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Asymptotic behaviour of ground states

Authors: Josephus Hulshof and Robertus C. A. M. van der Vorst
Journal: Proc. Amer. Math. Soc. 124 (1996), 2423-2431
MSC (1991): Primary 35J55; Secondary 34C37
MathSciNet review: 1363170
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Abstract: We derive the asymptotic behaviour of the ground states of a system of two coupled semilinear Poisson equations with a strongly indefinite variational structure in the critical Sobolev growth case.

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  • 1. Brezis, H. and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477. MR 84h:35059
  • 2. Carr, J., Applications of centre manifold theory, Appl. Math. Sc. 35, Springer Verlag, 1981. MR 83g:34039
  • 3. Chow, S.N. and J.K. Hale, Methods of Bifurcation Theorem, Grundl. der math. Wiss 251, Springer Verlag, 1982. MR 84e:58019
  • 4. Hulshof, J., Similarity solutions of the porous medium equation with sign changes, J. Math. Anal. Appl., 157, 1991, 75-111. MR 92f:35082
  • 5. Hulshof, J., Similarity solutions of the $k-\epsilon $ model for turbulence, Report W93-11, Leiden University, 1993. (To appear as Similarity solutions of Barenblatt's model for turbulence, SIAM J. Math. Anal.)
  • 6. Hulshof, J., A local analysis of similarity solutions of the thin film equation, Report W94-22, Leiden University, 1994.
  • 7. Hulshof, J, E. Mitidieri and R.C.A.M. van der Vorst, Strongly indefinite systems with critical Sobolev exponents, Report W95-15, Leiden University, 1995.
  • 8. Hulshof, J. and R.C.A.M. van der Vorst, Positive solutions of the equation $\Delta u+u^p=0$ on a bounded domain, course notes, Report W93-06, Leiden University, 1993.
  • 9. Hulshof, J. and R.C.A.M. van der Vorst, Differential Systems with Strongly Indefinite Variational Structure, J. Funct. Anal., 114, 1993, 32-58. MR 94g:35073
  • 10. Lieb, E., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), 349-374. MR 86j:42010
  • 11. Lions, P.L., The concentration-compactness principle in the calculus of variations, part 1, Rev. Mat. Iberoam. 1 (1985), 145-201. MR 87c:49007
  • 12. Mitidieri, E., A Rellich type identity and applications, Comm. PDE 18 (1993), 125-151. MR 94c:26016
  • 13. Van der Vorst, R.C.A.M., Best constant for the embedding of the space $H^2 \cap H_0^1$ into $L^{2N/(N-4)}$, Differential and Integral Equations 6 (1992), 259-276. MR 94b:46053
  • 14. Wang, X.J., Sharp constant in a Sobolev inequality, Nonlinear Anal. 20 (1993), 261-268. MR 94g:35035
  • 15. Felmer, P. and D. G. de Figueiredo, On super-quadratic elliptic systems, Trans. A. M. S. 343 (1994), 99--116.

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Additional Information

Josephus Hulshof
Affiliation: Mathematical Department of the Leiden University, Niels Bohrweg 1 2333 CA Leiden, The Netherlands

Robertus C. A. M. van der Vorst
Affiliation: Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30308-0190

Keywords: Systems, strongly indefinite variational structure, critical Sobolev growth, ground states, asymptotic behaviour, transformation to $3$- and $4$-dimensional quadratic systems, heteroclinic orbits, critical point analysis.
Received by editor(s): February 16, 1995
Additional Notes: We gratefully acknowledge the support by N.W.O., the Dutch Organisation for Scientific Research, and Enzo Mitidieri for his encouragement.
Communicated by: Jeffrey Rauch
Article copyright: © Copyright 1996 American Mathematical Society