Ground state and mountain pass solutions for discrete p( )-Laplacian

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1 Bereanu et al. Boundary Value Problems 202, 202:04 R E S E A R C H Open Access Ground state and mountain pass solutions for discrete p -Laplacian Cristian Bereanu *, Petru Jebelean 2 and Călin Şerban 2 * Correspondence: cristian.bereanu@imar.ro Institute of Mathematics Simion Stoilow, Romanian Academy, 2, Calea Griviţei, Sector, Bucharest, RO-00702, Romania Full list of author information is available at the end of the article Abstract In this paper we study the existence of solutions for discrete p -Laplacian equations subjected to a potential type boundary condition. Our approach relies on Szulkin s critical point theory and enables us to obtain the existence of ground state as well as mountain pass type solutions. MSC: 39A2; 39A70; 49J40; 65Q0 Keywords: discrete p -Laplacian operator; variational methods; critical point; Palais-Smale condition; Mountain Pass Theorem Introduction Let T be a positive integer, p :[0,T], andh pk : R R be defined by h pk x= x pk 2 x for all x R and k [0, T]. Here and below, for a, b N with a < b, weusethe notation [a, b]:={a, a +,...,b}. This paper is concerned with the existence of solutions for equations of the type pk xk =f k, xk, k [, T]. subjected to the potential boundary condition hp0 x0, hpt xt j x0, xt +,.2 where xk=xk + xk is the forward difference operator and p stands for the discrete p -Laplacian operator, that is, pk xk := h pk xk = hpk xk hpk xk. Here and hereafter, f :[,T] R R is a continuous function, while j : R R,+ ] is convex, proper i.e., Dj:={z R R : jz<+ }, lower semicontinuous in short, l.s.c. and j denotes the subdifferential of j. Recall, for z R R, theset jzisdefinedby jz= { ζ R R : jξ jz ζ ξ z, ξ R R },.3 where stands for the usual inner product in R R. 202 Bereanu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Bereanuetal. Boundary Value Problems 202, 202:04 Page 2 of 3 It should be noticed that the boundary condition.2 recovers the classical ones. For instance, denoting by I K the indicator function of a closed, nonempty and convex set K R R, the Dirichlet and Neumann boundary conditions are obtained by choosing j = I K with K = {0, 0} and K = R R,respectively.Ifp is T-periodic, taking K = {x, x, x R} K = {x, x, x R} andj = I K, we get the periodic antiperiodic conditions. For other choices of j yielding various boundary conditions, we refer the reader to Gasinski and Papageorgiou []andjebelean[2]. The study of boundary value problems with a discrete p-laplacian using variational approaches has captured attention in the last years. Most of the papers deal with classical boundary conditions such as Dirichlet see, e.g.,agarwalet al. [3],Cabada et al. [4],Neumann Candito and D Agui [5], Tian and Ge [6] and periodic He and Chen [7],Jebelean and Şerban [8]. Also, wenotetherecentpaperofmawhin[9] where variational techniques are employed to obtain the existence of periodic solutions for systems involving a general discrete φ-laplacian operator. Boundary value problems with the discrete p -Laplacian subjected to Dirichlet, Neumann or periodic boundary conditions were studied in recent time by Bereanu et al. [0], Galewski and Glab [, 2], Guiro et al. [3], Koné and Ouaro [4], Mashiyev et al. [5], Mihăilescu et al. [6, 7]. Here, we use a variational approach to obtain ground state and mountain pass solutions for problem.,.2. In this view, we employ some ideas originated in Jebelean and Moroşanu [8] alsosee Jebelean [2] combined with specific technicalities due to the discrete and anisotropic character of the problem. The main existence results are Theorem 3. and Theorem 4.2. These recover and generalize the similar ones for p = constant obtained in [9]. The rest of the paper is organized as follows. The functional framework and the variational approach of problem.,.2 arepresentedinsection2. InSection3, weobtain the existence of ground state solutions, while Section 4 is devoted to the existence of mountain pass type solutions. An example of application is given in Section 5. 2 The functional framework Our approach for the boundary value problem.,.2 relies on the critical point theory developed by Szulkin [20]. With this aim, we introduce the space X := { x :[0,T +] R }, which will be considered with the Luxemburg norm { T+ x η,p = inf ν >0: xk pk ν pk T + η xk pk ν for some η > 0. Also, we shall make use of the usual sup-norm x = max k [0,T+] xk. Next, let ϕ : X R be defined by T+ ϕx= xk pk, x X. 2. pk pk }

3 Bereanuetal. Boundary Value Problems 202, 202:04 Page 3 of 3 Standard arguments show that ϕ is convex, of class C. Using the summation by parts formula see, e.g.,[8, 9], one obtains that its derivative is given by ϕ x, y T+ = h pk xk yk, x, y X. 2.2 By means of j, we introduce the functional J : X,+ ]givenby Jx=j x0, xt +, x X. 2.3 Note that, as j is proper, convex and l.s.c., the same properties hold true for J. Then setting ψ = ϕ + J, 2.4 it is clear that ψ is proper, convex and l.s.c. on X. Further, we define F :[,T] R R by Fk, t= t 0 f k, τ dτ, k [, T], t R and T x= F k, xk, x X. 2.5 Itisasimplemattertocheckthat C X, Rand x, y T = f k, xk yk, x, y X. 2.6 The energy functional associated to problem.,.2is I = + ψ with ψ in 2.4and given by 2.5. Proposition 2. If x X is a critical point of the functional I in the sense that x, y x + ψy ψx 0, y X, 2.7 then x is a solution of problem.,.2. Proof In 2.7, we take y = x + sw, s > 0; then dividing by s and letting s 0 +,weget x, w + ϕ x, w + J x; w 0, w X,

4 Bereanuetal. Boundary Value Problems 202, 202:04 Page 4 of 3 where J x; w is the directional derivative of the convex function J at x in the direction of w.byvirtueof2.3, the above inequality becomes x, w + ϕ x, w + j x0, xt + ; w0, wt + 0, w X. Using 2.2, 2.6 and the summation by parts formula, a straightforward computation shows that T f k, xk wk+h pt xt wt + hp0 x0 w0 T pk xk wk+j x0, xt + ; w0, wt for all w X.Thus,weinfer T pk xk +f k, xk wk=0 for all w X with w0 = wt + = 0. This implies that pk xk =f k, xk, k [, T]. 2.9 To prove that x satisfies condition.2, we multiply the equality 2.9 by wk. Then summing from to T and using 2.8, one obtains j x0, xt + ; w0, wt + h pt xt wt ++hp0 x0 w0 for all w X. Takingw X with w0 = u and wt +=v, whereu, v R are arbitrarily chosen, we have j x0, xt + ;u, v h p0 x0 u hpt xt v, u, v R, which, by a standard result from convex analysis, means that hp0 x0, hpt xt j x0, xt + and the proof is complete. From now on, we will use the following notations: p = min k [0,T] pk, p+ = max pk and p = min pk, k [0,T] k [,T] p = max k [,T] pk. Remark 2.2 It is easy to check that for all x X and any η >0,wehave x η,p < x p+ η,p ϕx+η T x η,p > x p η,p ϕx+η T xk pk x p η,p pk, 2.0 xk pk p x + η,p pk. 2.

5 Bereanuetal. Boundary Value Problems 202, 202:04 Page 5 of 3 3 Ground state solutions We begin by a result which states that the energy functional I has a minimum point in X provided that the potential of the nonlinearity f lies asymptotically on the left of the first eigenvalue like constant { T+ λ := inf T xk pk pk xk pk pk : x X \{0}, x0, xt + } Dj. 3. Theorem 3. If lim sup t pkfk, t t pk < λ, k [, T], 3.2 then problem.,.2 has at least one solution which minimizes I on X. Proof By the continuity of and the lower semicontinuity of ψ, wehavethatthefunctional I is sequentially l.s.c. on X. It remains to prove that I is coercive on X.Then,bythe direct method in calculus of variations, I is bounded from below and attains its infimum at some x X,which,byvirtueof[20], Proposition. and Proposition 2.,isasolution of problem.,.2. From 3.2 there are constants σ >0andρ >0suchthat Fk, t λ σ pk t pk, k [, T], t R with t > ρ. If λ >0,wemayassumethatσ < λ. On the other hand, by the continuity of F,thereisa constant M ρ >0suchthat Fk, t M ρ, k [, T], t R with t ρ. Hence, we infer Fk, t M ρ + λ σ ρ pk + λ σ pk pk t pk, k [, T], t R. To prove the coercivity of I, fromthe above inequality, we obtain Ix ϕx M ρ T λ σ T pk ρpk λ σ ϕx M ρ T λ σ p ρ p + ρ p+ T λ σ = ϕx λ σ where C = M ρ T + λ σ p ρ p + ρ p+ T. T xk pk + Jx pk T xk pk + Jx pk T xk pk C + Jx, x X, pk

6 Bereanuetal. Boundary Value Problems 202, 202:04 Page 6 of 3 If λ =0,using2.fromRemark2.2,wehave Ix ϕx+σ T xk pk C + Jx pk x p σ,p C + Jx, x X, x σ,p >. 3.3 In the case λ >0,byvirtueof2.and3., for x λ,p >,oneobtains x p λ,p ϕx+λ T xk pk 2ϕx, 3.4 pk which, using again 3., implies Ix ϕx+σ λ T+ xk pk pk C + Jx λ = σ λ ϕx C + Jx σ 2λ x p λ,p C + Jx, x DJ. 3.5 In both cases, by virtue of 3.3and3.5, there exist constants η, C 2 >0suchthat Ix C 2 x p η,p C + Jx, x DJ, x η,p >. On the other hand, as j is convex and l.s.c., it is bounded from below by an affine functional. Therefore, on account of 2.3, there are positive constants k, k 2, k 3 such that Ix C 2 x p η,p C k x0 k2 xt + k3 C 2 x p η,p C 3 x C 4, x DJ with C 3 = k + k 2 and C 4 = C + k 3. Since any norm on X is equivalent to η,p,there exists C 5 >0suchthat Ix C 2 x p η,p C 5 x η,p C 4. Consequently, Ix +, as x η,p, meaning that I is coercive on X, η,p and the proof is complete. In order to give an application of Theorem 3., we consider the problem pk xk =λgk, xk, k [, T], h p0 x0, h pt xt jx0, xt +, 3.6 where g :[,T] R R is a continuous function and λ is a positive parameter.

7 Bereanuetal. Boundary Value Problems 202, 202:04 Page 7 of 3 Corollary 3.2 Assume that λ >0and g satisfies the growth condition gk, t a t qk + b, k [, T], t R, 3.7 where a >0,b R + are constants and q :[,T] [0,. The following hold true: i if p > q +, then problem 3.6 has a solution for any λ >0; ii if p = q +, then there is some λ >0such that for any λ 0, λ, problem 3.6 has a solution. Proof We apply Theorem 3. with f k, t=λgk, t for all k [, T] andt R.From3.7, we obtain Fk, t t t = f k, τ dτ λ aτ qk + b dτ Thus, we deduce pkfk, t t pk 0 0 = λa t qk+ λa + λ t b qk+ q + t qk+ + λ t b, k [, T], t R. pλa t qk+ + pλb t pλa t q+ q + t pk t pk q + t p + pλb t t p for all k [, T]andt R with t >.So,ifp > q +,then lim sup t Also, if p = q +, setting pkfk, t t pk 0<λ, k [, T]. λ = λ q +, 3.8 ap it is easy to see that condition 3.2 is fulfilled for any λ 0, λ. Remark 3.3 i Note that a valid λ in Corollary 3.2ii is given by formula 3.8. ii Theorem 5 in [] is an immediate consequence of Corollary 3.2 with j = I K, K = {0, 0}. 4 Mountain pass type solutions In this section, we deal with the existence of nontrivial solutions for the equation pk xk +rkh pk xk = f k, xk, k [, T], 4. associated with the potential boundary condition.2. Here, f and j are as in the case of the previous problem.,.2andr:[,t] [0, isagivenfunction.themaintool in obtaining such a result will be the Mountain Pass Theorem [20].

8 Bereanuetal. Boundary Value Problems 202, 202:04 Page 8 of 3 To treat problem 4.,.2, instead of ϕ,therewillbeϕ r : X R defined by T+ ϕ r x= xk T pk rk + xk pk, x X, 4.2 pk pk which is convex, of class C on X, and its derivative is given by ϕ r x, y T+ = h pk xk yk T + rkh pk xk yk, x, y X. 4.3 Then, setting ψ r = ϕ r + J,wedefine I r = ψ r +, 4.4 with J given by 2.3and in 2.5. By means of λ in 3., we define the constants η := λ + r and η := λ + r. 4.5 Lemma 4. If η >0and there exist constants θ > p + and C, ρ >0such that j z; z θjz+c, z Dj 4.6 and θfk, t tf k, t, k [, T], t R with t > ρ, 4.7 then the functional I r defined in 4.4 satisfies the Palais-Smale condition PS condition for short on X, η,p, i.e., every sequence {x n } XforwhichI r x n c R and x n, y x n + ψr y ψ r x n ε n y x n η,p, y X, 4.8 where ε n 0, possesses a convergent subsequence. Proof Let {x n } X be a sequence for which I r x n c R and 4.8 holdstruewith ε n 0. Since X is finite dimensional, it is sufficient to prove that {x n } is bounded. In order to show this, we may assume that {x n } DI r =DJand x n η,p >foralln N. By virtue of 2., 3.and4.5, we get T+ x n p η,p x n k pk T +λ + r xn k pk pk pk 2 T+ xn k T pk + rk xn k. pk 4.9 p

9 Bereanuetal. Boundary Value Problems 202, 202:04 Page 9 of 3 From 2.3and4.6, it follows Jy θ J y; y C 2, y DJ 4.0 with C 2 = C /θ.using4.7wededucethat,foralln N,itholds x n + θ x n, x n = θ θ T [ θf k, xn k x n kf k, x n k ] x n k ρ [ θf k, xn k x n kf k, x n k ] θ T max θfk, x xf k, x =: C3. 4. x ρ Clearly, there is a constant C 4 >0,suchthat I r x n C 4, n N. 4.2 Further, setting y = x n + sx n in 4.8, dividing by s > 0 and then letting s 0 +,weobtain x n, x n + ϕ r x n, x n + J x n ; x n ε n x n η,p, n N. 4.3 Using 4.2 and4.3, we deduce that C 4 + ε n θ x n η,p x n +ϕ r x n +Jx n + ε n θ x n η,p x n x n, x n + ϕr x n ϕ θ θ r x n, x n + Jxn θ J x n ; x n and by virtue of 4.0, 4., 4.3and4.9, we have C 2 + C 3 + C 4 + ε n θ x n η,p p T+ x + n k T pk + rk x n k pk θ p 2 p x + n p η,p θ. Since θ > p +,weinferthat{x n } is bounded and the proof is complete. Now, we can state the following result of Ambrosetti-Rabinowitz type [2]. Theorem 4.2 Assumethat η >0and, in addition, i 0, 0 j0, 0; pkfk,t ii lim sup t 0 < η, k [, T]; t pk iii there are constants θ > p + and C, ρ >0such that 4.6 holds true and 0<θFk, t tf k, t, k [, T], t R with t > ρ. 4.4 Then, problem 4.,.2 has a nontrivial solution.

10 Bereanu et al. Boundary Value Problems 202, 202:04 Page 0 of 3 Proof Without loss of generality, we may assume that j0, 0 = 0, 4.5 which implies that I r 0 = 0. From i, 2.3and4.5, we have Jx J0 = 0, x DJ. 4.6 From Lemma 4. and iii, the functional I r satisfies the PS condition on X, η,p. Next, we shall prove that I r has a mountain pass geometry: a there exist α, μ >0such that I r x α if x η,p = μ; b I r e 0 for some e X with e η,p > μ. By the equivalence of the norms on X,thereissomeC 2 >0suchthat x C 2 x η,p, x X. 4.7 Using ii we can find constants σ 0, ηandμ 0, such that Fk, t η σ pk t pk, k [, T], t R with t μc Let x X,with x η,p μ, be arbitrarily chosen. From 4.7and4.8, we have F k, xk η σ xk pk, pk k [, T], which implies x η σ T xk pk. pk Now, using 2.0and3., we get T+ x p+ η,p xk pk +λ + r pk T xk pk 2ϕr x. 4.9 pk By virtue of 3., 4.5, 4.2and4.9, we deduce x+ϕ r x ϕ r x+σ η ϕ r x+σ η T+ T xk pk pk pk xk pk + T rk pk xk pk η = σ η ϕ rx σ 2η x p+ η,p, x DJ, x η,p μ.

11 Bereanu et al. Boundary Value Problems 202, 202:04 Page of 3 On account of 4.6, we infer that I r x= x+ϕ r x+jx α, if x η,p = μ, 4.20 with σμp+ =: α > 0, and condition a is fulfilled. 2η Our next task is to prove that I r satisfies condition b. To this end, let us first observe that, by virtue of 4.4, there exist a, a 2 >0suchthat Fk, t a t θ a 2, k [, T], t R. 4.2 Let x 0 X \{0} be such that x 0 0 = x 0 T +=0and x 0 η,p >.Using2.and4.5, one obtains T+ x 0 p+ η,p x 0 k pk T +λ + r x0 k pk ϕ r x pk pk From 4.5, we have that Jsx 0 =0, s R, which, together with 4.2and4.22 for any s, gives I r sx 0 s p+ ϕ r x 0 + sx 0 s p+ x 0 p+ η,p sθ a T x 0 k θ + a 2 T, as s + because θ > p +.Hence,wecanchooses 0 large enough to satisfy I r s 0 x 0 0 and s 0 x 0 η,p > μ, withμ entering in This means that condition b is satisfied with e = s 0 x 0. 5 An application In this section, we show how Theorem 4.2 can be applied to derive the existence of nontrivial solutions for equation 4. associated with some concreteboundary conditions. Let g : R R R be a convex and Gâteaux differentiable function with dg0, 0 = 0, 0, where dg denotes the differential of g. Also, given a nonempty closed convex cone K R R,wedenotebyN K z the normal cone to K at z K, i.e., N K z= { ζ R R :ζ ξ z 0, ξ K }, z K. The equation 4. is considered to be associated with the boundary conditions x0, xt + K, hp0 x0, hpt xt dg x0, xt + NK x0, xt We set { T+ η K := r + inf xk pk pk T : x X \{0}, pk xk pk x0, xt + K }.

12 Bereanu et al. Boundary Value Problems 202, 202:04 Page 2 of 3 Theorem 5. If f :[,T] R R is continuous, η K >0and, in addition, we assume that pkfk,t i lim sup t 0 < η t pk K, k [, T]; ii there are constants θ > p + and C, ρ >0such that 4.4 holds true and dgz, z θgz+c, z K, 5.2 then problem 4., 5. has a nontrivial solution. Proof Since N K z= I K z for all z K,Theorem4.2 applies with jz=gz+i K z, z R R. Remark 5.2 Conditions 5. allow various possible choices of g and K,which,amongothers, recover classical boundary conditions. For instance, if g = 0, then the homogeneous boundary conditions x0 = 0 = xt + x0 = 0 = xt Dirichlet, Neumann are obtained by choosing K = {0, 0}, respectivelyk = R R. If, in addition, p is T- periodic, then taking K = {x, x, x R} and K = {x, x, x R},weget x0 xt +=0= x0 xt x0 + xt +=0= x0 + xt periodic, antiperiodic, respectively. If the T-periodicity condition is not assumed, then we only have h p0 x0 = hpt xt and h p0 x0 = hpt xt instead of x0 = xtand x0 = xt, respectively. As g = 0, in these four cases, condition 5.2 is automatically satisfied with any θ R and C =0. Also, if α, β > 0 are given, then with g defined by gz:= z p0 p0 α p0 + pt z 2 pt β pt, z =z, z 2 R R and K = R R,we deduce the Sturm-Liouville type boundary conditions x0 α x0 = 0, xt ++β xt=0. In this case, 5.2 is fulfilled with any θ min{p0, pt} and C =0. Therefore, sufficient conditions ensuring the existence of nontrivial solutions of 4. subjected to one of the above boundary conditions can be easily stated by means of Theorem 5.. Remark 5.3 It is worth pointing out that in the cases of Dirichlet and antiperiodic boundary conditions, r is allowed to be = 0, and hence, r may be 0on[,T]; while in the Neumann, periodic and Sturm-Liouville cases, r must be > 0, meaning r > 0 on[,t].

13 Bereanu et al. Boundary Value Problems 202, 202:04 Page 3 of 3 Competing interests The authors declare that they have no competing interests. Authors contributions The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript. Author details Institute of Mathematics Simion Stoilow, Romanian Academy, 2, Calea Griviţei, Sector, Bucharest, RO-00702, Romania. 2 Department of Mathematics, West University of Timişoara, 4, Blvd. V. Pârvan, Timişoara, RO , Romania. Acknowledgements Dedicated to Professor Jean Mawhin for his 70th anniversary. The research of CŞ was supported by the strategic grant POSDRU/CPP07/DMI.5/S/7842, Project ID , co-financed by the European Social Fund - Investing in People, within the Sectoral Operational Programme Human Resources Development Also, the support for CB and PJ from the grant TE-PN-II-RU-TE CNCS-Romania is gratefully acknowledged. Received: 25 July 202 Accepted: 5 September 202 Published: 9 September 202 References. Gasinski, L, Papageorgiou, NS: Nonlinear second-order multivalued boundary value problems. Proc. Indian Acad. Sci. Math. Sci. 3, Jebelean, P: Variational methods for ordinary p-laplacian systems with potential boundary conditions. Adv. Differ. Equ. 33-4, Agarwal, RP, Perera, K, O Regan, D: Multiple positive solutions of singular discrete p-laplacian problems via variational methods. Adv. Differ. Equ , Cabada, A, Iannizzotto, A, Tersian, S: Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl. 356, Candito, P, D Agui, G: Three solutions for a discrete nonlinear Neumann problem involving p-laplacian. Adv. Differ. Equ. 200, Tian, Y, Ge, W: The existence of solutions for a second-order discrete Neumann problem with a p-laplacian. J. Appl. Math. Comput. 26, He, T, Chen, W: Periodic solutions of second order discrete convex systems involving the p-laplacian. Appl. Math. Comput. 206, Jebelean, P, Şerban, C: Ground state periodic solutions for difference equations with discrete p-laplacian. Appl. Math. Comput. 27, Mawhin, J: Periodic solutions of second order nonlinear difference systems with φ-laplacian: a variational approach. Nonlinear Anal. 752, Bereanu, C, Jebelean, P, Şerban, C: Periodic and Neumann problems for discrete p -Laplacian. J. Math. Anal. Appl doi:0.06/j.jmaa Galewski, M, Gła b, S: On the discrete boundary value problem for anisotropic equation. J. Math. Anal. Appl. 386, Galewski, M, Gła b, S: Positive solutions for anisotropic discrete BVP. arxiv: v 3. Guiro, A, Nyanquini, I, Ouaro, S: On the solvability of discrete nonlinear Neumann problems involving the px-laplacian. Adv. Differ. Equ. 20, Koné, B, Ouaro, S: Weak solutions for anisotropic discrete boundary value problems. J. Differ. Equ. Appl. 62, Mashiyev, R, Yucedag, Z, Ogras, S: Existence and multiplicity of solutions for a Dirichlet problem involving the discrete px-laplacian operator. Electron. J. Qual. Theory Differ. Equ. 67, Mihăilescu, M, Rădulescu, V, Tersian, S: Eigenvalue problems for anisotropic discrete boundary value problems. J. Differ. Equ. Appl. 56, Mihăilescu, M, Rădulescu, V, Tersian, S: Homoclinic solutions of difference equations with variable exponents. Topol. Methods Nonlinear Anal. 38, Jebelean, P, Moroşanu, G: Ordinary p-laplacian systems with nonlinear boundary conditions. J. Math. Anal. Appl. 33, Şerban, C: Existence of solutions for discrete p-laplacian with potential boundary conditions. J. Differ. Equ. Appl., 202. doi:0.080/ Szulkin, A: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 32, Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 4, doi:0.86/ Cite this article as: Bereanu et al.: Ground state and mountain pass solutions for discrete p -Laplacian. Boundary Value Problems :04.