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Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 195376, 17 pages doi:10.1155/2010/195376 Research Article On Homoclinic Solutions of a Semilinear p-Laplacian Difference Equation with Periodic Coefficients Alberto Cabada, 1 Chengyue Li, 2 and Stepan Tersian 3 1 Departamento de An ´ alise Matem ´ atica, Fac u ltade de Matem ´ aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain 2 Department of Mathematics, Minzu U niversity of China, Beijing 100081, China 3 Department of Mathematical Analysis, University of Rousse, 7017 Rousse, Bulgaria Correspondence should be addressed to Alberto Cabada, alberto.cabada@usc.es Received 5 July 2010; Accepted 27 October 2010 Academic Editor: Jianshe Yu Copyright q 2010 Alberto Cabada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original w ork is properly cited. We study the existence of homoclinic solutions for semilinear p-Laplacian difference equations with periodic coefficients. The proof of the main result is based on Brezis-Nirenberg’s Mountain Pass Theorem. Several examples and remarks are given. 1. Introduction This paper is concerned with the study of the existence of homoclinic solutions for the p- Laplacian difference equation Δ 2 p u  k − 1  − V  k  u  k | u  k | q−2  λf  k, u  k   0,u  t  → 0, | t | →∞, 1.1 where uk,k ∈ is a sequence or real numbers, Δ is the difference operator Δukuk  1 −uk, Δ 2 p u  k − 1  Δu  k | Δu  k | p−2 − Δu  k − 1 | Δu  k − 1 | p−2 1.2 2AdvancesinDifference Equations is referred to as the p-Laplacian difference operator, and functions V k and fk, x are T- periodic in k and satisfy suitable conditions. In the theory of differential equations, a trajectory xt, which is asymptotic to a constant as |t|→∞is called doubly asymptotic or homoclinic orbit. The notion of homoclinic orbit is introduced by Poincar ´ e 1 for continuous Hamiltonian systems. Recently, there is a large literature on the use of variational methods to the existence of homoclinic or heteroclinic orbits of Hamiltonian systems; see 2–7 and the references therein. In the recent paper of Li 8 a unified approach to the existence of homoclinic orbits for some classes of ODE’s with periodic potentials is presented. It is based on the Brezis and Nirenberg’s mountain-pass theorem 9. In this paper we extend this approach to homoclinic orbits for discrete p-Laplacian type equations. Discrete boundary value problems have been intensively studied in the last decade. The studies of such kind of problems can be placed at the interface of certain mathematical fields, such as nonlinear differential equations and numerical analysis. On the other hand, they are strongly motivated by their applicability to mathematical physics and biology. The variational approach to the study of various problems for difference equations has been recently applied in, among others, the papers of Agarwal et al. 10, Cabada et al. 11, Chen and Fang 12, Fang and Zhao 13, Jiang and Zhou 14,MaandGuo15 ,Mih˘ailescu et al. 16,Krist ´ aly et al. 17. Along the paper, given two integer numbers a<b, we will denote a, b{a, ,b}. Moreover, for every p>1, we consider the following function ϕ p  t   t | t | p−2 , Φ p  t   | t | p p . 1.3 It is obvious that Φ  p tϕ p t for all t ∈ and p /  0. Moreover Δ 2 p u  k − 1  Δ  ϕ p  Δu  k − 1   . 1.4 Suppose that V : → is a T-periodic positive potential 1.5 0 <V 0  min { V  0  , ,V  T − 1 } ≤ max { V  0  , ,V  T − 1 }  V 1 . 1.6 Denote A  u    k∈ Φ p  Δu  k − 1    k∈ V  k  Φ q  u  k  . 1.7 Let us consider functions f satisfying the following assumptions. F 1  The function fk, t is continuous in t ∈ and T-periodic in k. Advances in Difference Equations 3 F 2  The potential function Fk, t of fk, t F  k, t    t 0 f  k, s  ds 1.8 satisfies the Rabinowitz’s type condition: There exist μ>p≥ q>1ands>0suchthat μF  k, t  ≤ tf  k, t  ,k∈ ,t /  0, F  k, t  > 0, ∀k ∈ , for t ≥ s>0. 1.9 F 3  fk, to|t| q−1  as |t|→0. Further we consider the semilinear eigenvalue p-Laplacian difference equation Δ 2 p u  k − 1  − V  k  u  k | u  k | q−2  λf  k, u  k   0, 1.10 where λ>0 and we are looking for its homoclinic solutions, that is, solutions of 1.10 such that uk → 0as|k|→∞. In order to obtain homoclinic solutions of 1.10, we will use variational approach and Brezis-Nirenberg mountain pass theorem 9 . To this end, consider the functional J :  q → ,definedas J  u   A  u  − λ  k∈ F  k, u  k  . 1.11 Our main result is the following. Theorem 1 .1. Suppose that the function V : → is positive and T-periodic and the functions fk, · : × → satisfy assumptions F 1 –F 3 .Then,foreachλ>0, 1.10 has a nonzero homoclinic solution u ∈  q , which is a critical point of the functional J :  q → . Moreover, given a nontrivial solution u of problem 1.10,thereexistk ± two integer numbers such that for all k>k  and k<k − , the sequence uk is strictly monotone. The paper is organized as follows. In Section 2,wepresenttheproofofthemainresult and discuss the optimality o f the condition F 2 .InSection 3 , we give some examples of equations modeled by this kind of problems and present some additional remarks. 4AdvancesinDifference Equations 2. Proof of the Main Result Let u  {uk : k ∈ }be a sequence, q>1and  q   u : | u | q q   k∈ | u  k | q < ∞  ,  ∞   u : | u | ∞  sup k∈ | u  k | < ∞  . 2.1 It is well known that if 0 <q≤ p,then q ⊆  p .Indeed,if  k∈ |uk| q < ∞,thereexists a positive integer number R,suchthatforallk satisfying |k| >Rit is verified that |uk| q < 1 and, as consequence, |uk| p ≤|uk| q and the series  k∈ |uk| p is convergent too. Consider now the functional J :  q → ,definedas J  u   A  u  − λ  k∈ F  k, u  k  , 2.2 with A given in 1.7 and F defined in 1.8. We have the following result. Lemma 2.1. The functional J :  q → is well defined, C 1 -differentiable, and its critical points are solutions of 1.10. Proof. By using the inequality for nonnegative a and b and p>1  a  b 2  p ≤ a p  b p 2 , 2.3 and the inclusion  q ⊆  p for 1 <q≤ p, it follows that  k∈ | Δu  k − 1 | p ≤ 2 p−1  k∈  | u  k | p  | u  k − 1 | p   2 p  k∈ | u  k | p < ∞. 2.4 Now, let us see that the series  k∈ Fk,uk is convergent: by using F 3 , it follows that there exist δ ∈ 0, 1 and sufficiently large N such that F  k, u  k  < | u  k | q for | u  k | q <δ<1, | k | >N. 2.5 Then, the series  k∈ Fk,uk is convergent and the functional J is well defined on  q . Advances in Difference Equations 5 It is G ˆ ateaux differentiable and for v ∈  q :  J   u  ,v   lim t →0 J  u  tv  − J  u  t   k∈ Δu  k − 1 | Δu  k − 1 | p−2 Δv  k − 1    k∈ V  k  u  k | u  k | q−2 v  k  − λ  k∈ f  k, u  k  v  k  2.6 and partial derivatives ∂J  u  ∂u  k   −Δ 2 p u  k − 1   V  k  u  k | u  k | q−2 − λf  k, u  k  , 2.7 are continuous functions. Moreover the functional J is continuously Fr ´ echet-differentiable in  q .Itisclear,by 2.7, that the critical points of J are solutions of 1.10. To obtain homoclinic solutions of 1.10 we will use mountain-pass theorem of Brezis and Nirenberg 9. Recall its statement. Let X be a Banach space with norm ·,andI : X → be a C 1 -functional. I satisfies the PS c condition if every sequence x k  of X such that I  x k  −→ c, I   x k  −→ 0, 2.8 has a convergent subsequence. A sequence x k  ⊂ X such that 2.8 holds is referred to as PS c -sequence. Theorem 2 .2 mountain-pass theorem, Brezis and Nirenberg 9. Let X be a Banach space with norm ·, I ∈ C 1 X,  and suppose that there exist r>0, α>0 and e ∈ X such that e >r i Ix ≥ α if x  r, ii Ie < 0. Let c  inf γ∈Γ {max 0≤t≤1 Iγt}≥α,where Γ  γ ∈ C  0, 1  ,X  : γ  0   0,γ  1   e  . 2.9 Then, there exists a PS c sequence for I.Moreover,ifI satisfies the PS c condition, then c is a critical value of I, that is, there e xists u 0 ∈ X such that Iu 0 c and I  u 0 0. Note that, by assumption 1.5, t he norm |·| q in  q is equivalent to  u  q q  1 q  k∈ V  k | u  k | q . 2.10 6AdvancesinDifference Equations Lemma 2.3. Suppose that F 1 –F 3  hold, then there exist ρ>0, α>0 and e ∈  q such that e q >ρ and 1 Ju ≥ α if u q  ρ, 2 Je < 0. Proof. By F 3 ,thereexistsδ ∈ 0, 1 such that F  k, t  ≤ V 0 2qλ | t | q if | t | ≤ δ. 2.11 Let ρ V 0 /q 1/q δ V 0 defined in 1.6, then, for u, u q  ρ, V 0 q δ q  ρ q   u  q q  1 q  k∈ V  k | u  k | q ≥ V 0 q | u  k | q for all k ∈ , 2.12 which implies that |uk|≤δ for all k ∈ . Hence, by 2.11  k∈ F  k, u  k  ≤ V 0 2qλ  k∈ | u  k | q ≤ 1 2qλ  k∈ V  k | u  k | q  1 2λ  u  q q J  u   A  u  − λ  k∈ F  k, u  k  ≥  u  q q − 1 2  u  q q  1 2  u  q q  ρ q 2 > 0. 2.13 By F 2 ,thereexistc 1 , c 2 > 0suchthatFk, t ≥ c 1 t μ − c 2 for all t>0andk ∈ . Take v ∈  q , v0a>0, vk0ifk /  0. Then, since μ>p≥ q J  κv   A  κv  − λ  k∈ F  k, κv  k  ≤ 2 p κ p a p  V  0  κ q a q q − λ  c 1 κ μ a μ − c 2  < 0, 2.14 if κ is sufficiently large. Then, we can take κ large enough, such that for e  κv, e q q  V 0κ q a q /q >ρ q and 2.14 holds. Advances in Difference Equations 7 Lemma 2.4. Suppose that the assumptions of Lemma 2.3 hold. Then, there exists c>0 and a  q - bounded PS c sequence for J. Proof. By Lemma 2.3 and Theorem 2.2 there exists a sequence u m  ⊂  q such that J  u m  −→ c, J   u m  −→ 0, 2.15 where c  inf γ∈Γ  max t∈0,1 J  γ  t    , Γ  γ ∈ C  0, 1  , q  : γ  0   0,γ  1   e  , 2.16 and e is d efined in the proof of Lemma 2.3. We will prove that the sequence u m  is bounded in  q .Wehaveforμ>p≥ q  J   u m  ,u m    k∈ | Δu m  k − 1 | p   k∈ V  k | u m  k | q − λ  k∈ f  k, u m  k  u m  k  , 2.17 and, by F 2 , μJ  u m  −  J   u m  ,u m    μ p − 1   k∈ | Δu m  k − 1 | p   μ q − 1   k∈ V  k | u m  k | q − λ  k∈  μF  k, u m  k  − f  k, u m  k  u m  k   ≥  μ q − 1  q  u m  q q   μ −q   u m  q q , 2.18 which implies that the sequence u m is bounded in  q . Now we are in a position to prove Theorem 1.1. Proof of Theorem 1.1. For any m ∈ ,thesequence{|u m k|,k∈ Z}, given in Lemma 2.4,is bounded in  q and, in consequence, |u m k|→0as|k|→∞.Let|u m k| takes its maximum at k m ∈ . There exists a unique j m ∈ ,suchthatj m T ≤ k m < j m  1T and let w m k u m kj m T.Then|w m k| takes its maximum at i m  k m −j m T ∈ 0,T−1.BytheT-periodicity of V and f·,t, it follows that  u m  q   w m  q , J  u m   J  w m  . 2.19 8AdvancesinDifference Equations Since u m  is bounded in  q ,thereexistsw ∈  q ,suchthatw m wweakly in  q .Theweak convergence in  q implies that w m k → wk for every k ∈ .Indeed,ifwetakeatest function v k ∈  q , v k k1, v k j0ifj /  k,then w m  k    w m ,v k  −→  w, v k   w  k  . 2.20 Moreover, for any v ∈  q    J   w m  ,v        J   u m  ,v  ·  j m T    ≤   J   u m    ∗   v  ·  j m T    q    J   u m    ∗  v  q −→ 0, 2.21 which implies that J  w m  → 0, which means that for every v ∈  q ,  k∈ ϕ p  Δw m  k − 1  Δv  k − 1    k∈ V  k  ϕ q  w m  k  v  k  − λ  k∈ f  k, w m  k  v  k  −→ 0, ∀k ∈ , as m −→ ∞. 2.22 Let us take v ∈  q with compact support, that is, there exist a, b ∈ , a<bsuch that vk0ifk ∈ \a, b and vk /  0ifk ∈{a1,b−1}. The set of such elements  q 0 is d ense in  q because if v ∈  q and v k ∈  q 0 is such that v k j0if|j|≥k  1, v k jvj if |j|≤k,then v − v k  q → 0ask →∞. Taking v ∈  q 0 in 2.22, due to the finite sums and the continuity of functions fk, ·, we obtain, passing to a limit, that  k∈ ϕ p  Δw  k − 1  Δv  k − 1    k∈ V  k  ϕ q  w  k  v  k  − λ  k∈ f  k, w  k  v  k   0, ∀v ∈ l q 0 . 2.23 From the density of l q 0 in  q , we deduce that the previous equality is fulfilled for all v ∈  q and, in consequence, w is a critical point of the functional J,thatis,w is a solution of 1.10. It remains to show that w /  0. Assuming, on the contrary, that w  0, we conclude that | u m | ∞  | w m | ∞  max {| w m  k | : k ∈ } −→ 0, as m −→ ∞. 2.24 By F 3 , for a given ε>0, there exists δ>0, such that if |x| <δthen, for every k ∈ 0,T − 1, the following inequalities holds: | F  k, x | ≤ ε | x | q ,   f  k, x  x   ≤ ε | x | q . 2.25 Advances in Difference Equations 9 By 2.24, for every k ∈ 0,T − 1, there exists a positive integer M k such that for all m>M k it follows that |w m k| <δ. Since the maximum value of |w m | is attained at i m ∈ 0,T − 1, it follows that for m>M max{M k : k ∈ 0,T − 1} and every k ∈ | w m  k | ≤ | w m  i m | ≤ δ. 2.26 Then, by 2.25,form>Mand every k ∈ : | F  k, w m  k | ≤ ε | w m  k | q ,   f  k, w m  k  w m  k    ≤ ε | w m  k | q , 2.27 which implies that 0 ≤ qJ  w m   q p  k∈ | Δw m  k − 1 | p   k∈ V  k | w m  k | q − λ  k∈ qF  k, w m  k  ≤  k∈ | Δw m  k − 1 | p   k∈ V  k | w m  k | q − λ  k∈ f  k, w m  k  w m  k  − λ  k∈  qF  k, w m  k  − f  k, w m  k  w m  k   ≤  J   w m  ,w m   λ  qε | w m | q q  ε | w m | q q  ≤   J   w m    ∗  w m  q  λε q  1 V 0  w m  q q . 2.28 Since w m  is bounded in  q , J  w m  → 0andε is arbitrary, by 2.28 we obtain a contradiction with Jw m Ju m  → c>0. The proof of the first part is complete. Now, let u be a nonzero homoclinic solution of problem 1.10. Assume that it attains positive local maximums and/or negative local minimums at infinitely many points k n .In particular we can assume that {|k n |} → ∞.InconsequenceΔ 2 p uk n −1uk n  ≤ 0anduk n  → 0. From this, multiplying in 1.10 by uk n /|uk n | q ,wehave λ f  k n ,u  k n  u  k n  | u  k n | q ≥ Δ 2 p u  k n − 1  u  k n  | u  k n | q  λ f  k n ,u  k n  u  k n  | u  k n | q  V  k n  ≥ V 0 > 0. 2.29 By means of condition F 3  we arrive at the following contradiction: 0  λ lim n →∞ f  k n ,u  k n  u  k n  | u  k n | q ≥ V 0 > 0. 2.30 Suppose now that function u vanishes at infinitely many points l n . From condition F 3  we conclude that Δ 2 p ul n − 10and,inconsequence,ul n − 1ul n  1 < 0. Therefore 10 Advances in Difference Equations it has an unbounded sequence of positive local maximums and negative local minimums, in contradiction with the previous assertion. As a direct consequence of the two previous properties, we deduce that, for |k| large enough, function u has constant sign and it is strictly monotone. To illustrate the optimality of the obtained results, we present in the sequel an example in which it is pointed out that condition F 2  cannot be removed to deduce the existence result proved in Theorem 1.1. Example 2.5. Let Wk > 0beaT-periodic sequence, W 1  max{Wk : k ∈ 0,T − 1}, p ≥ q>1andr>qbe fixed. Consider problem 1.10 with f  k, t   ⎧ ⎨ ⎩ W  k  ϕ r  t  if | t | ≤ 1, W  k  ϕ q  t  if | t | ≥ 1. 2.31 It is obvious that condition F 1  holds. Since r>qwe have that c o ndition F 3  is trivially fulfilled. Concerning to condition F 2 ,wehavethat F  k, t   ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ W  k  | t | r r if | t | ≤ 1, W  k   | t | q q  q −r qr  if | t | ≥ 1. 2.32 It is clear that Fk, t > 0forallt /  0andthatμFk, t ≤ tfk, t for all t ∈ −1, 1 if and only if 0 <μ≤ r. When |t|≥1, the inequality μFk, t ≤ tfk, t holds if and only if either μ  q or μ>q and | t | q ≤ μ  r −q  r  μ −q  < ∞. 2.33 As consequence, the inequality μFk, t ≤ tfk, t for all t /  0 is satisfied if and only if μ  q, that is, condition F 2  does not hold. Let us see that this problem has only the trivial solution for small values of the parameter λ. Since r>q,itisnotdifficult to verify that, for 0 <λ<q −1V 0 /r −1W 1 , the function λfk, t − V kϕ q t is strictly decreasing for every integer k.So,forλ in that situation, we have that  λf  k, t  − V  k  ϕ q  t   t<0forallt /  0andall k ∈ . 2.34 Suppose that there is a nontrivial solution u of the considered problem, and moreover it takes some positive values. Let k 0 be such that uk 0 max{uk; k ∈ }> 0. In such a case we deduce the following contradiction: 0 Δ 2 p u  k 0 − 1  − V  k 0  ϕ q  u  k 0   λf  k 0 ,u  k 0  < Δ 2 p u  k 0 − 1  ≤ 0. 2.35 [...]...Advances in Difference Equations 11 Analogously it can be verified that the solution u has no negative values on 3 Remarks and Examples In this section we will consider some examples and remarks on applications and extensions of Theorem 1.1 to the existence of homoclinic solutions of difference equations of following types: A Second-order discrete p-Laplacian equations of the form Δ2 u... 6 K Tanaka, Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits,” Journal of Differential Equations, vol 94, no 2, pp 315–339, 1991 7 S Tersian and J Chaparova, Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations,” Journal of Mathematical Analysis and Applications, vol 260, no 2, pp 490–506, 2001 8 C Li, A unified proof on existence... singular and nonsingular discrete problems via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol 58, no 1-2, pp 69–73, 2004 11 A Cabada, A Iannizzotto, and S Tersian, “Multiple solutions for discrete boundary value problems,” Journal of Mathematical Analysis and Applications, vol 356, no 2, pp 418–428, 2009 12 P Chen and H Fang, “Existence of periodic and subharmonic solutions. .. order ´ differential equations,” Journal of Mathematical Analysis and Applications, vol 240, no 1, pp 163–173, 1999 4 W Omana and M Willem, Homoclinic orbits for a class of Hamiltonian systems,” Differential and Integral Equations, vol 5, no 5, pp 1115–1120, 1992 5 P H Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,” Proceedings of the Royal Society of Edinburgh Section A, vol 114, no... existence of homoclinic orbits for some semilinear ordinary differential equations with periodic potentials,” Journal of Mathematical Analysis and Applications, vol 365, no 2, pp 510–516, 2010 9 H Brezis and L Nirenberg, “Remarks on finding critical points,” Communications on Pure and Applied Mathematics, vol 44, no 8-9, pp 939–963, 1991 10 R P Agarwal, K Perera, and D O’Regan, “Multiple positive solutions of. .. that um km Note that if T 2, then k∗ 0 or k∗ 1 Dedication This work is dedicated to Professor Gheorghe Morosanu on the occasion of his 60-th birthday ¸ Acknowledgment S Tersian is thankful to Department of Mathematical Analysis at University of Santiago de Compostela, Spain, where a part of this work was prepared during his visit A Cabada partially supported by Ministerio de Educacion y Ciencia, Spain,... for second-order p-Laplacian difference equations,” Advances in Difference Equations, vol 2007, Article ID 42530, 9 pages, 2007 13 H Fang and D Zhao, “Existence of nontrivial homoclinic orbits for fourth-order difference equations,” Applied Mathematics and Computation, vol 214, no 1, pp 163–170, 2009 Advances in Difference Equations 17 14 L Jiang and Z Zhou, “Three solutions to Dirichlet boundary value... problems for p-Laplacian difference equations,” Advances in Difference Equations, vol 2008, Article ID 345916, 10 pages, 2008 15 M Ma and Z Guo, Homoclinic orbits for second order self-adjoint difference equations,” Journal of Mathematical Analysis and Applications, vol 323, no 1, pp 513–521, 2006 16 M Mih˘ ilescu, V R˘ dulescu, and S Tersian, “Eigenvalue problems for anisotropic discrete boundary a a value... and concentration-compactness arguments As a consequence of Theorem 3.2 we obtain the following corollary Corollary 3.3 Suppose that a > 0, the function V : → Ê is positive and T -periodic and r > q > 1 Then, for each λ > 0, 3.2 has a nonzero homoclinic solution u ∈ q (C) Second-Order Difference Equations with Cubic and Quintic Nonlinearities Our next example is 3.3 , known as stationary Ginzubrg-Landau... functions f k, · : × Ê → Ê satisfy assumptions F1 – F3 and μ > pj ≥ q > 1, j 1, 2 Then, for each λ > 0, 3.8 has a nonzero homoclinic solution u ∈ q , which is a critical point of the functional J 1 : q → Ê A typical example of 3.8 is 3.2 , which is a discretization of a fourth-order extended Fisher-Kolmogorov equation Homoclinic solutions for fourth-order ODEs are studied in 7 using variational approach . Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 195376, 17 pages doi:10.1155/2010/195376 Research Article On Homoclinic Solutions of a Semilinear p-Laplacian. difference equations has been recently applied in, among others, the papers of Agarwal et al. 10, Cabada et al. 11, Chen and Fang 12, Fang and Zhao 13, Jiang and Zhou 14,MaandGuo15 ,Mih˘ailescu et. and remarks on applications and extensions of Theorem 1.1 to the existence of homoclinic solutions of difference equations of following types: A Second-order discrete p-Laplacian equations of the