MULTIPLICITY RESULTS FOR DISCRETE ANISOTROPIC EQUATIONS

. In this article we continue the study of discrete anisotropic equations and we will provide a new multiplicity results of the solutions for a discrete anisotropic equation. The procedure viewed here is according to variational methods and critical point theory. In fact, using a consequence of the local minimum theorem due Bonanno and mountain pass theorem we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by mingling two algebraic conditions on the nonlinear term employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

1. Introduction. In this paper we are mainly concerned with existence and multiplicity results for the following discrete problem −∆(α(k)|∆u(k − 1)| p(k−1)−2 ∆u(k − 1)) = λf (k, u(k)), k ∈ [1, T ], u(0) = u(T + 1) = 0 (P f λ ) where T ≥ 2 is a fixed positive integer number, [1, T ] is the discrete interval Discrete boundary value problems and nonlinear difference equations emerge from real world problems and are claimed to be employed as handy means for the description of the processes which are endowed with discrete intervals. 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 [28] 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 [2,3,8,10,11,12,24,27,29,30,31,36,43,45] and the references therein.
Anisotropic operators appear in several places in the literature. Recent relevant applications include models in physics [6,14,22,23], biology [4,5] and image processing (see, for instance, the monograph by Weickert [44]). We also refer to Fragala, Gazzola and Kawohl [16] and El Hamidi and Vétois [15] as basic references in the treatment of nonlinear anisotropic problems. Moreover, note that Mihailescu et al. (see [33,34]) were the first authors to study anisotropic elliptic problems with variable exponent, and after that, in recent years, a great deal of work has been done in the study of the existence of solutions for discrete anisotropic boundary value problems (BVPs). For background and recent results, we refer the reader to [18,20,21,35,37,42] and the references therein.
Our approach is variational method and the main tools are a local minimum theorem for differentiable functionals due Bonanno [7] and Mountain Pass Theorem. Two of the consequences of the local minimum theorem are here applied (see Theorems 2.1 and 2.2). Indeed, we investigate existence results for the problem (P f λ ) under some algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term (see [1]). Moreover, by combining two algebraic conditions on the nonlinear term employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin ([38]) we establish the existence of third solution for the problem (P f λ ). For some related results we would also like to mention [7].
Here, we state two special cases of our results.
For a through research on the subject, we also refer the reader to [13,25,26].
2. Preliminaries. Our main tools include the following theorems, consequences of the local minimum theorem due Bonanno [7, Theorem 3.1], which is in turn motivated by Ricceri's variational principle (see [41]), and is related to the celebrated three critical point theorem of Pucci and Serrin [38,39].
Here and in the sequel, we take the T -dimensional Hilbert space Put F (k, t) := t 0 f (k, ξ)dξ for all (k, t) ∈ [1, T ] × R. Remark 1. We recall that a map f :

MAREK GALEWSKI, SHAPOUR HEIDARKHANI AND AMJAD SALARI
We say that a vector u ∈ X is a solution of the problem (P f λ ) if and only if and Standard arguments show that Φ and Ψ are C 1 functionals whose derivatives at the point u ∈ X are given by Hence, a critical point of the functional Φ − λΨ, gives us a solution of (P f λ ). We need the following proposition in the proofs of our main results.
for every u, v ∈ X. Then, J admits a continuous inverse on X * .
Proof. Assume that u > 1, by [19, Section 2(A.1)] we have Since p − > 1, this follows that J is coercive. Following Lemma 4.2 in [32] we obtain that if p ≥ 2, then for for every u, v ∈ X, u = v, which means that J is strictly monotone. Since it is also coercive and continuous and the setting is finite dimensional Theorem 2.2 from [17] states that J is invertible with continuous inverse.
3. Main results. In this section, we formulate our basic conclusions. For a nonnegative constant γ and a constant δ > 1 with Theorem 3.1. Assume that p + ≥ 2, f (k, 0) = 0 for all k ∈ [1, T ] and suppose that there exist a non-negative constant γ 1 and two positive constants γ 2 and δ, with there exist ν > 2 and R > 0 such that for all |ξ| ≥ R and for all k ∈ [1, T ].
Then, for each λ ∈ admits at least two non-trivial solutions u 1 and u 2 in X, such that Moreover, there exists at least one another solution u 3 .
Proof. Put I λ = Φ − λΨ, where Φ and Ψ are given as in (2) and (3), respectively. Of course Φ is C 1 functional while Proposition 1 gives that its Gâteaux derivative admits a continuous inverse on X * . The functional Ψ : X → R is well defined and is continuously Gâteaux differentiable whose Gâteaux derivative is compact. Choose Thus . Now, arguing as above, we obtain sup Therefore, by (A 1 ) one has On the other hand, one has Hence, from (A 1 ), one has β(r 1 , r 2 ) < ρ 2 (r 1 , r 2 ). Therefore, from Theorem 2.1, , the functional I λ admits at least one non-trivial critical point u 1 such that Now, we prove the existence of the second local minimum distinct from the first one. To this goal, we verify the hypotheses of the mountain-pass theorem for the functional I λ . Clearly, the functional I λ is of class C 1 and I λ (0) = 0. The first part of the proof guarantees that u 1 ∈ X is a local nontrivial local minimum for I λ in X. We can assume that u 1 is a strict local minimum for I λ in X. Therefore, there is ρ > 0 such that inf u−u1 =ρ I λ (u) > I λ (u 1 ), so the condition [40, (I 1 ), Theorem 2.2] is verified. By integrating the condition (5), there exist constants a 1 , a 2 > 0 such that F (k, t) ≥ a 1 |t| ν − a 2 for all k ∈ [1, T ] and t ∈ R. Now, choosing any u ∈ X \ {0}, by the assumption p + > 2 and [19, Section 2(A.4)] one has as u → +∞, so condition [40, (I 2 ), Theorem 2.2] is satisfied; here ζ is some positive constant that for any u ∈ X, ν T k=1 |u(k)| ν ≥ ζ u . Thus, the functional I λ satisfies the mountain pass geometry. Moreover, I λ satisfies the Palais-Smale condition since it is ani-coercive. Hence, by the classical theorem of Ambrosetti and Rabinowitz we establish a critical point u 2 of I λ such that I λ (u 2 ) > I λ (u 1 ). Since f (k, 0) = 0 for all k ∈ [1, T ], u 1 and u 2 are two distinct non-trivial solutions of (P f λ ). Moreover, by direct maximization we obtain the third solution u 3 which is an argument of a maximum. Not that I λ (u 3 ) ≥ I λ (u 2 ) > I λ (u 1 ). If equality I λ (u 3 ) = I λ (u 2 ) holds, than we obtain in fact infinitely many solutions on the level I λ (u 3 ) = I λ (u 2 ). The proof is complete.
Then, for each λ ∈ , the problem (P f λ ) admits at least two non-trivial solutions u 1 and u 2 in X such that Moreover, there exists at least one another solution u 3 .
Proof. The conclusion follows from Theorem 3.1, by taking γ 1 = 0 and γ 2 = γ. Indeed, owing to the inequality (7), one has In particular, one has a γ (δ) Hence, Theorem 3.1 ensures the conclusion. Now we illustrate Theorem 3.2 by presenting the following example.
Remark 3. If f is non-negative then the solutions ensured in Theorems 3.1, 3.2 and 3.3 is non-negative. Here we reason exactly as in [20] supposing the solution is not positive and arriving at contradiction. Now, we point out some results in which the function f has separated variables. To be precise, consider the following problem where θ : [1, T ] → R is a non-negative and non-zero function and θ k := θ(k) for all k ∈ [1, T ] such that T k=1 θ k < ∞ and g : R → R is a non-negative continuous function. Put The following existence results are consequences of Theorems 3.1-3.3, respectively.
From (14) there exists a positive constant δ with δ < p + 2α +δp + . Therefore, we can use Theorem 3.2 and so the proof is complete. Remark 4. Theorem 1.1 immediately follows from Theorem 3.7. Now we illustrate Theorem 3.7 by presenting the following example.
In particular, u λ is a global minimum of the restriction of I λ to Φ −1 (−∞, r λ ). We will prove that the function u λ cannot be trivial. Let us show that lim sup Thanks to our assumptions at zero, we can fix a sequence {ξ n } ⊂ R + converging to zero and two constants ε, κ (with ε > 0) such that for every ξ ∈ [0, ε] Finally, fix M > 0 and consider a real positive number η with .