Three solutions for discrete anisotropic Kirchhoff-type problems

Abstract: In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class of double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate the applicability of our main theoretical findings.


Introduction
which Kirchhoff studied in 1883 (see [1]) and which extends d'Alembert's wave equation, by considering the effect during vibrations when the length of the string is varied. In (1.2), the parameter L denotes string length, h stands for the cross-sectional area, E is the material's Young modulus, ρ is the mass density, and ρ 0 is the initial tension. A special feature of the Kirchhoff equation is that (1.2) contains the nonlocal coefficient on [ ] L 0, , and therefore (1.2) is not a pointwise identity. On the other hand, the stationary analogue of (1.2) is given as follows: which was studied extensively only after Lions [2] initiated an abstract setting for this problem. Some related, interesting, and important results can be seen, e.g., in [3][4][5][6][7].
Difference equations are generally understood as the first theory to appear with the systematic growth of mathematics, and they can be found in biological neural networks, economy, signal processing, computer engineering, genetics, medicine, ecology, and digital control. In the last decades, many researchers around the globe have used variational methods and critical point theory to study the existence and multiplicity of solutions for discrete boundary value problems, as referenced in [8][9][10][11][12]. We also refer the reader to [13][14][15][16], where discrete Kirchhoff-type equations were studied. However, as to the problem (1.1), it contains the Kirchhoff term , which makes it much more complicated to work with, and there are some studies [17][18][19][20][21][22][23], that discuss the existence of solutions for some discrete boundary value problems of ( ) p k -Kirchhoff-type using variational methods and critical point theory. Inspired by the above results, in this article, we investigate the existence of three solutions for (1.1). In this case, we apply suitable conditions and create intervals for the two parameters λ and μ. We also give Example 3.3 to show the use of our proven theorems.

Preliminaries and basic notation
In this article, X denotes a finite-dimensional real Banach space and → I X : λ is a functional satisfying the following structure hypothesis: are two functions of class C 1 on X such that Φ is coercive, i.e., , and all > r 0 3 , we define Assume that there are three positive constants r r , 1 2 , and r 3 with < r r , the functional − λ Φ Ψ admits three distinct critical points u u , 1 2 , and u 3 such that For an application of Theorem 2.1 to the discrete case, see [18]. Furthermore, we refer the reader to [25][26][27][28] for situations of successful employment of results such as Theorem 2.1 in order to prove the existence of solutions for various boundary value problems. We introduce the N -dimensional Hilbert space

Main results
In this section, we formulate our main results based on the existence of at least three solutions for the problem (1.1). Set Then, for every Proof. Our aim is to apply Theorem 2.1 to the problem. We consider the auxiliary problem is a continuous function defined by , then u is a solution of (1.1). This estimate is obtained at the end of this proof. Therefore, for our goal, it is enough to show that our conclusion holds for (3.4). Let the functions Φ and Ψ, for every ∈ u X, be defined as follows: Furthermore, let us denote by I λ the energy functional associated with problem (1.1), i.e., for every ∈ u X. By standard arguments, Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse. On the other hand, Ψ is well defined, continuously Gâteaux differentiable and with compact derivative. More precisely, one has . By the definition of Φ and thanks to Lemma 2.2, we have so Φ is coercive. Therefore, the assumptions on Φ and Ψ, as requested in We set By first using the Cauchy-Schwarz inequality, then (3.5), and then the estimate ( ) < u r Φ 1 , we obtain From the definition of r 1 , we obtain By using the assumption (A 1 ), one has Therefore, since Therefore, the assumptions (a 3 ), (a 4 ), and (a 5 ) of Theorem 2.1 are verified. Since fˆand g are nonnegative, the solutions of the problem (1.1) are nonnegative. Indeed, let * u be a nontrivial solution of the problem (1.1). Then, * u is nonnegative. Arguing by contradiction, assume that the discrete interval . Clearly, ∈ v X , and one has M ρ u u n u n v n λ f n u n v n μ g n u n v n Δ 1 Δ 1Δ¯1ˆ,¯,¯. i.e., Thus, = * u 0 in , which is absurd. Hence, * u is nonnegative. Now, we show that the functional I λ satisfies the assumption (a 2 ) of Theorem 2.1. Let u 1 and u 2 be two local minima for I λ (see [24, proof of Theorem 3.3]). Then u 1 and u 2 are critical points for I λ , and so, they are nonnegative solutions to the problem (1.1). Then, we have ≥ u u , 0 1 2 . Thus, it follows that Hence, Theorem 2.1 implies that for every λ in the interval given in the statement and for every μ in the interval given in the statement, the functional I λ has three critical points ∈ u u u X , , 1 2 3 We now present the following example to illustrate Theorem 3.1. Here, and , there exists δ such that, for each [ ) ∈ μ δ 0, , the problem (3.7) has at least three nonnegative solutions u 1 , u 2 , and u 3 satisfying u n u n u n max 10, max 10 , max 10 .
, or both hold true, then the solutions from Theorem 3.1 are not trivial.
We now deduce the following consequence of Theorem 3.1. such that 1, , 4 4 , and for all   Hence, from (3.8), (3.9), and (A 6 ), the assumption (A 6 ) of Theorem 3.1 is satisfied, and it follows the conclusion. □ Here, we present a simple consequence of Theorem 3.5 in the case when f does not depend upon n.