Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Strongly indefinite functionals and multiple solutions of elliptic systems

Authors: D. G. De Figueiredo and Y. H. Ding
Journal: Trans. Amer. Math. Soc. 355 (2003), 2973-2989
MSC (2000): Primary 35J50; Secondary 58E99
Published electronically: March 14, 2003
MathSciNet review: 1975408
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study existence and multiplicity of solutions of the elliptic system

\begin{displaymath}\begin{cases}-\Delta u =H_u(x,u,v) & \text{in} \Omega,\\ -\D... ...\quad u(x) =v(x)=0 \quad\text{on} \partial \Omega ,\end{cases}\end{displaymath}

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

\begin{displaymath}H(x,u,v)\sim \vert u\vert^p + \vert v\vert^q + R(x,u,v) \ \te... ...ert\to\infty}\frac{R(x,u,v)}{\vert u\vert^p+\vert v\vert^q}=0, \end{displaymath}

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 [Enhancements On Off] (What's this?)

  • 1. T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. TMA, 20 (1993), 1205-1216. MR 94g:35093
  • 2. T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems, Progress in Nonlinear Differential Equations and Their Applications, vol. 35, Birkhäuser, Basel/Switzerland, 1999, pp. 51-67. MR 2000j:35072
  • 3. V. Benci and P. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), 241-273. MR 80i:58019
  • 4. K. C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. MR 94e:58023
  • 5. D. G. Costa and C. A. Magalhães, A variational approach to noncooperative elliptic systems, Nonlinear Anal. TMA 25 (1995), 699-715. MR 96g:35070
  • 6. D. G. Costa and C. A. Magalhães, A unified approach to a class of strongly indefinite functionals, J. Differential Equations 125 (1996), 521-547. MR 96m:58061
  • 7. D. G. De Figueiredo and C. A. Magalhães, On nonquadratic Hamiltonian elliptic systems, Adv. Differential Equations 1 (1996), 881-898. MR 97f:35049
  • 8. D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994), 97-116. MR 94g:35072
  • 9. Y. H. Ding, Infinitely many entire solutions of an elliptic system with symmetry, Topological Methods in Nonlinear Anal. 9 (1997), 313-323. MR 99a:35062
  • 10. P. L. Felmer, Periodic solutions of ``superquadratic" Hamiltonian systems, J. Differential Equations 102 (1993), 188-207. MR 94c:58160
  • 11. J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114 (1993), 32-58. MR 94g:35073
  • 12. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, C. B. M. S. vol. 65, Amer. Math. Soc., Providence, RI, 1986. MR 87j:58024
  • 13. E. A. B. Silva, Nontrivial solutions for noncooperative elliptic systems at resonance, Electronic J. Differential Equations 6 (2001), 267-283. MR 2001j:35097
  • 14. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser, Boston, 1996. MR 97h:58037

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J50, 58E99

Retrieve articles in all journals with MSC (2000): 35J50, 58E99

Additional Information

D. G. De Figueiredo
Affiliation: IMECC-UNICAMP, Caixa Postal 6065, 13083-970 Campinas S.P. Brazil

Y. H. Ding
Affiliation: Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100080 Beijing, People’s Republic of China

Keywords: Elliptic system, multiple solutions, critical point theory
Received by editor(s): June 18, 2001
Published electronically: March 14, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society