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
Let T be a positive integer, and be defined by for all and . Here and below, for with , we use the notation .
This paper is concerned with the existence of solutions for equations of the type
subjected to the potential boundary condition
where is the forward difference operator and stands for the discrete -Laplacian operator, that is,
Here and hereafter, is a continuous function, while is convex, proper (i.e., ), lower semicontinuous (in short, l.s.c.) and ∂j denotes the subdifferential of j. Recall, for , the set is defined by
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
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
for some . Also, we shall make use of the usual sup-norm .
Next, let be defined by
By means of j, we introduce the functional given by
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.
Further, we define by
It is a simple matter to check that and
The energy functional associated to problem (1.1), (1.2) is
with ψ in (2.4) and Φ given by (2.5).
Proposition 2.1If is a critical point of the functional in the sense that
thenxis a solution of problem (1.1), (1.2).
Proof In (2.7), we take , ; then dividing by s and letting , we get
where is the directional derivative of the convex function J at x in the direction of w. By virtue of (2.3), the above inequality becomes
Using (2.2), (2.6) and the summation by parts formula, a straightforward computation shows that
for all . Thus, we infer
for all with . This implies that
To prove that x satisfies condition (1.2), we multiply the equality (2.9) by . Then summing from 1 to T and using (2.8), one obtains
for all . Taking with and , where are arbitrarily chosen, we have
which, by a standard result from convex analysis, means that
and the proof is complete. □
From now on, we will use the following notations:
Remark 2.2 It is easy to check that for all and any , we have
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
then problem (1.1), (1.2) has at least one solution which minimizes onX.
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).
From (3.2) there are constants and such that
If , we may assume that . On the other hand, by the continuity of F, there is a constant such that
Hence, we infer
To prove the coercivity of , from the above inequality, we obtain
If , using (2.11) from Remark 2.2, we have
In the case , by virtue of (2.11) and (3.1), for , one obtains
which, using again (3.1), implies
In both cases, by virtue of (3.3) and (3.5), there exist constants such that
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 , , such that
with and . Since any norm on X is equivalent to , there exists such that
meaning that is coercive on and the proof is complete. □
In order to give an application of Theorem 3.1, we consider the problem
where is a continuous function and λ is a positive parameter.
Corollary 3.2Assume that andgsatisfies the growth condition
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.
Proof We apply Theorem 3.1 with for all and . From (3.7), we obtain
Thus, we deduce
for all and with . So, if , then
Also, if , setting
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
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 .
To treat problem (4.1), (1.2), instead of φ, there will be defined by
which is convex, of class on X, and its derivative is given by
Then, setting , we define
with J given by (2.3) and Φ in (2.5).
By means of in (3.1), we define the constants
Lemma 4.1If and there exist constants and such that
then the functional defined in (4.4) satisfies the Palais-Smale condition ((PS) condition for short) on , i.e., every sequence for which and
where , possesses a convergent subsequence.
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
with . Using (4.7) we deduce that, for all , it holds
Clearly, there is a constant , such that
Further, setting in (4.8), dividing by and then letting , we obtain
Using (4.12) and (4.13), we deduce that
and by virtue of (4.10), (4.11), (4.3) and (4.9), we have
Since , we infer that is bounded and the proof is complete. □
Now, we can state the following result of Ambrosetti-Rabinowitz type .
Theorem 4.2Assume that and, in addition,
(ii) , ;
(iii) there are constants and such that (4.6) holds true and
Then, problem (4.1), (1.2) has a nontrivial solution.
Proof Without loss of generality, we may assume that
which implies that . From (i), (2.3) and (4.15), we have
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 .
By the equivalence of the norms on X, there is some such that
Using (ii) we can find constants and such that
Let , with , be arbitrarily chosen. From (4.17) and (4.18), we have
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
with , and condition (a) is fulfilled.
Our next task is to prove that satisfies condition (b). To this end, let us first observe that, by virtue of (4.14), there exist such that
Let be such that and . Using (2.11) and (4.5), one obtains
From (4.15), we have that
which, together with (4.21) and (4.22) for any , gives
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.
Let be a convex and Gâteaux differentiable function with , where dg denotes the differential of g. Also, given a nonempty closed convex cone , we denote by the normal cone to K at , i.e.,
The equation (4.1) is considered to be associated with the boundary conditions
Theorem 5.1If is continuous, and, in addition, we assume that
(i) , ;
(ii) there are constants and such that (4.14) holds true and
then problem (4.1), (5.1) has a nontrivial solution.
Proof Since for all , Theorem 4.2 applies with , . □
Remark 5.2 Conditions (5.1) allow various possible choices of g and K, which, among others, recover classical boundary conditions. For instance, if , then the homogeneous boundary conditions
are obtained by choosing , respectively . If, in addition, p is T-periodic, then taking and , we get
respectively. If the T-periodicity condition is not assumed, then we only have
instead of and , respectively. As , in these four cases, condition (5.2) is automatically satisfied with any and .
Also, if are given, then with g defined by
and , we deduce the Sturm-Liouville type boundary conditions
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 .
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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