Strongly indefinite functionals and multiple solutions of elliptic systems
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- by D. G. De Figueiredo and Y. H. Ding PDF
- Trans. Amer. Math. Soc. 355 (2003), 2973-2989 Request permission
Abstract:
We study existence and multiplicity of solutions of the elliptic system \[ \begin {cases} -\Delta u =H_u(x,u,v) & \text {in $\Omega $}, \\ -\Delta v =-H_v(x,u,v) & \text {in $\Omega $}, \quad u(x) = v(x) = 0 \quad \text {on $\partial \Omega $}, \end {cases} \] where $\Omega \subset \mathbb {R}^N, N\geq 3$, is a smooth bounded domain and $H\in \mathcal {C}^1(\bar {\Omega }\times \mathbb {R}^2, \mathbb {R})$. We assume that the nonlinear term \[ H(x,u,v)\sim |u|^p + |v|^q + R(x,u,v) \ \ \text {with} \ \ \lim _{|(u,v)|\to \infty }\frac {R(x,u,v)}{|u|^p+|v|^q}=0, \] where $p\in (1, \ 2^*)$, $2^*:=2N/(N-2)$, and $q\in (1, \ \infty )$. So some supercritical systems are included. Nontrivial solutions are obtained. When $H(x,u,v)$ is even in $(u,v)$, we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if $p>2$ (resp. $p<2$). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.References
- Thomas Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), no. 10, 1205–1216. MR 1219237, DOI 10.1016/0362-546X(93)90151-H
- Thomas Bartsch and Djairo G. de Figueiredo, Infinitely many solutions of nonlinear elliptic systems, Topics in nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 35, Birkhäuser, Basel, 1999, pp. 51–67. MR 1725565
- Vieri Benci and Paul H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. MR 537061, DOI 10.1007/BF01389883
- Kung-ching Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1196690, DOI 10.1007/978-1-4612-0385-8
- David G. Costa and Celius A. Magalhães, A variational approach to noncooperative elliptic systems, Nonlinear Anal. 25 (1995), no. 7, 699–715. MR 1341522, DOI 10.1016/0362-546X(94)00180-P
- David G. Costa and Celius A. Magalhães, A unified approach to a class of strongly indefinite functionals, J. Differential Equations 125 (1996), no. 2, 521–547. MR 1378765, DOI 10.1006/jdeq.1996.0039
- D. G. De Figueiredo and C. A. Magalhães, On nonquadratic Hamiltonian elliptic systems, Adv. Differential Equations 1 (1996), no. 5, 881–898. MR 1392009
- Djairo G. de Figueiredo and Patricio L. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994), no. 1, 99–116. MR 1214781, DOI 10.1090/S0002-9947-1994-1214781-2
- Yanheng Ding, Infinitely many entire solutions of an elliptic system with symmetry, Topol. Methods Nonlinear Anal. 9 (1997), no. 2, 313–323. MR 1491850, DOI 10.12775/TMNA.1997.016
- Patricio L. Felmer, Periodic solutions of “superquadratic” Hamiltonian systems, J. Differential Equations 102 (1993), no. 1, 188–207. MR 1209982, DOI 10.1006/jdeq.1993.1027
- Josephus Hulshof and Robertus van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114 (1993), no. 1, 32–58. MR 1220982, DOI 10.1006/jfan.1993.1062
- Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785, DOI 10.1090/cbms/065
- Elves A. B. Silva, Nontrivial solutions for noncooperative elliptic systems at resonance, Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000) Electron. J. Differ. Equ. Conf., vol. 6, Southwest Texas State Univ., San Marcos, TX, 2001, pp. 267–283. MR 1804780
- Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1
Additional Information
- D. G. De Figueiredo
- Affiliation: IMECC-UNICAMP, Caixa Postal 6065, 13083-970 Campinas S.P. Brazil
- MR Author ID: 66760
- ORCID: 0000-0002-9902-244X
- Email: djairo@ime.unicamp.br
- Y. H. Ding
- Affiliation: Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100080 Beijing, People’s Republic of China
- MR Author ID: 255943
- Email: dingyh@math03.math.ac.cn
- Received by editor(s): June 18, 2001
- Published electronically: March 14, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 2973-2989
- MSC (2000): Primary 35J50; Secondary 58E99
- DOI: https://doi.org/10.1090/S0002-9947-03-03257-4
- MathSciNet review: 1975408