and let g be the function satisfying condition (74). Since λ¯<λ2, one has μg,λ¯>0. Fix μ¯∈]0,μg,λ¯[ and set ν3≔λ3 and ν4≔λ4/1+μ¯/λ¯λ4g0. If g0=0, clearly, ν3=λ3, ν4=λ4, and λ∈ν3,ν4. If g0≠0, since μ¯<μg,λ¯, one has (76)λ¯λ4+μ¯g0<1,and so (77)λ41+μ¯/λ¯λ4g0>λ¯;namely, λ¯<ν4. Hence, recalling that λ¯>λ3=ν3, one has λ¯∈ν3,ν4.
Now, putQ(k,t)=F(k,t)+μ¯/λ¯G(k,t) for all t∈R and k∈[1,T]. Since (78)∑k=1Tsupt≤ξ⁡Qk,tξp-≤∑k=1Tsupt≤ξ⁡Fk,tξp-+μ¯λ¯∑k=1Tsupt≤ξ⁡Gk,tξp-,taking (74) into account, one has(79)Tp-p+α-lim⁡infξ→0+⁡∑k=1Tsupt≤ξ⁡Qk,tξp-≤Tp-p+α-lim⁡infξ→0+⁡∑k=1Tsupt≤ξ⁡Fk,tξp-+μ¯λ¯g0.Moreover, since G is nonnegative, from Assumption (A7), we have(80)lim⁡supξ→0+⁡∑k=1TQk,ξξp+≥lim⁡supξ→0+⁡∑k=1TFk,ξξp+.Therefore, from (79) and (80), we obtain(81)λ¯∈ν3,ν4⊆2α+p-lim⁡supξ→0+⁡∑k=1TQk,ξ/ξp+,α-Tp-p+lim⁡infξ→0+⁡∑k=1Tsupt≤ξ⁡Qk,t/ξp-⊆λ3,λ4.We take X, Φ, Ψ, and Iλ¯ as in the proof of Theorem 6. We prove that δ<+∞. For this, let {ξn} be a sequence of positive numbers such that ξn→0+ as n→+∞ and (82)limn→∞⁡∑k=1Tsupt≤ξn⁡Fk,tξnp-<+∞.Put rn=α-/Tp-p+ξnp- for all n∈N. Let us show that the functional Iλ¯ has not a local minimum at zero. For this, let {γn} be a sequence of positive numbers and τ>0 such that γn→0+ as n→∞ and(83)1λ¯<τ<p-2α+∑k=1TFk,γnγnp+for each n∈N large enough. Let {wn} be a sequence in X defined by (40). So, owing to (42), (43), and (83), we obtain (84)Iλ¯wn=Φwn-λ¯Ψwn≤2α+p-γnp+-λ¯∑k=1TFk,γn<2α+1-λ¯τp-γnp+<0for every n∈N large enough. Since Iλ¯(0)=0, that means that 0 is not a local minimum of the functional Iλ¯. Hence, part (c) of Theorem 1 ensures that there exists a sequence un in X of critical points of Iλ¯ such that un→0 as n→∞, and the proof is complete.Remark 15.
Applying Theorem14, results similar to Remark 9 and Corollaries 10 and 12 can be obtained.We end this paper by giving the following example as an application of Theorem14.Example 16.
LetT=5, p(k)=k2+2 for all k∈[0,5], and α(k)=k2+1 for all k∈[1,6] and let f1:[1,T]×(R∖{0})→R be the function defined by (85)f1t=2tln⁡ln⁡1t2-ln-1⁡1t2sin2⁡ln⁡ln⁡ln⁡1t2-4tln-1⁡1t2sin⁡ln⁡ln⁡ln⁡1t2+8tln-2⁡1t21+ln⁡1t2,fk,t=ekf1tifk,t∈1,5×R∖0,0ifk,t∈1,5×0.Let g(k,t)=t2ln⁡k2 for all (k,t)∈[1,5]×R. Direct calculations show p-=2, p+=27, α-=2, α+=26, G(k,t)=t3ln⁡k2/3 for all k,t∈1,5×R, and (86)Fk,t=ekt2ln⁡ln⁡1t2sin2⁡ln⁡ln⁡ln⁡1t2+4t2ln-1⁡1t2,ifk,t∈1,T×R∖0,0ifk,t∈1,T×0. Thus (87)lim⁡infξ→0+⁡∑k=1Tsupt≤ξ⁡Fk,tξp-=lim⁡infξ→0+⁡∑k=15supt≤ξ⁡Fk,tξ2=0,lim⁡supξ→0+⁡∑k=1TFk,ξξp+=lim⁡supξ→0+⁡∑k=15Fk,ξξ27=+∞. Hence, using Theorem 14, since (88)g0≔Tp-p+α-limξ→0+⁡∑k=1Tsupt≤ξ⁡Gk,tξp+=0,the problem (89)-Δk2+1Δuk-1k2Δuk-1=λfk,uk+μln⁡k2u2k,k∈1,5,u0=u6=0for every (λ,μ)∈(0,+∞)×[0,∞) has a sequence of pairwise distinct solutions which strongly converges to 0 in the space (90)E5≔u:0,6⟶R:u0=u6=0.Remark 17.
Iff and g are nonnegative then the nontrivial solutions of the problem Pλf,g are positive. Indeed, let u∗ be a nontrivial solution of the problem Pλf,g; assume that the discrete interval A=k∈1,T;u∗k<0 is nonempty. Put v¯(k)=min⁡u∗k,0 for k∈1,T. Clearly, v¯∈E and one has (91)∑k=1T+1αkΔu∗k-1pk-1-2Δu∗k-1Δv¯k-1-λ∑k=1Tfk,u∗kv¯k-μ∑k=1Tgk,u∗kv¯k=0.Since f and g are nonnegative, by choosing v¯=u∗, one has(92)0≤α-∑k∈AΔu∗k-1pk-1≤0;that is,(93)u∗A≤0,which contradicts with the fact that u∗ is a nontrivial solution. Hence, u∗ is positive.



---

*Source: 5676138-2016-11-20.xml*