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No. 43/2015 pp. 45-51 doi: 10.17114/j.aua.2015.43.04 Acta Universitatis Apulensis ISSN: 1582-5329 http://www.uab.ro/auajournal/ EXISTENCE OF SOLUTION TO A SEMILINEAR DISCRETE PROBLEM INVOLVING P −LAPLACIAN A. Ourraoui Abstract. In this work, under appropriate assumptions, we present a result on the existence of solutions of difference equation −∆ |∆u(k − 1)|p−2 ∆u(k − 1) = f (k, u(k)) + g(k)|u(k)|q−2 u, k ∈ [1, T ]. Our method is based on the Ekeland’s variational principle. 2010 Mathematics Subject Classification: 47A75, 35B38, 35P30, 34L05, 34L30. Keywords: Variational method; Discrete boundary value problem. 1. Introduction Many papers devoted to study of the discrete equations have increased in recent years, by using various methods and techniques as fixed point theorems, lower and upper solutions, and variational approach, see for example [1, 2, 3, 4, 5, 6, 7, 10, 11, 12]... Let T ≥ 2 be a fixed positive integer, [a, b] be the discrete interval {a, a + 1, ..., b} where a and b are integers and a < b. Motivated by the above mentioned papers, we deal with the following discrete boundary value problem −∆ |∆u(k − 1)|p−2 ∆u(k − 1) = f (k, u(k)) + g(k)|u(k)|q−2 u, k ∈ [1, T ]. (1) u(0) = u(T + 1) = 0, where ∆u(k) = u(k + 1) − u(k) is the forward difference operator, 1 < q < p < ∞ while f : [1, T ] × R → R is a continuous function. Denoting by F : [1, T ] → R the primitive of f i.e., Z u F (k, u) = f (k, s)ds, u ∈ R. 0 45 A. Ourraoui – Existence of solution to a semilinear discrete problem . . . There is by now a rich literature on problems like (1), which began to receive much attention and are known to be mathematical models of various phenomena arising in the study of elastic mechanics, control systems, artificial or biological neural networks, computing, electrical circuit analysis, dynamical systems, economics,...For more detail, we may refer to [8, 13]. Here, we are concerned with investigating nonlinear discrete boundary value problems by using variational approach. The rest of this paper is arranged as follows. In section 2, we recall some basic definitions and tools in order to prove our main result and at after all we give an example. 2. Preliminaries and statement of main results In this section, we recall some basic properties which will be used in the proof of the precise result. Solutions to (1) will be investigated in a space n o W = u : [0, T + 1] → R s.t u(0) = u(T + 1) = 0 , which is a T -dimensional Hilbert space, see [2], with the inner product (u, v) = T +1 X ∆u(k − 1)∆v(k − 1), for all u, v ∈ W. k=1 The associated norm is defined by kuk = +1 TX |∆u(k − 1)|p 1 p . k=1 Moreover, it is useful to introduce other norms on W , namely |u|m = T X |u(k)|m 1 m , ∀u ∈ W and m ≥ 2. k=1 It can be verified (see [6]) that T 2−m 2m 1 |u|2 ≤ |u|m ≤ T m |u|2 , ∀u ∈ W and m ≥ 2. Next, we list some inequalities that will be are used later. 46 (2) A. Ourraoui – Existence of solution to a semilinear discrete problem . . . Lemma 1. ([10]) For every u ∈ W , we have (a) T X m |u(k)| ≤ T (T + 1) m−1 k=1 (b) T +1 X T +1 X |∆u(k − 1)|m , ∀m ≥ 2. k=1 p(k−1) |∆u(k − 1)| ≤2 m k=1 T X |u(k)|m , ∀m > 1. k=1 We say that u ∈ W is a weak solution of problem (1) if T +1 X p(k−1)−2 |∆u(k − 1)| T T X X ∆ϕ(k − 1) = f (k, u(k))ϕ(k) + µ g(k)|u(k)|p ϕ(k), k=1 k=1 k=1 for any ϕ ∈ W . Related to problem (1), define the functional φ : W → R given by φ(u) = T +1 X k=1 T T k=1 k=1 X 1X 1 F (k, u(k)) − |∆u(k − 1)|p − g(k)|u(k)|q . p q We assume the following conditions. (g1 ) g ∈ L∞ ([1, T ]), g ≥ (6≡)0. (f1 ) f (x, t) ∈ C([1, k] × R), f 6≡ 0. f (k, t) f (k, t) < λ1 and λ1 < l2 := lim p−1 < ∞ p−1 t→∞ t t→0 |t| uniformly in [1, T ] where λ1 > 0 is the smallest positive eigenvalue of −∆ |∆u(k − 1)|p−2 ∆u(k − 1) = λ|u(k)|p−2 u, k ∈ [1, T ], (f2 ) 0 ≤ l1 := lim (3) u(0) = u(T + 1) = 0, see [2] . The following theorem is the main result of this paper. Theorem 2. If (g1 ), (f1 ) and (f2 ) hold true, then the problem (1) has at least one nontrivial weak solution. Proof of Theorem 2. It is clear that the functional φ is differentiable and its derivative is given by 0 φ (u).v = T +1 X |∆u(k−1)| p−2 T T X X ∆u(k−1)∆v(k−1)− f (k, u(k))v(k)− g(k)|u(k)|p−2 u(k)v(k), k=1 k=1 47 k=1 A. Ourraoui – Existence of solution to a semilinear discrete problem . . . for all u, v ∈ W. For simplicity, we give an auxiliary result about f : In view of (f1 ) and (f2 ) with l2 < ∞ we have lim t→∞ f (k, t) = 0, for p < r < ∞, uniformly in tr−1 [1, T ]. Therefore, for any ε > 0 there exists C(ε) > 0 such that f (k, t) ≤ (l1 + ε)tp−1 + C(ε)tr−1 for all t ≥ 0, k ∈ [1, T ], and thus r F (k, t) ≤ l1p+ε tp + C(ε) r t , for all t ≥ 0, k ∈ [1, T ]. On the other hand, according to (f2 ) and the fact that l1 < λ1 , we may find ε1 > 0 such that l1 + ε1 < λ1 , F (k, t) ≤ l 1 + ε1 p t + C0 tr , for all t ≥ 0, k ∈ [1, T ] p with C0 > 0 is nonnegative constant. We present the following Lemma: Lemma 3. There exist ρ > 0 and α > 0 such that φ(u) ≥ α > 0, for every u ∈ W, with kuk = ρ. Proof of Lemma 3. By using of Lemma 1 we have φ(u) = T +1 X k=1 ≥ T T k=1 k=1 X 1 1X F (k, u(k)) − |∆u(k − 1)|p − g(k)|u(k)|q p q T T T 1 C(ε) X |g|∞ X l1 + ε X p r p |u(k)| − |u(k)| − |u(k)|q kuk − p p r q k=1 ≥ k=1 1 |g|∞ kukp − T (T + 1)q−1 p q −C(ε)T (T + 1)r−1 T +1 X T +1 X k=1 k=1 T +1 X l1 + ε1 |∆u(k − 1)| − T (T + 1)p−1 |∆u(k − 1)|p p q k=1 |∆u(k − 1)|r k=1 ≥ C1 kukp − |g|∞ C2 kukq − C3 kukr > 0, 1 > 0 and C2 , C3 are positive for kuk is small enough, with C1 = p1 1 − l1λ+ε 1 constants. 48 (4) A. Ourraoui – Existence of solution to a semilinear discrete problem . . . Define B ρ = {u ∈ E : kuk ≤ ρ} endowed with metric d(u, v) = ku − vk, u, v ∈ B ρ , and ∂Bρ = {u ∈ W : kuk = ρ}. By the previous Lemma 3, we know that φ(u)/∂Bρ ≥ α > 0 and it is easy to see that the functional φ is bounded from below. Let c∗ = minB ρ φ(u), so we have c∗ < 0. In fact, fix a nonnegative function v ∈ W \ {0}, and note that for t > 0 we have φ(tv) = T +1 X k=1 ≤ tp p T T X 1 1X |∆tv(k − 1)|p − F (k, tv(k)) − g(k)|tv(k)|q p q kvkp − k=1 tp l1 p T X v(k)p − k=1 tq q k=1 T X g(k)v(k)q . k=1 Since q < p, the last inequality implies that for some t∗ > 0 sufficiently small, φ(t∗ v) < 0, and t∗ v ∈ B ρ (0). By the Ekeland’s variational principle in [9], we may find a sequence (un )n ⊂ B ρ (0) such that φ(un ) → c∗ i.e c∗ ≤ φ(un ) ≤ c∗ + n1 , ∀n > 0 and 1 φ(v) ≥ φ(un ) − kun − vk, ∀v 6= un . n Once that φ is differentiable, it follows from the last inequality that φ0 (un ) → 0. Indeed, let u ∈ W such that kuk = 1, putting ωn = un + λu. Fix n > 1 we have kωn k ≤ kun k + λ < ρ with λ > 0 is small enough. It yields φ(un + λu) ≥ φ(un ) − 49 λ kuk n A. Ourraoui – Existence of solution to a semilinear discrete problem . . . then φ(un + λu) − φ(un ) 1 ≥ − kuk, λ n tending λ → 0 so we get φ0 (un ).u ≥ − 1 n that is, |φ0 (un ).u| ≤ 1 n with kuk = 1. So φ0 (un ) → 0. Since the sequence (un ) is bounded in W, there exists u1 ∈ W such that, up to a subsequence, (un ) converges to u1 in W. Therefore, φ(u1 ) = c∗ , and φ0 (u1 ) = 0, and thus u1 is a non trivial solution of problem (1). Example 1. Taking β > 0 and l2 > λ1 , the function f such that tβ+1 f (x, t) = l21+t β when t ≥ 0 and f (x, t) = 0 if t ≤ 0, satisfies the conditions (f1 ) and (f2 ) provided l2 < ∞. f (k, t) < λ1 , k ∈ [1, T ] then, it is easy to p−2 t |t|→∞ |t| check that there exists 0 ≤ λ < λ1 such that 1 λ φ(u) ≥ 1− kukp − C4 kuk − C5 kukq , (5) p λ1 Remark 1. If we assume that lim sup where C4 and C5 are positive constants. Thus φ is coercive and bounded from below so it has a global minimizer. References [1] R.P. Agarwal, D. O’Regan, Boundary value problems for discrete equations, Appl. Math. Lett 10 (1997), 83-89. [2] R. P. Agarwal, K. Perera, D. 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Anass Ourraoui Department of Mathematics, ENSAH, University of Mohamed I, Oujda, Morocco email: anas.our@hotmail.com 51