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On Ground States of Discrete -Laplacian Systems in Generalized Orlicz Sequence Spaces
Using the critical point theory, we establish sufficient conditions on the existence of ground states for discrete -Laplacian systems. Our results considerably generalize some existing ones.
1. Introduction and Main Results
The aim of this paper is to study the existence of ground state for discrete -Laplacian system where , for all , and are real valued on . is continuous in the second variable. Moreover, is the forward difference operator defined by .
We may think of (1) as being a discrete analogue of the following differential system:
For the case , system (2) is a -Laplacian system, which has been widely studied; to mention a few, see [1, 2]. Even for the special case , system (2) can be regarded as the more general form of Emden-Fowler equation appearing in the study of astrophysics, fluid mechanics, gas dynamics, nuclear physics, relativistic mechanics, and chemically reacting system in terms of various special forms of (see, e.g., ). The more general differential operator (2), namely, the so-called -Laplacian, has been studied by Fan et al. [4–7]. The -Laplacian operator can be used to describe the physical phenomena with “pointwise different properties.” The -Laplacian operator has more complicated properties than that of the -Laplacian; for example, it is not homogeneous, and this makes some classic theories and methods, such as the theory of Sobolev spaces, not applicable.
With the theory of nonlinear discrete dynamical systems being widely used to study discrete models appearing in many fields such as economics, ecology, computer science, neural networks, and cybernetics , the existence of solutions of discrete dynamical systems has become a hot topic; to mention a few, see [9–15]. For the case , Iannizzotto and Tersian  obtained multiple homoclinic solutions for system (1) by using the critical point theorem, and for the special case , Ma and Guo [13, 14] provided some sufficient conditions on the existence of homoclinic solutions for system (1). For the more general case—-Laplacian system (1)—Chen et al.  established some existence criteria to guarantee that the system has at least one or infinitely many homoclinic orbits. Motivated by Liu , which discussed the existence of ground state for -Laplacian system, in this paper, we will consider the existence of ground state for the -Laplacian system (1).
Now we are in a position to state our main results.
Theorem 1. Assume the following conditions hold:(a) for all ,(q) for all and as ,(p) for all ,(f) is continuous in for all , and for . Moreover,
(f1)there exists a constant such that
where for ;(f2)there exists a constant such that
where for .
Then (1) has a ground state solution.
Remark 2. (q) implies that there exists such that for all .
Remark 3. We extend Theorem 3.1 in  to the more general case—-Laplacian system. Furthermore, we obtain the existence of the ground state.
The rest of this paper is organized as follows. In Section 2, we establish the variational structure associated with (1). Some preliminary results are also provided in this section. In Section 3, we give the proof of the main result.
2. Variational Structure and Some Preliminary Results
In this section, we establish a variational structure which enables us to reduce the existence of solutions for (1) to the existence of critical points of the corresponding functional.
Let be the set of all two-sided sequences; that is, Then is a vector space with for .
We define as the set of all functions such that with the norm We also define with the norm We call the space a sequence space; it is a special kind of generalized Orlicz sequence space. For the general theory of generalized Orlicz spaces, see [18, 19].
Consider the functional on defined by Using the similar arguments as , we have the following lemmas.
Lemma 4. is a reflexive Banach space. Let and one has (1)if , then ;(2)if , then .
Lemma 5. is a reflexive Banach space. Let and one has(1)if , then ;(2)if , then .
Lemma 6. and the Fréchet derivative is given by for all . Moreover, the nonzero critical points of the functional on are the nontrivial solutions of (1).
Lemma 7. (f1) and (f2) imply that Moreover, for every and , is nondecreasing on and is nonincreasing on .
Let Then with is said to be a ground state solution of (1).
As usual, we make use of the following basic notations. Let be a Hilbert space and denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on .
Definition 8. Let . A sequence is called a Palais-Smale sequence (P.S. sequence) for if is bounded and as . We say that satisfies the Palais-Smale condition (P.S. condition) if any P.S. sequence for possesses a convergent subsequence.
Let be the open ball in with radius and center , and let denote its boundary.
Lemma 9 (mountain pass lemma). Let be a real Hilbert space and satisfies the P.S. condition. Assume that and the following two conditions hold.()There exist constants and such that .()There exists an such that .Then possesses a critical value . Moreover, can be characterized as where
3. Proof of Main Result
In order to prove Theorem 1, we first prove the following lemmas.
Lemma 10. The embedding is compact.
Proof. Let be a bounded sequence in ; that is, there exists such that for all . By reflexivity, passing to a subsequence we have in for some . We may assume , in particular as for all . For all , we can find such that By continuity of the finite sum, there exists such that So for all we have Since letting , we have Thus, in , and the proof is completed.
Lemma 11. Assume that and . Moreover, satisfies condition () and for all . Then where and .
Proof. Let then The proof is completed.
Lemma 12. Assume that all the conditions of Theorem 1 hold. Then the functional satisfies the P.S. condition.
Proof. Assume that is a sequence such that is a bounded and as . Then there exists a positive constant such that for all .
First, we show that is bounded. Now we may assume that ; otherwise, is bounded obviously. When is large enough, we have It follows from and that there exists a constant such that By Lemma 10, we can choose a subsequence, still denoted by , such that for some .
Next, we prove that
By (f), for any , there exists a positive constant with such that Since , there exists a positive integer such that we have By (30), there exists such that This, combined with (33) and (32), gives us Then for , where for all .
Let and , and , and , and and .
It is easy to check that and . Then using Lemma 11, for , we have Now we show that is bounded. We may assume that ; otherwise, is bounded obviously. By Lemmas 4 and 5, we have Let ; then ; that is, is bounded. Using the similar arguments as above, we obtain that , , and are bounded; that is, there exist three positive constants , , and such that This, combined with (38), gives us By continuity of the finite sum and (30), there exists such that Let . Combining (41) and (42) together, we have Thus, (31) holds.
Finally, we show that possesses a convergent subsequence. Since and , it follows at once that This, combined with (31), gives us
The following two inequalities are taken from  and will play an important role in the proof of our main result: for every and in . We define This, combined with (45), produces at once Now we show that as . That is, Let us first prove (50). Since and are bounded in , there exists a constant such that and for all . We denote By (46), we have
Let Then it is easy to check that and . By Lemma 11 and (53), we have Since It is easy to see that as and is bounded for all . This, combined with (52) and (55), gives us Using the similar arguments, we have (49). So as . By Lemma 5, it follows that as , and the proof is completed.
Proof of Theorem 1. The proof consists of two steps.
Step 1. We use Lemma 9 to show that (1) has a nontrivial solution in .
First we prove that satisfies () of Lemma 9. It follows from (32) that for and . Then, let , for all , we have and and hence satisfies () of Lemma 9.
Next, we prove that satisfies () of Lemma 9. Let and Then By Lemma 7, for , we have Then Since and , are smaller than , we can choose large enough such that . So we have verified all assumptions of Lemma 9; we know that possesses a critical value , where A critical point of corresponding to is nonzero as .
Step 2. We prove that (1) has a ground state in .
Let be the critical set of . Obviously, is a nonempty set. Denote Since , we have Then .
Suppose that such that . Obviously, is a P.S. sequence. By Lemma 12, we can choose a subsequence, still denoted by , such that Then and . Now we prove that is nonzero. If , then there exists a positive integer such that for all , we have By (32), it follows that which is in contradiction to . Thus, is the ground state solution of (1). The proof is completed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (no. 2013LYM-0093.), by Research Fund for the Doctoral Program of Higher Education of China (no. 20124410110001), and by Science Foundation for Young Teachers of Wuyi University (no. 2013zk02.).
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