- Open Access
Existence of homoclinic orbits for a class of p-Laplacian systems in a weighted Sobolev space
© Shi et al.; licensee Springer. 2013
- Received: 5 December 2012
- Accepted: 7 May 2013
- Published: 24 May 2013
By applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, the existence of at least one or infinitely many homoclinic solutions is obtained for the following p-Laplacian system:
where , , , , and are not periodic in t.
MSC:34C37, 35A15, 37J45, 47J30.
- homoclinic solutions
- variational methods
- weighted space
- p-Laplacian systems
where , , , , , . As usual, we say that a solution u of (1.1) is a nontrivial homoclinic (to 0) if such that , as .
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 .
Theorem A 
- (A)Let , is a continuous, positive function on ℝ such that for all
(W2) as uniformly with respect to .
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.
Theorem B 
Assume that a and W satisfy (A), (W1)-(W3) and the following condition:
(W4) , .
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 1.1 Suppose that a and W satisfy the following conditions:
uniformly in .
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.2 Suppose that a and W satisfy (A)′, (W6) and the following conditions:
uniformly in .
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.3 Suppose that a and W satisfy (A)′ and (W4)-(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.
Theorem 1.4 Suppose that a and W satisfy (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.
Furthermore, the critical points of φ in E are classical solutions of (1.1) with .
To prove our results, we need the following generalization of the Lebesgue dominated convergence theorem.
Lemma 2.1 
The following lemma is an improvement result of  in which the author considered the case .
where from (A)′, . Then (2.3) holds.
(2.4) holds by taking .
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.
Lemma 2.3 
There exist constants such that .
There exists an such that .
where and is an open ball in E of radius ρ centered at 0.
Lemma 2.4 
For each finite dimensional subspace , there is such that for , is an open ball in E of radius r centered at 0.
Then I possesses an unbounded sequence of critical values.
is nondecreasing on ;
is nonincreasing on .
The proof of Lemma 2.5 is routine and we omit it.
Hence, by (3.12) and (3.13), in E. This shows that φ satisfies (PS)-condition.
Therefore, we can choose a constant depending on ρ such that for any with .
The function is a desired solution of problem (1.1). Since , is a nontrivial homoclinic solution. The proof is complete. □
Therefore, we can choose a constant depending on ρ such that for any with . The proof of Theorem 1.2 is complete. □
This contradicts the fact that is unbounded, and so is unbounded. The proof is complete. □
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. □
Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with and . Hence, problem (4.1) has an unbounded sequence of homoclinic solutions.
Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with and . Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of homoclinic solutions.
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).
- Poincaré H: Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris; 1897–1899.Google Scholar
- Alves CO, Carriao PC, Miyagaki OH: Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. Appl. Math. Lett. 2003, 16(5):639–642. 10.1016/S0893-9659(03)00059-4MathSciNetView ArticleGoogle Scholar
- Carriao PC, Miyagaki OH: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. Math. Anal. Appl. 1999, 230(1):157–172. 10.1006/jmaa.1998.6184MathSciNetView ArticleGoogle Scholar
- Coti ZV, Rabinowitz PH: Homoclinic orbits for second-order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 1991, 4(4):693–727.View ArticleGoogle Scholar
- Chen CN, Tzeng SY: Existence and multiplicity results for homoclinic orbits of Hamiltonian systems. Electron. J. Differ. Equ. 1997., 1997: Article ID 7Google Scholar
- Ding YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 1995, 25(11):1095–1113. 10.1016/0362-546X(94)00229-BMathSciNetView ArticleGoogle Scholar
- Izydorek M, Janczewska J: Homoclinic solutions for a class of the second-order Hamiltonian systems. J. Differ. Equ. 2005, 219(2):375–389. 10.1016/j.jde.2005.06.029MathSciNetView ArticleGoogle Scholar
- Korman P, Lazer AC: Homoclinic orbits for a class of symmetric Hamiltonian systems. Electron. J. Differ. Equ. 1994., 1994: Article ID 1Google Scholar
- Korman P, Lazer AC, Li Y: On homoclinic and heteroclinic orbits for Hamiltonian systems. Differ. Integral Equ. 1997, 10(2):357–368.MathSciNetGoogle Scholar
- Lu YF, Li CY, Zhong SZ, Zhang WJ: Homoclinic orbits for a class of Hamiltonian systems with potentials changing sign. Ann. Differ. Equ. 2005, 21(3):370–372.MathSciNetGoogle Scholar
- Lv X, Lu SP, Yan P: Existence of homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal. 2010, 72(1):390–398. 10.1016/j.na.2009.06.073MathSciNetView ArticleGoogle Scholar
- Omana W, Willem M: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 1992, 5(5):1115–1120.MathSciNetGoogle Scholar
- Rabinowitz PH: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb. A 1990, 114(1–2):33–38. 10.1017/S0308210500024240MathSciNetView ArticleGoogle Scholar
- Rabinowitz PH, Tanaka K: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 1991, 206(3):473–499.MathSciNetView ArticleGoogle Scholar
- Tang XH, Xiao L: Homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal. 2009, 71(3–4):1140–1152. 10.1016/j.na.2008.11.038MathSciNetView ArticleGoogle Scholar
- Tang XH, Xiao L: Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 2009, 351(2):586–594. 10.1016/j.jmaa.2008.10.038MathSciNetView ArticleGoogle Scholar
- Tang XH, Xiao L: Homoclinic solutions for ordinary p -Laplacian systems with a coercive potential. Nonlinear Anal. 2009, 71(3–4):1124–1132. 10.1016/j.na.2008.11.027MathSciNetView ArticleGoogle Scholar
- Tang XH, Lin XY: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Proc. R. Soc. Edinb. A 2011, 141: 1103–1119. 10.1017/S0308210509001346MathSciNetView ArticleGoogle Scholar
- Zhang QF, Tang XH: Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems. Math. Nachr. 2012, 285(5–6):778–789. 10.1002/mana.201000096MathSciNetView ArticleGoogle Scholar
- Zhang XY, Tang XH: A note on the minimal periodic solutions of nonconvex superlinear Hamiltonian system. Appl. Math. Comput. 2013, 219: 7586–7590. 10.1016/j.amc.2013.01.044MathSciNetView ArticleGoogle Scholar
- Chen P, Tang XH: New existence of homoclinic orbits for a second-order Hamiltonian system. Comput. Math. Appl. 2011, 62(1):131–141. 10.1016/j.camwa.2011.04.060MathSciNetView ArticleGoogle Scholar
- Salvatore A: On the existence of homoclinic orbits for a second-order Hamiltonian system. Differ. Integral Equ. 1997, 10(2):381–392.MathSciNetGoogle Scholar
- Salvatore A: Homoclinic orbits for a class of strictly convex Hamiltonian systems. Dyn. Syst. Appl. 1997, 6: 153–164.MathSciNetGoogle Scholar
- Benci V, Fortunato D: Weighted Sobolev space and the nonlinear Dirichlet problem in unbounded domains. Ann. Mat. Pura Appl. 1979, 121: 319–336. 10.1007/BF02412010MathSciNetView ArticleGoogle Scholar
- Royden HL: Real Analysis. 2nd edition. Macmillan Co., New York; 1968.Google Scholar
- Rabinowitz PH CBMS Regional Conf. Ser. in Math. 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.Google Scholar
- Adams RA, Fournier JJF: Sobolev Spaces. 2nd edition. Academic Press, Amsterdam; 2003.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.