Electronic Journal of Differential Euation, Vol. 017 (017), No. 5,. 1 1. ISSN: 107-6691. URL: htt://ejde.math.txtate.edu or htt://ejde.math.unt.edu POSITIVE SOLUTIONS OF DISCRETE BOUNDARY VALUE PROBLEMS WITH THE (, )-LAPLACIAN OPERATOR ANTONELLA NASTASI, CALOGERO VETRO, FRANCESCA VETRO Communicated by Mokhtar Kirane Abtract. We conider a dicrete Dirichlet boundary value roblem of euation with the (, )-Lalacian oerator in the rincial art and rove the exitence of at leat two oitive olution. The aumtion on the reaction term enure that the Euler-Lagrange functional, correonding to the roblem, atifie an abtract two critical oint reult. 1. Introduction Let N be a oitive integer and denote by [1, N] the dicrete et {1,..., N}. We tudy the dicrete boundary value roblem u(z 1) u(z 1) + α(z)φ (u(z)) + β(z)φ (u(z)) = λg(z, u(z)), for all z [1, N], u(0) = u(n + 1) = 0, (1.1) where r u(z 1) := (φ r ( u(z 1))) = φ r ( u(z)) φ r ( u(z 1)) i the dicrete r-lalacian, φ r : R R i the homomorhim given a φ r (u) = u r u with u R (z [1, N]), u(z 1) = u(z) u(z 1) i the forward difference oerator, g : [1, N + 1] R R i a continuou function with g(n + 1, t) = 0 for all t R, α, β : [1, N + 1] R, 1 < < < + and λ ]0, + [. Here, we conider the following hyothee: (H1) g(z, 0) 0 for all z [1, N], and g(z, t) = g(z, 0) for all t < 0 and for all z [1, N]; (H) α(z), β(z) 0 for all z [1, N]. The olvability of general differential roblem, with variou boundary value condition, wa a maximum interet field of reearch over the lat decade. It attract ure and alied mathematician and ha it trong motivation in the oibility to model the dynamical behaviour of real henomena in hyic, economic, engineering and o on (ee, for examle, Diening-Harjulehto-Hätö-Rŭzĭcka [5]). There i a rich literature on thi ubject, which collect and exlain the rincial techniue of calculu of variation, critical and fixed oint theorie, Lyaunov-Schmidt reduction method, critical grou (More theory), and many other. We refer, for 010 Mathematic Subject Claification. 34B18, 39A10, 39A1. Key word and hrae. Difference euation; (, )-Lalacian oerator; (PS)-condition; oitive olution. c 017 Texa State Univerity. Submitted July 1, 017. Publihed Setember 0, 017. 1
A. NASTASI, C. VETRO, F. VETRO EJDE-017/5 examle, to the book of Motreanu-Motreanu-Paageorgiou [9] (for roblem with the -Lalacian oerator). So, one can olve other abtract tye of boundary value roblem, by combining and extending thee aroache. For examle, the exitence and multilicity of olution for roblem with the (, )-Lalacian oerator are conidered in Marano-Moconi-Paageorgiou [8], Mugnai-Paageorgiou [11] (euation), and Motreanu-Vetro-Vetro [10] (ytem of euation). On the other hand, the increaing of comuter erformance ha allowed reearcher (in alied mathematic) to focu on the olution of difference euation, and, in articular, of the dicrete verion of variou continuou differential roblem. We refer to the book of Agarwal [1], Kelly-Peteron [7] (difference euation), and the article of Cabada-Iannizzotto-Terian [3], D Aguì-Mawhin-Sciammetta [4], Jiang-Zhou [6] (dicrete -Lalacian oerator). Here, we ue the critical oint theory to rove the exitence of two oitive olution for dicrete (, )-Lalacian euation ubjected to Dirichlet tye boundary condition. Indeed, the idea i to reduce the roblem of exitence of olution in variational form, which mean to conider the roblem of finding critical oint for the Euler-Lagrange functional correonding to roblem (1.1). The aumtion on the reaction term enure that the involved Euler-Lagrange functional atifie an abtract two critical oint reult of Bonanno-D Aguì [].. Mathematical background We fix the notation a follow. By X and X we mean a Banach ace and it toological dual, reectively. Here, we conider the N-dimenional Banach ace and define the norm X d = {u : [0, N + 1] R uch that u(0) = u(n + 1) = 0}, u r,h := ( [ u(z 1) r + h(z) u(z) r ] ) 1/r, where h : [1, N + 1] R, with h(z) 0 for all z [1, N], and r ]1, + [. By u := max z [1,N] u(z) we denote the uual u-norm o that we conider the ineuality r 1 (N + 1) r u u r,h for all u X d (.1) (ee [4] and [6, Lemma.]). Prooition.1. Let h = N h(z). The following ineualitie hold N + 1 u u r,h ( r N + h) 1/r u. Proof. The left ineuality follow by (.1). On the other hand, ince u r r,h = [ u(z 1) r + h(z) u(z) r ] N N u r + r u r + u r h(z) z= we deduce eaily the right ineuality. = [ r (N 1) + + h] u r [ r N + h] u r,
EJDE-017/5 POSITIVE SOLUTIONS OF DISCRETE BOUNDARY VALUE PROBLEMS 3 Now, let X d be endowed with the norm u = u,α + u,β, where α and β (atifying (H)) are the coefficient of φ and φ in (1.1), reectively. We conider the function G : [1, N + 1] R R given a G(z, t) = t and the functional B : X d R given a 0 B(u) = g(z, ξ)dξ, for all t R, z [1, N + 1], It i clear that B C 1 (X d, R) and B (u), v = G(z, u(z)), for all u X d. g(z, u(z))v(z), for all u, v X d. Alo, define the functional A 1, A : X d R by A 1 (u) = 1 u,α and A (u) = 1 u,β, for all u X d. Clearly, A 1, A C 1 (X d, R) and (by the ummation by art formula) we have the following Gâteaux derivative at the oint u X d : A 1(u), v = A (u), v = for all u, v X d. Now, for r ]1, + [, we obtain Then, we have φ ( u(z 1)) v(z 1) + α(z)φ (u(z))v(z), φ ( u(z 1)) v(z 1) + β(z)φ (u(z))v(z), φ r ( u(z 1)) v(z 1) = [φ r ( u(z 1))v(z) φ r ( u(z 1))v(z 1)] N N = φ r ( u(z 1))v(z) φ r ( u(z))v(z) = φ r ( u(z 1))v(z). A 1(u), v = [ φ ( u(z 1)) + α(z)φ (u(z))]v(z), A (u), v = [ φ ( u(z 1)) + β(z)φ (u(z))]v(z),
4 A. NASTASI, C. VETRO, F. VETRO EJDE-017/5 for all u, v X d. Next, we conider the functional I λ : X d R given a I λ (u) = A 1 (u) + A (u) λb(u), for all u X d. We oint out that I λ (0) = 0. Alo, we have I λ(u), v = [ φ ( u(z 1)) φ ( u(z 1)) + α(z)φ (u(z)) + β(z)φ (u(z)) λg(z, u(z))]v(z), for all u, v X d. Thu, u X d i a olution of roblem (1.1) if and only if u i a critical oint of I λ. We recall the following notion. Definition.. Let X be a real Banach ace and X it toological dual. Then, I λ : X R atifie the Palai-Smale condition if any euence {u n } uch that (i) {I λ (u n )} i bounded; (ii) lim n + I λ (u n) X = 0, ha a convergent ubeuence. Our firt reult i the following auxiliary lemma, which characterize the functional I λ. G(z,t) Lemma.3. Let m (z) := lim inf t + t and m := min z [1,N] m (z). If m > 0, and (H1)-(H) hold, then I λ atifie the (PS)-condition and it i unbounded from below for all λ Λ :=] ( + )N+α+β m and β = N β(z)., + [, where α = N α(z) Proof. A m > 0, let λ > ( + )N+α+β m and m R uch that m > m > ( + )N+α+β λ. We conider a euence {u n } X d uch that I λ (u n ) c R and I λ (u n) 0 in Xd, a n +. Let u+ n = max{u n, 0} and u n = max{ u n, 0} for all n N. We how that the euence {u n } i bounded. So, we have u n (z 1) u n (z 1) u n (z 1) u n (z 1) for all z [1, N + 1]. Alo, we obtain u n (z 1) u n (z 1) u n (z 1) = φ ( u n (z 1)) u n (z 1), α(z) u n (z) = α(z) u n (z) u n (z)u n (z) = α(z)φ (u n (z))u n (z), for all z [1, N + 1]. Coneuently, we have u n,α = [ u n (z 1) + α(z) u n (z) ] [φ ( u n (z 1)) u n (z 1) + α(z)φ (u n (z))u n (z)] = A 1(u n ), u n.
EJDE-017/5 POSITIVE SOLUTIONS OF DISCRETE BOUNDARY VALUE PROBLEMS 5 In a imilar fahion, one ha u n,β A (u n ), u n. On the other hand, we obtain So, we obtain B (u n ), u n = g(z, u n (z))u n (z) 0 (by (H1)). u n,α u n,α + u n,β A 1(u n ), u n A (u n ), u n + λ B (u n ), u n = I λ(u n ), u n, for all n N, which lead to u n 1,α 0 a n +. Similarly, we deduce that u n 1,β 0 a n +, and hence u n 0 a n +. It follow that there exit ρ > 0 uch that u n ρ u n ρ + ρn := γ, for all n N. Next, we uoe that the euence {u n } i unbounded, which mean that {u + n } i unbounded. We may uoe without any lo of generality (aing to a ubeuence if neceary) that u n + a n +. By the aumtion on m at the beginning of the roof, we deduce that there i δ z max{γ, 1} uch that G(z, t) > mt for all t > δ z. Now, for all z [1, N], a G(z, ) i a continuou function, there exit a contant C(z) 0 uch that G(z, t) m t C(z) for all t [ γ, δ z ]. Thi imlie that G(z, t) m t C(z) for all t γ, all z [1, N]. It follow eaily that B(u n ) = G(z, u n (z)) N m u n (z) C m u n C, for all n N, where C = N C(z). For all u n uch that u n 1, we conclude that So, ince ( + )N+α+β I λ (u n ) = A 1 (u n ) + A (u n ) λb(u n ) = 1 u n,α + 1 u n,β λb(u n) ( N + α + N + β [ ( + )N + α + β ) u n λm u n + λc ] λm u n + λc. λm < 0, we obtain I λ (u n ) a n + ( u n + u n + ). Thi i an aburd entence and o {u n } i a bounded euence. Thi enure that I λ atifie the (PS)-condition. On the other hand, again reaoning on a euence {u n } X d uch that {u n } i bounded and u n + a n +, we deduce eaily that I λ (u n ) a n + and hence I λ i unbounded from below. A in D aguì-mawhin-sciammetta [4], our key-theorem i the following two nonzero critical oint reult of Bonanno-D Aguì [], which we arrange according to our notation and further ue.
6 A. NASTASI, C. VETRO, F. VETRO EJDE-017/5 Theorem.4. Let X d = {u : [0, N + 1] R uch that u(0) = u(n + 1) = 0} and A 1, A, B C 1 (X d, R) three functional uch that inf u Xd (A 1 (u) + A (u)) = A 1 (0) + A (0) = B(0) = 0. Aume that (i) there are R and û X d, with 0 < A 1 (û) + A (û) <, uch that B(û) A 1 (û) + A (û) > u u (A 1+A ) 1 (],]) B(u) ; (ii) the functional I λ : X d R given a I λ (u) = A 1 (u) + A (u) λb(u) for all u X d atifie the (PS)-condition and it i unbounded from below for all λ Λ := ] A 1(bu)+A (bu) [ B(bu), u. u (A1 +A ) 1 (],]) B(u) Then I λ admit two non-zero critical oint u λ,1, u λ, X d uch that I λ (u λ,1 ) < 0 < I λ (u λ, ), for all λ Λ. 3. Main reult We tart thi ection with an ueful obervation. Let h : [0, N + 1] [0, + [ and r ]1, + [. If (φ r ( u(z 1))) + h(z)φ r (u(z)) 0 and u(z) 0, then { 0 if u(z 1) 0; u(z) (3.1) < 0 if u(z 1) < 0. Indeed, if u(z) 0 then φ r (u(z)) 0 and hence (φ r ( u(z 1))) 0. So, we have φ r ( u(z)) φ r ( u(z 1)), which imlie that (3.1) hold. We denote by C + := {u X d : u(z) > 0 for all z [1, N]}. A olution u of roblem (1.1) i oitive if u C +. Now, we are ready to etablih the following trong maximum rincile reult tye. Theorem 3.1. Let u X d be fixed o that one of the following ineualitie hold for each z [1, N]: (a) u(z) > 0; (b) (φ ( u(z 1))) + α(z)φ (u(z)) 0; (c) (φ ( u(z 1))) + β(z)φ (u(z)) 0. Then, either u C + or u 0, rovided that (H) hold too. Proof. Let u X d \ {0} and J = {z [1, N] : u(z) 0}. If J =, then u C +. Proceeding by aburd, we aume that J. Now, if min J = 1, then from (3.1) we deduce that u(1) 0, which imlie u() 0. By iterating thi argument, we obtain eaily 0 = u(n + 1) u(n) u() u(1) 0, which lead to contradiction (i.e., u 0). On the other hand, if min J = j [, N], then u(j 1) = u(j) u(j 1) < 0 (note that u(j 1) > 0). By (3.1), we obtain u(j) < 0 u(j + 1) < u(j) 0. By iterating thi argument, we obtain eaily u(n + 1) < u(n) < < u(j + 1) < u(j) 0, which lead to a contradiction (i.e., u(n + 1) < 0). u C +. Then, J = and hence In the euel, let ξ + = max{0, ξ} and we denote with g + : [1, N + 1] R R the function defined by g + (z, ξ) = g(z, ξ + ) for all z [1, N], all ξ R.
EJDE-017/5 POSITIVE SOLUTIONS OF DISCRETE BOUNDARY VALUE PROBLEMS 7 Remark 3.. If the function g : [1, N + 1] R R i uch that g(z, 0) 0 for all z [1, N], then g + atifie the condition (H1). Now, conider the function G + : [1, N + 1] R R given a G + (z, t) = t and the functional B + : X d R defined by 0 B + (u) = g + (z, ξ)dξ, for all t R, z [1, N + 1], G + (z, u(z)), for all u X d. It i clear that B + C 1 (X d, R). Alo, the functional I + λ : X d R given a I + λ (u) = A 1(u) + A (u) λb + (u), for all u X d, ha a critical oint the olution of the roblem u(z 1) u(z 1) + α(z)φ (u(z)) + β(z)φ (u(z)) = λg + (z, u(z)), for all z [1, N], u(0) = u(n + 1) = 0. (3.) Remark 3.3. It i immediate to check that Lemma.3 hold for the functional I + λ, if we aume that g(z, 0) 0 for all z [1, N]. In fact, thi enure that (H1) hold for g + (by Remark 3.). The roof of the following rooition i an immediate coneuence of Theorem 3.1 (ee alo [4]). Prooition 3.4. If the function g : [1, N + 1] R R i uch that g(z, 0) 0 for all z [1, N], then each non-zero critical oint of I + λ i a oitive olution of (1.1), rovided that (H) hold. Proof. We note that each oitive olution u X d of (3.) i a oitive olution of (1.1), ince g + (z, u(z)) = g(z, u(z)) for all z [1, N]. So, we rove that the non-zero olution of (3.) are oitive. Aume that u X d \ {0} i a olution of (3.). If for ome z [1, N] we have u(z) 0, then u(z 1) u(z 1) + α(z)φ (u(z)) + β(z)φ (u(z)) = λg(z, u + (z)) = λg(z, 0) 0. Thi enure that either (b) or (c) hold for each z [1, N] uch that u(z) 0. So, by an alication of Theorem 3.1, we conclude that u C +. It follow that the non-zero olution of (3.) are oitive and hence are oitive olution of (1.1). Invoking Theorem.4, we have the following reult concerning roblem (1.1). We etablih it with reect to the functional I + λ. Theorem 3.5. Let g : [1, N + 1] R R be a continuou function uch that g(z, 0) 0 for all z [1, N] and g(n + 1, t) = 0 for all t R. Aume that (H) hold, and there exit c, d ]0, + [ with c > d uch that the following ineuality i
8 A. NASTASI, C. VETRO, F. VETRO EJDE-017/5 atified: c < max G(z, ξ) 0 ξ c (N + 1)1 min { (3.3) G(z, d) d 1 ( + α) + d 1 ( + β), m }, ( + )N + α + β where m > 0 i a in Lemma.3. Then roblem (1.1) ha at leat two oitive olution, for each λ Λ with Λ being the oen interval ] max { d 1 ( + α) + d 1 ( + β) G(z, d), ( + )N + α + β }, m 1 (N + 1) 1 c [ max. 0 ξ c G(z, ξ) Proof. We how that there are R and û X d, with 0 < A 1 (û) + A (û) <, uch that B + (û) A 1 (û) + A (û) > u u (A 1+A ) 1 (],]) B + (u). Let c := (N + 1) 1. For all u (A 1 + A ) 1 (], ]), we have 1 u,α + 1 u,β, 1 u,α, u,α () 1, 1 1 (N + 1) (N + 1) u u,α () 1/ < c (by (.1)). Since G + (z, t) G + (z, 0) = G(z, 0) for all t < 0 and z [1, N], we have B + (u) = G + (z, u(z)) max 0 ξ c for all u X d with u (A 1 + A ) 1 (], ]), and hence G(z, ξ), u u (A1+A ) 1 (],]) B + (u) (N + 1) 1 max 0 ξ c G(z, ξ) c. (3.4) Next, let û X d be given a û(z) = d for all z [1, N]. We have imlie A 1 (û) + A (û) = ( + α)d + ( + β)d B + (û) A 1 (û) + A (û) = G(z, d) d 1 ( + α) + d 1 ( + β) = d 1 ( + α) + d 1 ( + β) > (N + 1) 1 max 0 ξ c G(z, ξ) c,
EJDE-017/5 POSITIVE SOLUTIONS OF DISCRETE BOUNDARY VALUE PROBLEMS 9 which imlie B + (û) A 1 (û) + A (û) > u u (A 1+A ) 1 (],]) B + (u) by (3.4). We oberve that 0 < d < c imlie So, by (3.3), we obtain Alo, we have G(z, d) max 0 ξ c 0 < d 1 ( + α) + d 1 ( + β) < G(z, ξ). 0 < A 1 (û) + A (û) = d 1 ( + α) + d 1 ( + β) < c (N + 1) 1. c =. (N + 1) 1 By an alication of Theorem.4, ince the functional I + λ atifie Lemma.3, we conclude that the roblem (3.) ha at leat two non-zero olution, for each λ Λ. Finally, Prooition 3.4 imlie that the two olution are oitive and hence they are oitive olution of the roblem (1.1). Now, we aume that g : [1, N + 1] R R i a continuou function uch that g(z, 0) 0 for all z [1, N], g(n + 1, t) = 0 for all t R, and G(z, ξ) G(z, ξ) lim u ξ 0 ξ + = +, lim ξ + ξ = + for all z [1, N]. (3.5) Note that the econd limit in (3.5) enure that m = +. On the other hand, the firt limit in (3.5) enure that So, we ut λ = max G(z, ξ) > 0 for all z [1, N], all c > 0. 0 ξ c 1 u (N + 1) 1 c>0 c max 0 ξ c G(z, ξ) > 0. It follow that for all λ < λ there exit c > 0 uch that λ < 1 (N + 1) 1 c max 0 ξ c G(z, ξ) > 0. By the firt limit in (3.5), we obtain that there i d ]0, c[ uch that G(z, d) d 1 ( + α) + d 1 ( + β) > 1 λ. Coneuently c max G(z, ξ) < 1 0 ξ c (N + 1) 1 G(z, d) d 1 ( + α) + d 1 ( + β). By uing Theorem 3.5, we obtain the following corollary.
10 A. NASTASI, C. VETRO, F. VETRO EJDE-017/5 Corollary 3.6. Let g : [1, N + 1] R R be a continuou function uch that g(z, 0) 0 for all z [1, N] and g(n + 1, t) = 0 for all t R. Alo, aume that (H), (3.5) hold. Then roblem (1.1) ha at leat two oitive olution, for each λ ]0, λ[. Along the line of Theorem 3.5, in the cae c, d ]0, 1], we have the following reult. Corollary 3.7. Let g : [1, N +1] R R be a continuou function with g(n +1, t) = 0 for all t R. Aume alo that (H1) (H) hold, and there exit c, d ]0, 1] with c > d uch that the following ineuality i atified: { c max G(z, ξ) < 0 ξ c (N + 1) 1 G(z, d) min (4 + α + β)d, m } ( +, )N + α + β (3.6) where m > 0 i a in Lemma.3. Then roblem (1.1) ha at leat two oitive olution, for each λ Λ with Λ being the oen interval ] max { 4 + α + β d ( + )N + α + β } 1 (N + 1) 1 c, G(z, d), m max 0 ξ c G(z, ξ) Proof. Arguing a in the roof of Theorem 3.5, we can how that there are R and û X d, with 0 < A 1 (û) + A (û) <, uch that Let B(û) A 1 (û) + A (û) > u u (A 1+A ) 1 (],]) B(u). = c (N + 1) 1. For all u (A 1 + A ) 1 (], ]), we have 1 u,α + 1 u,β, 1 u,β, u,β () 1, 1 (N + 1) 1 (N + 1) u u,β Since G(z, t) G(z, 0) for all t < 0, we obtain B(u) = G(z, u(z)) max 0 ξ c () 1 c (by (.1)). G(z, ξ), for all u X d with u (A 1 + A ) 1 (], ]). Then, we have u u (A1+A ) 1 (],]) B(u) (N + 1) 1 max 0 ξ c G(z, ξ) c. (3.7) Moreover, let û X d be given a û(z) = d for all z [1, N]. So, we obtain that A 1 (û) + A (û) = ( + α)d + ( + β)d < (4 + α + β)d [.
EJDE-017/5 POSITIVE SOLUTIONS OF DISCRETE BOUNDARY VALUE PROBLEMS 11 imlie B(û) A 1 (û) + A (û) G(z, d) (4 + α + β)d > which imlie (N + 1) 1 B(û) A 1 (û) + A (û) > u u (A 1+A ) 1 (],]) B(u) max 0 ξ c G(z, ξ) c, (by (3.7)). Now, 0 < d < c imlie G(z, d) max 0 ξ c G(z, ξ), and by (3.6) we obtain c 0 < d <, [(4 + α + β)(n + 1) 1 ] 1 and hence we deduce that 0 < A 1 (û) + A (û) <. By an alication of Theorem.4 we conclude that roblem (1.1) ha at leat two non-zero olution, for each λ Λ. Now, the aumtion that (H1) hold for the function g, enure that the olution of roblem (1.1) are alo olution of roblem (3.). By Prooition 3.4, roblem (1.1) ha at leat two oitive olution, for each λ Λ. The next reult i a articular cae of Theorem 3.5. That i, we deal with the roblem u(z 1) u(z 1) + α(z)φ (u(z)) + β(z)φ (u(z)) = λω(z)f(u(z)), for all z [1, N], u(0) = u(n + 1) = 0, (3.8) where f : R [0, + [, and ω : [1, N + 1] [0, + [ with ω(z) > 0 for all z [1, N], and w(n + 1) = 0. Let W = N ω(z), F (t) = t f(ξ)dξ for all t R 0 and m F (t) := min z [1,N] ω(z) lim inf t + t > 0. Then, we have the following reult. Corollary 3.8. Let f : R [0, + [ be a continuou function. Aume that (H) hold, and that there exit c, d ]0, + [ with c > d uch that the following ineuality i atified: c F (c)w < (N + 1)1 { min F (d)w d 1 ( + α) + d 1 ( + β), m } ( +. )N + α + β Then roblem (3.8) ha at leat two oitive olution, for each λ Λ with Λ being the oen interval ] max { d 1 ( + α) + d 1 ( + β) F (d)w, ( + )N + α + β m Proof. Conider the function g : [1, N + 1] R R given a o that g(z, ξ) = ω(z)f(ξ), for all z [1, N + 1], all ξ R, }, 1 (N + 1) 1 c max G(z, ξ) = F (c)w and G(z, d) = F (d)w 0 ξ c F (c)w Then, all the aumtion of Theorem 3.5 hold and o we conclude that roblem (3.8) ha at leat two oitive olution, for each λ Λ. [.
1 A. NASTASI, C. VETRO, F. VETRO EJDE-017/5 Reference [1] R. P. Agarwal; Difference Euation and Ineualitie: Method and Alication, Second Edition, Revied and Exanded, M. Dekker Inc., New York, Bael, 000. [] G. Bonanno, G. D Aguì; Two non-zero olution for ellitic Dirichlet roblem, Z. Anal. Anwend., 35 (016), 449 464. [3] A. Cabada, A. Iannizzotto, S. Terian; Multile olution for dicrete boundary value roblem, J. Math. Anal. Al., 356 (009), 418 48. [4] G. D Aguì, J. Mawhin, A. Sciammetta; Poitive olution for a dicrete two oint nonlinear boundary value roblem with -Lalacian, J. Math. Anal. Al., 447 (017), 383 397. [5] L. Diening, P. Harjulehto, P. Hätö, M. Rŭzĭcka; Lebegue and Sobolev Sace with Variable Exonent, Lecture Note in Math., vol. 017, Sringer-Verlag, Heidelberg, 011. [6] L. Jiang, Z. Zhou; Three olution to Dirichlet boundary value roblem for -Lalacian difference euation, Adv. Difference Eu., 008: 345916 (007). [7] W. G. Kelly, A. C. Peteron; Difference Euation: An Introduction with Alication, Academic Pre, San Diego, New York, Bael, 1991. [8] S. A. Marano, S. J. N. Moconi, N. S. Paageorgiou; Multile olution to (, )-Lalacian roblem with reonant concave nonlinearity, Adv. Nonlinear Stud., 16 (016), 51 65. [9] D. Motreanu, V. V. Motreanu, N. S. Paageorgiou; Toological and variational method with alication to nonlinear boundary value roblem, Sringer, New York, 014. [10] D. Motreanu, C. Vetro, F. Vetro; A arametric Dirichlet roblem for ytem of uailinear ellitic euation with gradient deendence, Numer. Func. Anal. Ot., 37 (016), 1551 1561. [11] D. Mugnai, N.S. Paageorgiou; Wang multilicity reult for uerlinear (, )-euation without the Ambroetti-Rabinowitz condition, Tran. Amer. Math. Soc., 366 (014), 4919 4937. Antonella Natai Deartment of Mathematic and Comuter Science, Univerity of Palermo, Via Archirafi 34, 9013, Palermo, Italy E-mail addre: ella.natai.93@gmail.com Calogero Vetro (correonding author) Deartment of Mathematic and Comuter Science, Univerity of Palermo, Via Archirafi 34, 9013 Palermo, Italy E-mail addre: calogero.vetro@unia.it Franceca Vetro Deartment of Energy, Information Engineering and Mathematical Model (DEIM), Univerity of Palermo, Viale delle Scienze, 9018 Palermo, Italy E-mail addre: franceca.vetro@unia.it