Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Strongly indefinite functionals and multiple solutions of elliptic systems

Author(s): D. G. De Figueiredo; Y. H. Ding
Journal: Trans. Amer. Math. Soc. 355 (2003), 2973-2989.
MSC (2000): Primary 35J50; Secondary 58E99
Posted: March 14, 2003
MathSciNet review: 1975408
Retrieve article in: PDF
This article is available free of charge

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 $p\in (1, 2^*)$ ,

, and $q\in (1, \infty)$ . So some supercritical systems are included. Nontrivial solutions are obtained. When

is even in

, 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

(resp.

). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.

References:

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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|