Classification of positive solutions of semilinear elliptic equations

Jérôme Busca; Messoud Efendiev; S. Zelik

doi:10.1016/j.crma.2003.10.038

Partial Differential Equations

Classification of positive solutions of semilinear elliptic equations

Presented by: Haïm Brezis

Jérôme Busca ¹; Messoud Efendiev ²; S. Zelik ³

¹ CNRS and Ceremade, Université Paris Dauphine, place Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
² Lehrstuhl für Analysis und Modellierung, Universität zu Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
³ CNRS and laboratoire de mathématiques, Université de Poitiers, SP2MI, téléport 2, boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope cedex, France

Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 7-11.

Abstract
Résumé

We give a classification of all solutions of a general semilinear PDE in the positive quadrant of $ℝ^{2}$ .

Nous donnons une classification de toutes les solutions d'une EDP semi-linéaire générale dans le quadrant positif de $ℝ^{2}$ .

Received: 2003-10-24
Accepted: 2003-10-24
Published online: 2003-12-19

DOI: 10.1016/j.crma.2003.10.038

Author's affiliations:

Jérôme Busca ¹; Messoud Efendiev ²; S. Zelik ³

¹ CNRS and Ceremade, Université Paris Dauphine, place Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
² Lehrstuhl für Analysis und Modellierung, Universität zu Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
³ CNRS and laboratoire de mathématiques, Université de Poitiers, SP2MI, téléport 2, boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope cedex, France

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     author = {J\'er\^ome Busca and Messoud Efendiev and S. Zelik},
     title = {Classification of positive solutions of semilinear elliptic equations},
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     pages = {7--11},
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     number = {1},
     year = {2004},
     doi = {10.1016/j.crma.2003.10.038},
     language = {en},
}

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Jérôme Busca; Messoud Efendiev; S. Zelik. Classification of positive solutions of semilinear elliptic equations. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 7-11. doi : 10.1016/j.crma.2003.10.038. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.038/

References
Cited by

[1] A. Babin; M. Vishik Attractors of Evolutionary Equations, Stud. Math. Appl., vol. 25, North-Holland, Amsterdam, 1992

[2] H. Berestycki; L. Caffarelli; L. Nirenberg Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., Volume 50 (1997) no. 11, pp. 1089-1111

[3] H. Berestycki; L. Caffarelli; L. Nirenberg Further quantitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm Sup. Pisa Cl. Sci., Volume 25 (1998) no. 1–2, pp. 69-94

[4] H. Berestycki, M. Efendiev, S. Zelik, Dynamical approach for positive solutions of semilinear elliptic problems in unbounded domains, Preprint 01-11, Universitat Stuttgart, Mathematishes Institut, 2001

[5] H. Berestycki; P. Lions Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., Volume 82 (1983), pp. 313-345

[6] J. Busca; P. Felmer Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. Partial Differential Equations, Volume 13 (2001) no. 2, pp. 191-211

[7] J. Busca; B. Sirakov Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, Volume 163 (2000) no. 1, pp. 41-56

[8] M. Esteban; P.-L. Lions Existence and non-existence results for semilinear elliptic problems in unbounded domains, Coll. Progress in PDE, Pitman Res. Notes, vol. 249, 1991, pp. 1-14

[9] B. Gidas; W. Ni; L. Nirenberg Symmetry and related properties via the maximum principle, Comm. Math. Phys., Volume 6 (1981), pp. 883-901

[10] B. Gidas; W. Ni; L. Nirenberg Symmetry of positive solutions of nonlinear elliptic equations in $ℝ^{n}$ , Math. Anal Appl. Part A, Academic Press, New York, 1981, pp. 369-402

[11] M. Kwong Uniqueness of positive solutions of Δu−u+u^p=0 in $ℝ^{n}$ , Arch. Rational Mech. Anal., Volume 105 (1983), pp. 243-266

[12] R. Temam Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New-York, 1988

[13] A. Volpert; S. Khudyaev Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Nijhoff, Dordrecht, 1985

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