| Title:
|
On the existence of multiple periodic solutions for the vector $p$-Laplacian via critical point theory (English) |
| Author:
|
Lü, Haishen |
| Author:
|
O'Regan, Donal |
| Author:
|
Agarwal, Ravi P. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
50 |
| Issue:
|
6 |
| Year:
|
2005 |
| Pages:
|
555-568 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the vector $p$-Laplacian \[ \left\rbrace \begin{array}{ll}-(| u^{\prime }| ^{p-2}u^{\prime })^{\prime }=\nabla F(t,u) \quad \text{a.e.}\hspace{5.0pt}t\in [0,T], u(0) =u(T),\quad u^{\prime }(0)=u^{\prime }(T),\quad 1<p<\infty . \end{array}\right. \qquad \mathrm{(*)}\] We prove that there exists a sequence $(u_n)$ of solutions of ($*$) such that $u_n$ is a critical point of $\varphi $ and another sequence $(u_n^{*}) $ of solutions of $(*)$ such that $u_n^{*}$ is a local minimum point of $\varphi $, where $\varphi $ is a functional defined below. (English) |
| Keyword:
|
$p$-Laplacian equation |
| Keyword:
|
periodic solution |
| Keyword:
|
critical point theory |
| MSC:
|
34B15 |
| MSC:
|
34C25 |
| idZBL:
|
Zbl 1099.34021 |
| idMR:
|
MR2181026 |
| DOI:
|
10.1007/s10492-005-0037-8 |
| . |
| Date available:
|
2009-09-22T18:24:10Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134623 |
| . |
| Reference:
|
[1] J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems.Springer-Verlag, New York-Berlin-Heidelberg-London-Paris-Tokyo, 1989. MR 0982267 |
| Reference:
|
[2] J. Mawhin: Some boundary value problems for Hartman-type perturbations of the ordinary vector $p$-Laplacian.Nonlinear. Anal., Theory Methods Appl. 40A (2000), 497–503. Zbl 0959.34014, MR 1768905, 10.1016/S0362-546X(00)85028-2 |
| Reference:
|
[3] M. Del Pino, R. Manasevich, and A. Murua: Existence and multiplicity of solutions with prescribed period for a second order O.D.E.Nonlinear. Anal., Theory Methods Appl. 18 (1992), 79–92. MR 1138643, 10.1016/0362-546X(92)90048-J |
| Reference:
|
[4] C. Fabry, D. Fayyad: Periodic solutions of second order differential equations with a $p$-Laplacian and asymmetric nonlinearities.Rend. Ist. Mat. Univ. Trieste 24 (1992), 207–227. MR 1310080 |
| Reference:
|
[5] Z. Guo: Boundary value problems of a class of quasilinear ordinary differential equations.Differ. Integral Equ. 6 (1993), 705–719. Zbl 0784.34018, MR 1202567 |
| Reference:
|
[6] H. Dang, S. F. Oppenheimer: Existence and uniqueness results for some nonlinear boundary value problems.J. Math. Anal. Appl. 198 (1996), 35–48. MR 1373525, 10.1006/jmaa.1996.0066 |
| Reference:
|
[7] E. Zeidler: Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators.Springer-Verlag, New York-Berlin-Heidelberg, 1990. Zbl 0684.47029, MR 1033498 |
| Reference:
|
[8] P. Habets, R. Manasevich, and F. Zanolin: A nonlinear boundary value problem with potential oscillating around the first eigenvalue.J. Differ. Equations 117 (1995), 428–445. MR 1325805, 10.1006/jdeq.1995.1060 |
| Reference:
|
[9] J. Mawhin: Periodic solutions of systems with $p$-Laplacian-like operators.In: Nonlinear Analysis and Applications to Differential Equations. Papers from the Autumn School on Nonlinear Analysis and Differential Equations, Lisbon, September 14–October 23, 1998. Progress in Nonlinear Differential Equations and Applications, Birkhäuser-Verlag, Boston, 1998, pp. 37–63. MR 1800613 |
| Reference:
|
[10] K. Deimling: Nonlinear Functional Analysis.Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985. Zbl 0559.47040, MR 0787404 |
| . |
