Abstract
We show how monotonicity methods combined with infinite dimensional sandwich pairs can be used to solve very general systems of equations that are not semibounded.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Costa, D.G.: On a class of elliptic systems in R N. Electron. J. Differ. Equ., No. 07, approx. 14 pp. (electronic) (1994)
Costa D.G., Magalhães C.A.: A variational approach to subquadratic perturbations of elliptic systems. J. Differ. Equ. 111(1), 103–122 (1994)
Costa D.G., Magalhães C.A.: A variational approach to noncooperative systems. Nonlinear Anal. TMA 25, 699–715 (1995)
de Figueiredo D.G., Felmer P.L.: On superquadratic elliptic systems. Trans. Am. Math. Soc. 343(1), 99–116 (1994)
de Figueiredo D.G., Mitidieri E.: A maximim principle for an elliptic system and applications to semilinear problems. SIAM J. Math. Anal. 17, 836–849 (1986)
Furtado M.F., Maia L.A., Silva E.A.B.: Solutions for a resonant elliptic system with coupling in \({\mathbb R^ N}\). Commun. Partial Differ. Equ. 27(7–8), 1515–1536 (2002)
Hulshof J., Vander Vorst R.C.A.M.: Differential systems with strongly indefinite variational structure. J. Funct. Anal. 114, 32–58 (1993)
Jeanjean L.: On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on R N. Proc. R. Soc. Edinburgh A 129, 787–809 (1999)
Jeanjean L.: Local conditions insuring bifurcation from the continuous spectrum. Math. Z. 232, 651–664 (1999)
Kelley J.L.: General Topology. Van Nostrand Reinhold, UK (1955)
Kryszewski W., Szulkin A.: An infinite dimensional Morse theory with applications. Trans. Am. Math. Soc. 349, 3181–3234 (1997)
Li G., Yang J.: Asymptotically linear elliptic systems (English summary). Commun. Partial Differ. Equ. 29(5–6), 925–954 (2004)
Lazer A., McKenna P.: On steady-state solutions of a system of reaction-diffusion equations from biology. Nonlinear Anal. TMA 6, 523–530 (1982)
Schechter M.: New Saddle Point Theorems, Generalized Functions and Their Applications (Varanasi, 1991), pp. 213–219. Plenum, New York (1991)
Schechter M.: A generalization of the saddle point method with applications. Ann. Polon. Math. 57(3), 269–281 (1992)
Schechter M.: New linking theorems. Rend. Sem. Mat. Univ. Padova 99, 255–269 (1998)
Schechter M.: Linking Methods in Critical Point Theory. Birkhauser, Boston (1999)
Schechter M.: Sandwich pairs in critical point theory. Trans. Am. Math. Soc. 360(6), 2811–2823 (2008)
Schechter M.: Minimax Systems and Critical Point Theory. Birkhauser, Boston (2009)
Schechter M.: Infinite-dimensional linking. Duke Math. J. 94(3), 573–595 (1998)
Schechter M., Zou W.: Super-linear problems. Pacif. J. Math. 214(1), 145–160 (2004)
Schechter M., Zou W.: Weak linking. Nonlinear Anal. 55(6), 695–706 (2003)
de Silva E.A.B.: Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal. TMA 16, 455–477 (1991)
de Silva E.A.B.: Nontrivial solutions for noncooperative elliptic systems at resonance. In: Proceedings of the USA-Chile Workshop on Nonlinear Analysis (pp. 267–283). Via del Mar-Valparaiso (2000) (electronic). Electron. J. Differ. Equ. Conf., 6, Southwest Texas State University, San Marcos (2001)
de Silva E.A.B.: Existence and multiplicity of solutions for semilinear elliptic systems. NoDEA 1, 339–363 (1994)
Struwe M.: The existene of surfaces of constant mean curvature with free boundaries. Acta Math. 160, 19–64 (1988)
Struwe M.: Variational Methods, 2nd edn. Springer, Berlin (1996)
Tintarev, K.: Solutions to elliptic systems of Hamiltonian type in R N. Electron. J. Differ. Equ. 29, 11 pp. (1999)
Willem, M., Zou, W.: On a semilinear Dirichlet problem and a nonlinear Schrödinger equation with periodic potential (preprint)
Zhao P., Zhou W., Zhong C.: The existence of three nontrivial solutions of a class of elliptic systems. Nonlinear Anal. 49(3), 431–443 (2002) Ser. A: Theory Methods
Zou W.: Multiple solutions for asymptotically linear elliptic systems. J. Math. Anal. Appl. 255(1), 213–229 (2001)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Schechter, M. Noncooperative elliptic systems. Z. Angew. Math. Phys. 62, 649–666 (2011). https://doi.org/10.1007/s00033-010-0108-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-010-0108-x