GIOVANNI MOLICA BISCI AND DUŠAN REPOVŠ

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1 EXISTENCE OF SOLUTIONS FOR p-laplacian DISCRETE EQUATIONS arxiv:168.77v1 [math.ap] 7 Aug 16 GIOVANNI MOLICA BISCI AND DUŠAN REPOVŠ Abstract. This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete p-laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a positive real parameter α, whose prototype is given by { u(k 1) = αf(k,u(k)), k Z[1,T] u() = u(t +1) =. Our approach is based on variational methods in finite-dimensional setting. 1. Introduction We are interested in investigating nonlinear discrete boundary value problems by using variational methods. This approach has been recently adopted, for instance, in [, 3, 4] and [14,, 31]. More precisely, for every a,b Z, such that a < b, set Z[a,b] := {a,a+1,...,b} and let T be a positive integer. The aim of this paper is to prove some existence results for the following discrete problem { (φp ( u(k 1))) = f(k,u(k)), k Z[1,T] (D f ) where p > 1, φ p : IR IR is given by φ p (s) := s p s, for every s IR, f : Z[1,T] IR IR is a continuous function, and u(k 1) := u(k) u(k 1) is the forward difference operator. In recent years equations involving the discrete p-laplacian operator, subject to different boundary conditions, have been widely studied by many authors and several approaches. Inparticular,Problem(D f )hasbeenpreviouslystudied, forinstance, in[3,8,15] by using various methods. See the recent papers [9, 5] for the discrete anisotropic case. Motivations for this interest arose in by different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and others. Moreover, the main background in the real world for the discrete p-laplacian operator are the problems on the boundary between different substances. Key words and phrases. Discrete boundary value problem; existence result; discrete p- Laplacian; critical point theory. 1 Mathematics Subject Classification: Primary 39A1; Secondary 47J3, 58E5. Typeset by LATEX. 1

2 GIOVANNI MOLICA BISCI AND DUŠAN REPOVŠ Set c(p,t) := 1 p [ ( T p p(t +1) p 1 ) p 1 ( ) ] p 1 + T + if T is even if T is odd. Via variational approach, we are able to prove the existence of a solution for problem (D f ) by requiring that max ξ ξ ε ε p f(k,s)ds < c(p,t), for some ε >. See condition () in Theorem 3.. Next, by using Theorem 3., we study a parametric version of problem (D f ), defined as follows { (Dα) f (φp ( u(k 1))) = αf(k,u(k)), k Z[1,T] where α is a positive real parameter. In this case, requiring a suitable behaviour of the potentials at zero and at infinity, we obtain, for sufficiently large α, the existence of at least one positive solution for problem (D f α), see Theorem 4.. This result can be achieved exploiting Theorem 3. together with the well-known variational characterization of the first eigenvalue of the p-laplacian operator in the finite-dimensional context (see [3]). The simplest example we can deal with is a second-order boundary value problem. Theorem 1.1. Let f : Z[1, T] [, + ) [, + ) be a continuous function satisfying the following hypotheses: and lim ξ + γ κ := liminf ξ + for every k Z[1,T]. Then for every α > min k Z[1,T] γ k ξ ξ f(k,t)dt ξ = f(k,t)dt ξ >, ( sin π (T +1) the following second-order discrete problem { (S α ) u(k 1) = αf(k,u(k)), k Z[1,T] has at least one positive solution. ),

3 EXISTENCE OF SOLUTIONS... 3 We remark that the results obtained for second-order discrete equations in [4, 1] and our theorems are mutually independent. Moreover, the approach adopted here can be used studying the discrete counterpart of the following problem u x + u +f((x,y),u(x,y)) =, y u(x,) = u(x,n+1) =, x (,m+1) u(,y) = u(m+1,y) =, y (,n+1), where m,n IN\{} and f is a suitable continuous function. See [1] and [13] for details. We refer to the monograph of Cheng [7] for a geometrical interpretation of this equations. The plan of the paper is as follows. Section is devoted to our abstract framework and preliminaries. Successively, in Section 3 we prove our main result (see Theorem 3.). The parametric case is discussed in the last section (see Theorem 4.), where, a concrete example of an application is also presented (see Example 4.5). On the T-dimensional Banach space endowed by the norm. Abstract Framework H := {u : Z[,T +1] IR : u() = u(t +1) = }, u := ( T+1 ) 1/p u(k 1) p, we define the functional J : H IR given by J(u) := 1 T+1 u(k 1) p p u(k) f(k,t)dt, for every u H. We recall that a solution of problem (D f ) is a function u H such that T+1 φ p ( u(k 1)) v(k 1) = f(k,u(k))v(k), for every v H. We observethatproblem(d f ) hasavariationalstructure. Indeed, thefunctional J is differentiable in u H and one has for every v H. T+1 J (u),v = φ p ( u(k 1)) v(k 1) f(k,u(k))v(k),

4 4 GIOVANNI MOLICA BISCI AND DUŠAN REPOVŠ Thus critical points of J are solutions to problem (D f ). In order to find these critical points, we will make use of the following local minimum result due to Ricceri (see [6]) recalled here on the finite-dimensional setting. Theorem.1. Let (E, ) be a finite-dimensional Banach space and let Φ,Ψ : X IR be two lower semicontinuous functionals, with Ψ coercive and Φ( E ) = Ψ( E ) =. Further, set J µ := µψ+φ. Then for each σ > inf Ψ(u) and each µ satisfying u X inf Φ(u) u Ψ µ > 1 ((,σ]) σ the restriction of J µ to Ψ 1 ((,σ)) has a global minimum. See [9, 7, 8] for related abstract critical points results. We also mention the monograph [19] for some topics on variational methods adopted in this paper and [1] for general facts on finite difference equations. By [6, Lemma 4] one has that 3. The Main Result (1) u := max k Z[1,T] u(k) 1 κ u, for every u H, where [ ( T κ := Remark 3.1. Note that [ ( ) p 1 + T ) p 1 ( ) ] p 1 1/p + if T is even T + (T +1) (p 1)/p if T is odd. ( ) ] p 1 1/p < T + (T +1)(p 1)/p. Indeed, since the continuous function θ : (,T +1) IR defined by θ(s) := 1 (T s+1) p s p 1, p T +1 attains its minimum at s =, one has (T +1) p 1 Then (T +1) (p 1)/p < p < θ(t/). (T +1) p 1 [ ( and the conclusion is achieved. T ) p 1 ( ) ] p 1 1/p + = θ(t/) 1/p, T +

5 EXISTENCE OF SOLUTIONS... 5 Set F k (ξ) := ξ f(k,s)ds, for every k Z[1,T] and ξ IR. With the above notations our result reads as follows. Theorem 3.. Let f : Z[1,T] IR IR be a continuous function and assume that there exists ε > such that max F k(ξ) ξ ε () ε p < κp p. Then problem (D f ) has at least one solution such that u < ε. Proof. Let us apply Theorem.1 by choosing E := H, and Φ(u) := F k (u(k)), Ψ(u) := u p, for every u E. Taking σ := κ p ε p, clearly σ > inf Ψ(u). Moreover, let us estimate from the u E above the following quantity Inequality (1) yields Thus one has that ϕ(σ) := Hence it follows, by (), that sup Φ(u) u Ψ 1 ((,σ]). σ Ψ 1 ((,σ]) {u E : u ε}. ϕ(σ) sup max F k(ξ) ξ ε u Ψ 1 ((,σ]) κ p ε p. F k (u(k)) < 1 σ p, that is, inf 1 Φ(u) p > u Ψ 1 ((,σ]). σ Therefore, the assertion of Theorem.1 follows and the existence of one solution u Ψ 1 ((,σ)) to our problem is established. Remark 3.3. If in Theorem 3. the function f is nonnegative, hypothesis () assumes a simpler form F k (ε) ε p < κp p.

6 6 GIOVANNI MOLICA BISCI AND DUŠAN REPOVŠ Moreover, if for some k Z[1,T], f( k,), the obtained solution is clearly non-zero. 4. A parametric case In this section we shall study the following discrete parametric problem (D f α ) { (φp ( u(k 1))) = αf(k,u(k)), k Z[1,T] where α is a real positive parameter. For our goal, in order to obtain positive solutions to problem (Dα f ), i.e. u(k) > for each k Z[1,T], we shall need the following consequence of the strong comparison principle, see [3, Lemma.3]. Lemma 4.1. If then either u > in Z[1,T], or u. (φ p ( u(k 1))), k Z[1,T] u(), u(t +1), Moreover,letλ 1,p,ϕ 1 > bethefirsteigenvalueandeigenfunctionoftheproblem { (φp ( u(k 1))) = λφ (D λ,p ) p (u(k)), k Z[1,T] u() = u(t +1) =. As observed in [3], the following variational characterization (3) λ 1,p = min E\{ H} T+1 u(k 1) p, u(k) p holds. Taking into account the above facts, an important consequence of Theorem 3. is the following. Theorem 4.. Let f : Z[1, T] [, + ) [, + ) be a continuous function satisfying the following hypotheses: F k (ξ) and lim ξ + for every k Z[1,T]. Then for every ξ p =, F k (ξ) γ k := liminf ξ + ξ p >, α > λ 1,p p min γ, k k Z[1,T] problem (D f α) has at least one positive solution.

7 EXISTENCE OF SOLUTIONS... 7 Proof. Let α be as in the conclusion, and define f(k,t) := { f(k,t) if t f(k,) if t <. for every k Z[1,T]. Consider now the following problem (D f α ) { (φp ( u(k 1))) = α f(k,u(k)), k Z[1,T] u() = u(t +1) =. ByLemma4.1, everynon-zerosolutionofproblem(d f α )ispositive. Furthermore, every positive solution of (D f α ) also solves our initial problem (D f α). Now, since there exists ε > such that lim ξ + F k (ξ) F k (ε) ε p ξ p =, < κp p. Hence, bearing in mind Remark 3.3, condition () of Theorem 3. holds. Thus problem (D f α ) admits a solution u α H with u α < ε. In conclusion, we shall prove that H is not a local minimum of the functional J α (u) := 1 T+1 u(k 1) p α p u(k) f(k,t)dt, i.e. the obtained solution u α is non-zero. For this purpose, let us observe that the first eigenfunction ϕ 1 H is positive and it follows by (3) that T (4) ϕ 1 p = λ 1,p ϕ 1 (k) p. Since γ k > min k Z[1,T] γ k > λ 1,p pα, for every k Z[1,T], there exists δ > such that (5) F k (ξ) > λ 1,p pα ξp, for every k Z[1,T] and ξ (,δ). Hence, we can define θ ζ (k) := ζϕ 1 (k), for every k Z[,T +1], where δ ζ Λ δ :=, max ϕ. 1(k) k Z[1,T]

8 8 GIOVANNI MOLICA BISCI AND DUŠAN REPOVŠ Taking into account (5) and (4), we easily get α F k (θ ζ (k)) > λ 1,p p θ ζ (k) p = 1 p θ ζ p, that is, J α (θ ζ ) = 1 p θ ζ p α F k (θ ζ (k)) <, for every ζ Λ δ. The proof is thus complete. Remark 4.3. In Theorem 4., looking at the behaviour of the function h(ξ) := F k (ξ) ξ p, ( ξ > ) at infinity, the existence of one positive solution has been proved. On the other hand, if the function f(k, ) : IR IR has a s-sublinear potential F k with s < p, for every k Z[1,T], the behaviour at zero of the map max F k(ξ) ξ ε χ(ε) := ε p, ( ε > ) influences the existence of multiple solutions. More precisely, requiring that (6) lim ε +χ(ε) =, by using variational arguments, one can prove that there exists a real interval of parameters Λ such that, for every α Λ, the problem (Dα) f admits at least three solutions. If, instead of (6), we assume that (7) χ(c) < p 1 (T +1) p 1 for some positive constants c < d, then for every α ), p (h(d) cp d pχ(c) (h(d) cp d pχ(c) ), p pχ(c)(t +1) p 1, there exist at least three distinct solutions of the problem (Dα f ). Clearly condition (7) is technical and quite involved. Finally, we also note that a more precise result can be obtained if T is even. Remark 4.4. It is easy to see that Theorem 1.1 in Introduction is a consequence of Theorem 4. bearing in mind that the first eigenvalue of the problem { (D λ ) u(k 1) = λu(k), k Z[1,T]

9 EXISTENCE OF SOLUTIONS... 9 is given by ( ) λ 1 := 4sin π, (T +1) see, for instance, [5, p. 15] and [3]. More precisely, as is well-known, the eigenvalues λ k, for k Z[1,T], of problem (D λ ) are exactly the eigenvalues of the positive-definite matrix Thus it follows that A := T T ( ) kπ λ k = 4sin, k Z[1,T]. (T +1) A direct application of this result yields the following. Example 4.5. For every α (λ 1,+ ), the following second-order discrete problem (S α ) u(k 1) = α u(k) 1+u(k), k Z[1,T] has at least one positive solution. Remark 4.6. In Example 4.5, for every α sufficiently large, our approach ensures the existence of at least one positive solution u α H for the discrete problem (S α ). A more delicate problem is to find a concrete expression of the function u α that one may hope to be exploited by numerical methods. Remark 4.7. We refer to the paper of Galewski and Orpel [1] for several multiplicity results on discrete partial difference equations. See also the papers [11, 17, 18, 1, 3, 4] for recent contributions to discrete problems. Acknowledgements. This paper was written when the first author was visiting professor at the University of Ljubljana in 13. He expresses his gratitude to the host institution for warm hospitality. The manuscript was realized within the auspices of the GNAMPA Project 13 entitled Problemi non-locali di tipo Laplaciano frazionario and the SRA grants P and J The authors warmly thank the anonymous referees for their useful comments on the manuscript. References [1] R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York-Basel,. [] R.P. Agarwal, K. Perera and D. O Regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (4), [3] R.P. Agarwal, K. Perera and D. O Regan, Multiple positive solutions of singular discrete p-laplacian problems via variational methods, Advance in Difference Equations 5 () (5),

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11 EXISTENCE OF SOLUTIONS (G. Molica Bisci) Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 8914 Reggio Calabria, Italy address: gmolica@unirc.it (D. Repovš) Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, POB 964, Ljubljana, Slovenia 11 address: dusan.repovs@guest.arnes.si