Keywords:homoclinic solutions; variational methods; weighted space; p-Laplacian systems
Consider homoclinic solutions of the following p-Laplacian system:
The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by Poincaré . Up to the year of 1990, a few of isolated results can be found, and the only method for dealing with such a problem was the small perturbation technique of Melnikov.
Recently, the existence and multiplicity of homoclinic solutions and periodic solutions for Hamiltonian systems have been extensively studied by critical point theory. For example, see [2-20] and references therein. However, few results [21,22] have been obtained in the literature for system (1.2). In , by introducing a suitable Sobolev space, Salvatore established the following existence results for system (1.2) when .
Assume thataandWsatisfy the following conditions:
Then problem (1.2) has one nontrivial homoclinic solution.
When is an even function in x, Salvatore  obtained the following existence theorem of an unbounded sequence of homoclinic orbits for problem (1.2) by the symmetric mountain pass theorem.
Assume thataandWsatisfy (A), (W1)-(W3) and the following condition:
Then problem (1.2) has an unbounded sequence of homoclinic solutions.
In , Chen and Tang improved Theorem A and Theorem B by relaxing conditions (W1) and (W2) and removing condition (W3). Motivated mainly by the ideas of [18,21-23], we will consider homoclinic solutions of (1.1) by the mountain pass theorem and symmetric mountain pass theorem. Precisely, we obtain the following main results.
Theorem 1.1Suppose thataandWsatisfy the following conditions:
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.2Suppose thataandWsatisfy (A)′, (W6) and the following conditions:
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.3Suppose thataandWsatisfy (A)′ and (W4)-(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.
Theorem 1.4Suppose thataandWsatisfy (A)′, (W4), (W5)′, (W6), (W7)′. Then problem (1.1) has an unbounded sequence of homoclinic solutions.
Remark 1.1 When , condition (A)′ reduces to condition (A). Obviously, Theorem 1.1-Theorem 1.4 generalize and improve Theorem A, Theorem B and the corresponding results in . It is easy to see that our results hold true even if . To the best of our knowledge, similar results for problem (1.1) cannot be seen in the literature; from this point, our results are new.
Remark 1.2 If and , then problem (1.1) reduces to problem (1.3). As pointed out in , Theorem A can be proved by replacing (A) with the more general assumption: as .
The rest of this paper is organized as follows. In Section 2, some preliminaries are presented and we establish an embedding result. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.
with the usual norms
be the Sobolev space with the norm given by
with the norm given by
Clearly, it follows from (W5) or (W5)′ that . By Theorem 2.1 of , we can deduce that the map
To prove our results, we need the following generalization of the Lebesgue dominated convergence theorem.
The following lemma is an improvement result of  in which the author considered the case .
Lemma 2.2Ifasatisfies assumption (A)′, then
Moreover, there exists a Sobolev spaceZsuch that
Finally, as is the weighted Sobolev space , it follows from  that (2.5) holds. □
The following two lemmas are the mountain pass theorem and symmetric mountain pass theorem, which are useful in the proofs of our theorems.
ThenIpossesses an unbounded sequence of critical values.
The proof of Lemma 2.5 is routine and we omit it.
3 Proofs of theorems
It is well known  that there exists a constant such that
From (2.1), (2.2), (3.1), (W6) and (W7), we have
Hence, from (3.5), we have
It follows from Lemma 2.1, (3.6) and the above inequalities that
This shows that
From (2.2), we have
Inequality (3.9) implies that
By (3.14), we have
It follows from (W7), (3.15), (3.17) that
Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition in (W7) is only used in the proofs of (3.3) and Step 2. Therefore, we only need to prove that (3.3) and Step 2 still hold if we use (W5)′ and (W7)′ instead of (W5) and (W7). We first prove that (3.3) holds. From (W6), (W7)′, (2.1), (2.2) and (3.1), we have
By (3.23), we have
Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that and satisfies (PS)-condition and assumptions (i) of Lemma 2.3. Now, we prove that (iii) of Lemma 2.4. Let be a finite dimensional subspace of E. Since all norms of a finite dimensional space are equivalent, there exists such that
where δ is given in (3.24). Let
Then by (3.25)-(3.28), (3.30) and (3.31), we have
Therefore, from (3.33), (3.36) and (3.37), we have
From (3.29) and (3.30), we have
It follows that
By (3.2) and (3.41), we get
which implies that
Hence, by (2.1) and (3.43), we have
which, together with (3.41), implies that
Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □
Example 4.1 Consider the following system:
Example 4.2 Consider the following system:
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
XS and QZ are supported by the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093), Guangxi Natural Science Foundation (Nos. 2013GXNSFBA019004 and 2012GXNSFBA053013) and the Scientific Research Foundation of Guilin University of Technology. QMZ is supported by the NNSF of China (No. 11201138) and the Scientific Research Fund of Hunan Provincial Education Department (No. 12B034).
Alves, CO, Carriao, PC, Miyagaki, OH: Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. Appl. Math. Lett.. 16(5), 639–642 (2003). Publisher Full Text
Ding, YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal.. 25(11), 1095–1113 (1995). Publisher Full Text
Izydorek, M, Janczewska, J: Homoclinic solutions for a class of the second-order Hamiltonian systems. J. Differ. Equ.. 219(2), 375–389 (2005). Publisher Full Text
Lv, X, Lu, SP, Yan, P: Existence of homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal.. 72(1), 390–398 (2010). Publisher Full Text
Rabinowitz, PH: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb. A. 114(1-2), 33–38 (1990). Publisher Full Text
Tang, XH, Xiao, L: Homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal.. 71(3-4), 1140–1152 (2009). Publisher Full Text
Tang, XH, Xiao, L: Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl.. 351(2), 586–594 (2009). Publisher Full Text
Tang, XH, Xiao, L: Homoclinic solutions for ordinary p-Laplacian systems with a coercive potential. Nonlinear Anal.. 71(3-4), 1124–1132 (2009). Publisher Full Text
Tang, XH, Lin, XY: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Proc. R. Soc. Edinb. A. 141, 1103–1119 (2011). Publisher Full Text
Zhang, QF, Tang, XH: Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems. Math. Nachr.. 285(5-6), 778–789 (2012). Publisher Full Text
Zhang, XY, Tang, XH: A note on the minimal periodic solutions of nonconvex superlinear Hamiltonian system. Appl. Math. Comput.. 219, 7586–7590 (2013). Publisher Full Text
Chen, P, Tang, XH: New existence of homoclinic orbits for a second-order Hamiltonian system. Comput. Math. Appl.. 62(1), 131–141 (2011). Publisher Full Text
Benci, V, Fortunato, D: Weighted Sobolev space and the nonlinear Dirichlet problem in unbounded domains. Ann. Mat. Pura Appl.. 121, 319–336 (1979). Publisher Full Text