Ground state and mountain pass solutions for discrete -Laplacian
© Bereanu et al.; licensee Springer 2012
Received: 25 July 2012
Accepted: 5 September 2012
Published: 19 September 2012
In this paper we study the existence of solutions for discrete -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: 39A12, 39A70, 49J40, 65Q10.
Keywordsdiscrete -Laplacian operator variational methods critical point Palais-Smale condition Mountain Pass Theorem
Let T be a positive integer, and be defined by for all and . Here and below, for with , we use the notation .
where stands for the usual inner product in .
It should be noticed that the boundary condition (1.2) recovers the classical ones. For instance, denoting by the indicator function of a closed, nonempty and convex set , the Dirichlet and Neumann boundary conditions are obtained by choosing with and , respectively. If p is T-periodic, taking () and , we get the periodic (antiperiodic) conditions. For other choices of j yielding various boundary conditions, we refer the reader to Gasinski and Papageorgiou  and Jebelean .
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., Agarwal et al. , Cabada et al. ), Neumann (Candito and D’Agui , Tian and Ge ) and periodic (He and Chen , Jebelean and Şerban ). Also, we note the recent paper of Mawhin  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 -Laplacian subjected to Dirichlet, Neumann or periodic boundary conditions were studied in recent time by Bereanu et al. , Galewski and Glab [11, 12], Guiro et al. , Koné and Ouaro , Mashiyev et al. , Mihăilescu et al. [16, 17].
Here, we use a variational approach to obtain ground state and mountain pass solutions for problem (1.1), (1.2). In this view, we employ some ideas originated in Jebelean and Moroşanu  (also see Jebelean ) combined with specific technicalities due to the discrete and anisotropic character of the problem. The main existence results are Theorem 3.1 and Theorem 4.2. These recover and generalize the similar ones for p= constant obtained in .
The rest of the paper is organized as follows. The functional framework and the variational approach of problem (1.1), (1.2) are presented in Section 2. In Section 3, we obtain 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
for some . Also, we shall make use of the usual sup-norm .
it is clear that ψ is proper, convex and l.s.c. on X.
with ψ in (2.4) and Φ given by (2.5).
then x is a solution of problem (1.1), (1.2).
and the proof is complete. □
3 Ground state solutions
then problem (1.1), (1.2) has at least one solution which minimizes on X.
Proof By the continuity of Φ and the lower semicontinuity of ψ, we have that the functional is sequentially l.s.c. on X. It remains to prove that is coercive on X. Then, by the direct method in calculus of variations, is bounded from below and attains its infimum at some , which, by virtue of (, Proposition 1.1) and Proposition 2.1, is a solution of problem (1.1), (1.2).
meaning that is coercive on and the proof is complete. □
where is a continuous function and λ is a positive parameter.
where , are constants and . The following hold true:
(i) if , then problem (3.6) has a solution for any ;
(ii) if , then there is some such that for any , problem (3.6) has a solution.
it is easy to see that condition (3.2) is fulfilled for any . □
(i) Note that a valid in Corollary 3.2(ii) is given by formula (3.8).
(ii) Theorem 5 in  is an immediate consequence of Corollary 3.2 with , .
4 Mountain pass type solutions
associated with the potential boundary condition (1.2). Here, f and j are as in the case of the previous problem (1.1), (1.2) and is a given function. The main tool in obtaining such a result will be the Mountain Pass Theorem .
with J given by (2.3) and Φ in (2.5).
where , possesses a convergent subsequence.
Since , we infer that is bounded and the proof is complete. □
Now, we can state the following result of Ambrosetti-Rabinowitz type .
Theorem 4.2 Assume that and, in addition,
(ii) , ;
Then, problem (4.1), (1.2) has a nontrivial solution.
From Lemma 4.1 and (iii), the functional satisfies the (PS) condition on .
Next, we shall prove that has a ‘mountain pass’ geometry:
(a) there exist such that if ;
(b) for some with .
with , and condition (a) is fulfilled.
as because . Hence, we can choose large enough to satisfy and , with μ entering in (4.20). This means that condition (b) is satisfied with . □
5 An application
In this section, we show how Theorem 4.2 can be applied to derive the existence of nontrivial solutions for equation (4.1) associated with some concrete boundary conditions.
Theorem 5.1 If is continuous, and, in addition, we assume that
(i) , ;
then problem (4.1), (5.1) has a nontrivial solution.
Proof Since for all , Theorem 4.2 applies with , . □
instead of and , respectively. As , in these four cases, condition (5.2) is automatically satisfied with any and .
In this case, (5.2) is fulfilled with any and .
Therefore, sufficient conditions ensuring the existence of nontrivial solutions of (4.1) subjected to one of the above boundary conditions can be easily stated by means of Theorem 5.1.
Remark 5.3 It is worth pointing out that in the cases of Dirichlet and antiperiodic boundary conditions, is allowed to be =0, and hence, r may be ≥0 on ; while in the Neumann, periodic and Sturm-Liouville cases, must be >0, meaning on .
Dedicated to Professor Jean Mawhin for his 70th anniversary.
The research of CŞ was supported by the strategic grant POSDRU/CPP107/DMI1.5/S/78421, Project ID 78421 (2010), co-financed by the European Social Fund - Investing in People, within the Sectoral Operational Programme Human Resources Development 2007-2013. Also, the support for CB and PJ from the grant TE-PN-II-RU-TE-2011-3-0157 (CNCS-Romania) is gratefully acknowledged.
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