In this paper, the solutions to second order Dirichlet boundary value problems of $p$-Laplacian difference equations are investigated. By using critical point theory, existence and multiplicity results are obtained. The proof is based on the Mountain Pass Lemma in combination with variational techniques

Liu, Zhongzhi

Shi, Haiping

Wang, Zhigang

OpenstarTs

Rend. Istit. Mat. Univ. TriesteVolume 42 (2010), 19–29.Dirichlet Boundary Value Problemsfor Second Order p-LaplacianDifference Equations1Haiping Shi, Zhongzhi Liu and Zhigang WangAbstract. In this paper, the solutions to second order Dirichletboundary value problems of p-Laplacian difference equations are in-vestigated. By using critical point theory, existence and multiplicityresults are obtained. The proof is based on the Mountain Pass Lemmain combination with variational techniques.Keywords: Discrete Dirichlet Boundary Value Problem, Mountain Pass Lemma, p-Laplacian, Critical point theory.MS Classification 2010: 65L10, 35J051. IntroductionLet N, Z and R denote the sets of all natural numbers, integers and realnumbers respectively. For a, b ∈ Z, define Z(a) = {a, a + 1, · · · }, Z(a, b) ={a, a + 1, · · · , b} when a ≤ b. Let k be a positive integer and * denote thetranspose of a vector.In this paper, we consider the second order difference equation∆(ϕp(∆un−1)) + f(n, un+1, un, un−1) = 0, n ∈ Z(1, k) (1)with Dirichlet boundary value conditionsu0 = uk+1 = 0, (2)where ∆ is the forward difference operator ∆un = un+1−un, ∆2un = ∆(∆un),ϕp(s) is the p-Laplacian operator, that is, ϕp(s) = |s|p−2s(1 < p < ∞), f ∈C(Z×R3,R).We may think of Eq. (1) as being a discrete analogue of the second orderfunctional differential equation[ϕp(u′)]′+ f(t, u(t+ 1), u(t), u(t− 1)) = 0, t ∈ R (3)1This work is supported by the Scientific Research Fund of Hunan Provincial EducationDepartment, ♯08C118.20 H. SHI ET AL.which includes the following equationc2u′′(t) = V ′(u(t+ 1)− u(t))− V ′(u(t)− u(t− 1)), t ∈ R (4)where f ∈ C(R4,R). Eq. (3) has been studied extensively by many scholars.For example, Smets and Willem have obtained the existence of solitary wavesof lattice differential equations, see [30] and the references cited therein.The theory of nonlinear difference equations has been widely used to studydiscrete models appearing in many fields such as computer science, economics,neural network, ecology, cybernetics, etc. Since the last decade, there has beenmuch literature on qualitative properties of difference equations, those studiesover many of the branches of difference equations, such as [1, 3, 10, 12, 13,18–21, 23–25, 27, 28].In recent years, the study of boundary value problems for differential equa-tions has received much attention. By using various methods and techniques,such as Schauder fixed point theorem, the cone theoretic fixed point theorem,the method of upper and lower solutions, and coincidence degree theory, aseries of results showing the existence of nontrivial solutions for differentialequations have been obtained in the literatures, we refer to [2, 4–8, 17, 32,36]. It is well known that critical point theory is an powerful tool to deal withdifferential equations [22, 26, 36]. Starting in 2003, critical point theory hasbeen employed to establish sufficient conditions on the existence of periodicsolutions of difference equations. In particular, Yu, Guo, Chen, Shi, Tian andtheir collaborators considered second order nonlinear difference equations [9,14-16,29,31,33-35]. However, to our best knowledge, the results on the boundaryvalue problem (BVP) of p-Laplacian difference equations are very scarce inthe literature.Our aim in this paper is to use the critical point theory to give some suf-ficient conditions for the existence and multiplicity of the BVP (1) with (2).The main idea in this paper is to transfer the existence of the BVP (1) with(2) into the existence of the critical points of some functional.Our main results are as follows.Theorem 1.1. Assume that the following hypotheses are satisfied:(F1) there exists a functional F (n, v1, v2) ∈ C1(Z × R2,R) with F (0, ·) = 0such that for any n ∈ Z(1, k),∂F (n− 1, v2, v3)∂v2+∂F (n, v1, v2)∂v2= f(n, v1, v2, v3);(F2) there exist constants δ > 0, α ∈(0, 12p/2p(c1c2)pλp2min)such thatF (n, v1, v2) ≤ α(√v21 + v22)p, for n ∈ Z(1, k) and v21 + v22 ≤ δ2 ;DIRICHLET BOUNDARY VALUE PROBLEMS 21(F3) there exist constants ρ > 0, γ > 0, β ∈(1p(c2c1)pλp2max,+∞)such thatF (n, v1, v2) ≥ β(√v21+v22)p−γ, for n ∈ Z(1, k) and v21 + v22 ≥ ρ2 ,where c1, c2 are constants which can be referred to (8), and λmin, λmax areconstants which can be referred to (10).Then the BVP (1) with (2) possesses at least two nontrivial solutions.Remark 1.2. By (F3) it is easy to see that there exists a constant γ′ > 0such that(F ′3) F (n, v1, v2) ≥ β(√v21 + v22)p− γ′, ∀(n, v1, v2) ∈ Z(1, k)×R2.As a matter of fact, letγ1 = max{∣∣∣∣F (n, v1, v2)− β(√v21 + v22)p+ γ∣∣∣∣: n ∈ Z(1, k)v21 + v22 ≤ ρ2},γ′ = γ + γ1 ,we can easily get the desired result.The rest of the paper is organized as follows. In Sect. 2 we shall establishthe variational framework for the BVP (1) with (2) in order to apply the criticalpoint method and give some useful lemmas. In Sect. 3 we shall complete theproof of the main results and give an example to illustrate the result.2. Variational Structure and some LemmasIn order to apply the critical point theory, we shall establish the correspondingvariational framework for the BVP (1) with (2) and give some basic notationand useful lemmas.Let Rk be the real Euclidean space with dimension k. Define the innerproduct on Rk as follows:〈u, v〉 =k∑j=1ujvj , ∀u, v ∈ Rk, (5)by which the norm ‖ · ‖ can be induced by‖u‖ =k∑j=1u2j12, ∀u ∈ Rk. (6)22 H. SHI ET AL.On the other hand, we define the norm ‖ · ‖r on Rk as follows:‖u‖r =k∑j=1|uj |r1r, (7)for all u ∈ Rk and r > 1.Since ‖u‖r and ‖u‖2 are equivalent, there exist constants c1, c2 such thatc2 ≥ c1 > 0, andc1‖u‖2 ≤ ‖u‖r ≤ c2‖u‖2, ∀u ∈ Rk. (8)Clearly, ‖u‖ = ‖u‖2. For the BVP (1) with (2), consider the functional Jon Rk as follows:J(u) =k∑n=0[1p|∆un|p− F (n, un+1, un)], (9)∀u = (u1, u2, · · · , uk)∗ ∈ Rk, u0 = uk+1 = 0 and F is as in assumption (F1).Clearly, J ∈ C1(Rk,R) and for any u = {un}n∈Z(1,k) ∈ Rk, by usingu0 = uk+1 = 0, we can compute the partial derivative as∂J∂un= − [∆(ϕp(∆un−1)) + f(n, un+1, un, un−1)] , n ∈ Z(1, k).Thus, u is a critical point of J on Rk if and only if∆(ϕp(∆un−1)) + f(n, un+1, un, un−1) = 0, ∀n ∈ Z(1, k).We reduce the existence of the BVP (1) with (2) to the existence of criticalpoints of J on Rk. That is, the functional J is just the variational frameworkof the BVP (1) with (2).Let P is a k × k matrix defined byP =2 −1 0 · · · 0 0−1 2 −1 · · · 0 00 −1 2 · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · 2 −10 0 0 · · · −1 2.Clearly, P is positive definite. Let λ1, λ2, · · · , λk be the eigenvalues of P .Applying matrix theory, we know λj > 0, j = 1, 2, · · · , k.DIRICHLET BOUNDARY VALUE PROBLEMS 23Denote0 < λmin = min {λ1, λ2, · · · , λk} , λmax = max {λ1, λ2, · · · , λk} , (10)Let E be a real Banach space, J ∈ C1(E,R), i.e., J is a continuouslyFréchet-differentiable functional defined on E. J is said to satisfy the Palais-Smale condition (P.S. condition[11] for short) if any sequence{u(k)}⊂ Efor which{J(u(k))}is bounded and J ′(u(k))→ 0 as k → ∞ possesses aconvergent subsequence in E.Let Bρ denote the open ball in E about 0 of radius ρ and let ∂Bρ denoteits boundary.Lemma 2.1. (Mountain Pass Lemma [26]). Let E be a real Banach space andsuppose J ∈ C1(E,R) satisfies the P.S. condition. If J(0) = 0 and(J1) there exist constants ρ, a > 0 such that J |∂Bρ ≥ a, and(J2) there exists e ∈ E \Bρ such that J(e) ≤ 0.Then J possesses a critical value c ≥ a given byc = infg∈Γmaxs∈[0,1]J(g(s)), (11)whereΓ = {g ∈ C([0, 1], E)|g(0) = 0, g(1) = e}. (12)Lemma 2.2. Suppose that (F1) − (F3) are satisfied. Then the functional Jsatisfies the P.S. condition.Proof. Let u(l) ∈ Rk, l ∈ Z(1) be such that{J(u(l))}is bounded. Then thereexists a positive constant M2 such that−M2 ≤ J(u(l))≤ M2, ∀l ∈ N.By (F ′3), we have−M2 ≤ J(u(l))=1pk∑n=0∣∣∣∆u(l)n∣∣∣p−k∑n=1F(n, u(l)n+1, u(l)n)=1p[( k∑n=0∣∣∣∆u(l)n∣∣∣p)1p]p−k∑n=1F(n, u(l)n+1, u(l)n)≤1p[c2( k∑n=0∣∣∣∆u(l)n∣∣∣2)12]p−k∑n=1[β(√(u(l)n+1)2+(u(l)n)2)p− γ′]=1pcp2((u(l))∗Pu(l))p2− β[( k∑n=1(√(u(l)n+1)2+(u(l)n)2)p) 1p]p+ kγ′24 H. SHI ET AL.≤1pcp2λp2max∥∥∥u(l)∥∥∥p2− βcp1[ k∑n=1((u(l)n+1)2+(u(l)n)2)]p2+ kγ′≤1pcp2λp2max∥∥∥u(l)∥∥∥p2− βcp1∥∥∥u(l)∥∥∥p2+ kγ′=(1pcp2λp2max − βcp1)∥∥∥u(l)∥∥∥p2+ kγ′.That is,(βcp1 −1pcp2λp2max)∥∥∥u(l)∥∥∥p2≤ M2 + kγ′.Since β > 1p(c2c1)pλp2max, there exists a constant M3 > 0 such that∥∥∥u(l)∥∥∥2≤ M3, ∀l ∈ N.Therefore,{u(l)}is bounded on Rk. As a consequence,{u(l)}possesses aconvergence subsequence in Rk. And thus the P.S. condition is verified.3. Proof of the Main ResultsIn this section, we shall complete the proof of Theorem 1.1.3.1. Proof of Theorem 1.1Proof. For any n ∈ Z(1, k), by (F2) and (8),J(u) =k∑n=0[1p|∆un|p− F (n, un+1, un)]=1p[( k∑n=0|∆un|p)1p]p−k∑n=1F (n, un+1, un)≥1pcp1[( k∑n=0|∆un|2)12]p−k∑n=1F (n, un+1, un)=1pcp1 (u∗Pu)p2 −k∑n=1F (n, un+1, un)≥1pcp1λp2min‖u‖p2 − αk∑n=1(√u2n+1 + u2n)pDIRICHLET BOUNDARY VALUE PROBLEMS 25=1pcp1λp2min‖u‖p2 − α[( k∑n=1(√u2n+1 + u2n)p ) 1p]p≥1pcp1λp2min‖u‖p2 − αcp2[ k∑n=1(u2n+1 + u2n)]p2≥1pcp1λp2min‖u‖p2 − αcp2(2‖u‖22)p2≥(1pcp1λp2min− 2p2 cp2α)‖u‖p2.Take a ,(1pcp1λp2min− 2p2 cp2α)δp. Therefore,J(u) ≥ a > 0, ∀u ∈ ∂Bρ.At the same time, we have also proved that there exist constants a > 0 andρ > 0 such that J |∂Bρ ≥ a. That is to say, J satisfies the condition (J1) of theMountain Pass Lemma.For our setting, clearly J(0) = 0. In order to exploit the Mountain PassLemma in critical point theory, we need to verify other conditions of the Moun-tain Pass Lemma. By Lemma 2.2, J satisfies the P.S. condition. So it sufficesto verify the condition (J2).From the proof of the P.S. condition, we knowJ(u) ≤(1pcp2λp2max − βcp1)‖u‖p2+ kγ′.Since β > 1p(c2c1)pλp2max, we can choose ū large enough to ensure that J(ū) < 0.By the Mountain Pass Lemma, J possesses a critical value c ≥ a > 0, wherec = infh∈Γsups∈[0,1]J(h(s)),andΓ = {h ∈ C([0, 1],Rk) | h(0) = 0, h(1) = ū}.Let ũ ∈ Rk be a critical point associated to the critical value c of J , i.e.,J(ũ) = c. Similar to the proof of the P.S. condition, there exists u ∈ Rksuch that J(u) ≥ a. Hence, there exists û ∈ Rk such that J(û) = cmax =maxu∈RkJ(u) ≥ a > 0.Clearly, û 6= 0. If ũ 6= û, then the conclusion of Theorem 1.1 holds. Other-wise, ũ = û. Then c = J(ũ) = J(û) = cmax. That is,supu∈RkJ(u) = infh∈Γsups∈[0,1]J(h(s)).26 H. SHI ET AL.Therefore,cmax = maxs∈[0,1]J(h(s)), ∀h ∈ Γ.By the continuity of J(h(s)) with respect to s, J(0) = 0 and J(ū) < 0 implythat there exists s0 ∈ (0, 1) such thatJ (h (s0)) = cmax.Choose h1, h2 ∈ Γ such that {h1(s) | s ∈ (0, 1)} ∩ {h2(s) | s ∈ (0, 1)} isempty, then there exists s1, s2 ∈ (0, 1) such thatJ (h1 (s1)) = J (h2 (s2)) = cmax.Thus, we get two different critical points of J on Rk denoted byu1 = h1 (s1) , u2 = h2 (s2) .The above argument implies that the BVP (1) with (2) possesses at least twonontrivial solutions ũ, û or ũ, u1, u2. The proof of Theorem 1.1 is finished.Remark 3.1. As an application of Theorem 1.1, finally, we give an example toillustrate our result.For n ∈ Z(1, k), assume that∆(ϕp(∆un−1))+µun[(en−1)(u2n+1+u2n)µ2−1+(en−1−1) (u2n+u2n−1)µ2−1]= 0,(13)with boundary value conditionsu0 = uk+1 = 0, (14)where 1 < p < +∞, µ > p.We havef(n, v1, v2, v3) = µv2[(en − 1)(v21 + v22)µ2−1+(en−1 − 1) (v22 + v23)µ2−1]andF (n, v1, v2) = (en − 1)(v21 + v22)µ2 .Then∂F (n− 1, v2, v3)∂v2+∂F (n, v1, v2)∂v2= µv2[(en − 1)(v21 + v22)µ2−1+(en−1 − 1) (v22 + v23)µ2−1].It is easy to verify all the assumptions of Theorem 1.1 are satisfied and thenthe BVP (13) with (14) possesses at least two nontrivial solutions.DIRICHLET BOUNDARY VALUE PROBLEMS 27References[1] R.P. Agarwal, Difference equations and inequalities: theory, methods and ap-plications, Marcel Dekker, New York (1992).[2] R.P. Agarwal, K. Perera and D. O’Regan, Existence criteria for singu-lar boundary value problems with sign changing nonlinearities, J. Diff. Eq. 183(2002), 409–433.[3] R.P. Agarwal, K. Perera and D. 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Dirichlet Boundary Value Problems for Second Order $p$-Laplacian Difference Equations

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Dirichlet Boundary Value Problems for Second Order $p$ -Laplacian Difference Equations

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Dirichlet Boundary Value Problems for Second Order ppp-Laplacian Difference Equations

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Dirichlet Boundary Value Problems for Second Order $p$ -Laplacian Difference Equations