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.
Keywords:discrete -Laplacian operator; variational methods; critical point; Palais-Smale condition; Mountain Pass Theorem
This paper is concerned with the existence of solutions for equations of the type
subjected to the potential boundary condition
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
Our approach for the boundary value problem (1.1), (1.2) relies on the critical point theory developed by Szulkin . With this aim, we introduce the space
which will be considered with the Luxemburg norm
Note that, as j is proper, convex and l.s.c., the same properties hold true for J. Then setting
it is clear that ψ is proper, convex and l.s.c. on X.
The energy functional associated to problem (1.1), (1.2) is
with ψ in (2.4) and Φ given by (2.5).
thenxis a solution of problem (1.1), (1.2).
Using (2.2), (2.6) and the summation by parts formula, a straightforward computation shows that
which, by a standard result from convex analysis, means that
and the proof is complete. □
From now on, we will use the following notations:
3 Ground state solutions
We begin by a result which states that the energy functional 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
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).
Hence, we infer
which, using again (3.1), implies
In order to give an application of Theorem 3.1, we consider the problem
Thus, we deduce
(ii) Theorem 5 in  is an immediate consequence of Corollary 3.2 with , .
4 Mountain pass type solutions
In this section, we deal with the existence of nontrivial solutions for the equation
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).
Proof Let be a sequence for which and (4.8) holds true with . Since X is finite dimensional, it is sufficient to prove that is bounded. In order to show this, we may assume that and for all . By virtue of (2.11), (3.1) and (4.5), we get
From (2.3) and (4.6), it follows
Using (4.12) and (4.13), we deduce that
and by virtue of (4.10), (4.11), (4.3) and (4.9), we have
Now, we can state the following result of Ambrosetti-Rabinowitz type .
Then, problem (4.1), (1.2) has a nontrivial solution.
Proof Without loss of generality, we may assume that
Now, using (2.10) and (3.1), we get
By virtue of (3.1), (4.5), (4.2) and (4.19), we deduce
On account of (4.16), we infer that
From (4.15), we have that
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.
The equation (4.1) is considered to be associated with the boundary conditions
then problem (4.1), (5.1) has a nontrivial solution.
respectively. If the T-periodicity condition is not assumed, then we only have
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 .
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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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