A Variational Approach to Perturbed Discrete Anisotropic Equations

and Applied Analysis 3 (c) If δ < +∞, then, for each λ ∈ ]0, 1/δ[, the following alternative holds: either (c1) there is a global minimum of Φ which is a local minimum of Iλ or (c2) there is a sequence of pairwise distinct critical points (local minima) of Iλ which weakly converges to a global minimum ofΦ. 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 E fl {u : [0, T + 1] 󳨀→ R : u (0) = u (T + 1) = 0} , (5) endowed with the norm


Introduction
The aim of this paper is to investigate the existence of infinitely many solutions for the following perturbed discrete anisotropic problem:  () . ( 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 -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][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][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 points' theorems, investigated different sets of assumptions which guarantee the existence and multiplicity of solutions for difference equations involving the discrete -Laplacian operator.In [4] Bian et al. by using critical point theory studied a class of discrete -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][22][23][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, Mihȃilescu 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][33][34][35][36][37][38] and the references therein.For example, Mihȃilescu 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 ( ,  ) 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: where  is a positive parameter,   : R → R is a continuous function for every  ∈ Z[1, ] (with  ≥ 2), and Δ( − 1) fl () − ( − 1) is the forward difference operator, assuming that the map  : Z[0, ] → R satisfies  − fl min Z[0,] () > 1 as well as  + fl max Z[0,] () > 1. Stegliński in [38], based on critical point theory, obtained the existence of infinitely many solutions for the parametric version of the problem ( ,  ), in the case where  = 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  and  in which the problem ( ,  ) admits a sequence of solutions which is unbounded in the space  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).

Preliminaries
Our main tool to ensure the existence of infinitely many solutions for the problem ( ,  ) 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.Let  be a reflexive real Banach space and let Φ, Ψ :  → 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  > inf  Φ, let one put Then, one has the following: (a) For every  > inf  Φ and every  ∈ ]0, 1/()[, the restriction of the functional We refer the reader to the paper [41][42][43][44][45][46][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 the -dimensional Banach space endowed with the norm Remark 2. We consider that whenever  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  * .
A special case of our main result is the following theorem.
Theorem 5. Let  : R → R be a continuous function and put Then, for every continuous function  : R → R whose () = ∫  0 ()d, for every  ∈ R, is a nonnegative function satisfying the condition and for every  ∈ [0, ⋆, [, where  ⋆, fl has an unbounded sequence of solutions.

Main Results
We present our main result as follows.
Theorem 6. Assume that for every continuous function  : and for every  ∈ [0,  , [, where the problem (  ( Since  is a finite dimensional Banach space, Ψ is a Gâteaux differentiable functional and sequentially weakly upper semicontinuous whose Gâteaux derivative at the point  ∈  is the functional Ψ  () ∈  * , given by for every V ∈ , and Ψ  :  →  * is a compact operator.Moreover, Φ is a Gâteaux differentiable functional of which Gâteaux derivative at the point  ∈  is the functional Φ  () ∈  * , given by for every V ∈ .Furthermore, Φ is sequentially weakly lower semicontinuous (see [ Put   = ( − /  −  + )  −  for all  ∈ N. Let  0 ∈ N be such that ( + / − )  > 1 for all  >  0 .We claim that Then, for every  ∈ [1, 𝑇+1].Consequently, since  ∈ , we deduce by easy induction that for every  ∈ [1, ] and this gives (24).Hence, taking into account the fact that Φ(0) = Ψ(0) = 0, for every  large enough, one has Moreover, it follows from Assumption (A1) that lim inf which concludes that lim Then, in view of ( 18) and ( 30 taking ( 18) into account, one has Moreover, since  is nonnegative, we have lim sup Therefore, from ( 35) and ( 36) and from Assumption (A1) and ( 33), one has For fixed , inequality (33) we can consider a real sequence {  } with   > 1 for all  ∈ N and a positive constant  such that   → +∞ as  → ∞ and for each  ∈ N large enough.Thus, we consider a sequence {  } in  defined by setting Thus On the other hand, since  is nonnegative, we observe So, from (39), (42), and (43), we conclude that for every  ∈ N large enough.Hence, the functional   is unbounded from below, and it follows that   has no global minimum.Therefore, Theorem 1 assures that there is a sequence {  } ⊂  of critical points of   such that lim →∞ Φ(  ) = +∞, which from Lemma 3 follows that lim →∞ ‖  ‖ = +∞.Hence, we have the conclusion.
Remark 7.Under the conditions lim inf Theorem 6 assures that for every  > 0 and for each  ∈ [0,1/ ∞ [ the problem ( ,  ) admits infinitely many solutions.Moreover, if  ∞ = 0, the result holds for every  > 0 and  ≥ 0. Now, we give an application of Theorem 6 as follows.
Example 8. Let  = 10, let () = 2 + /5 for all  ∈ [0, 10], let () = 1 + 1/ for all  ∈ [1,11], and let   be a sequence defined by and let   be a sequence such that where (, ) =   ℎ() for all (, ) ∈ [1, 10] × R with where Indeed, clearly, by choosing   = 0 for all  ∈ N, from (A2) we obtain (A1).Moreover, if we assume (A2) instead of (A1) and choose   =  −   −  /  −  + for all  ∈ N, applying the same argument in the proof of Theorem 6, we obtain where   () is the same as (40) but   is replaced by   .We have the same conclusion as in Theorem 6 with the interval ] 1 ,  2 [ replaced by the interval Here, we point out a simple consequence of Theorem 6. (18) and for every  ∈ [0,  ,1 [, where We here give the following consequence of the main result.
A direct calculation shows  − = 2,  + = 4, and Then Hence, all assumptions of Corollary 12 with  = 0 are satisfied.So, for every  ∈ (0, +∞), problem (66) has an unbounded sequence of solutions in the space Arguing as in the proof of Theorem 6 but using conclusion (c) of Theorem 1 instead of (b), one establishes the following result.for every  ∈ N large enough.Since   (0) = 0, that means that 0 is not a local minimum of the functional   .Hence, part (c) of Theorem 1 ensures that there exists a sequence {  } in  of critical points of   such that ‖  ‖ → 0 as  → ∞, and the proof is complete.
Remark 15.Applying Theorem 14, 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 Theorem 14.