Hostname: page-component-5447f9dfdb-ltchl Total loading time: 0 Render date: 2025-07-29T02:02:29.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Liouville theorems for elliptic inequalities and applications

Published online by Cambridge University Press:  14 November 2011

Isabeau Birindelli  and
Enzo Mitidieri
Isabeau Birindelli
Affiliation:
Dipartimento di Matematica, Università di Roma ‘La Sapienza’, Piazzale Aldo Moro 5, 00185 Rome, Italy
Enzo Mitidieri
Affiliation:
Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, Piazzale Europa 1, 34100 Trieste, Italy
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we prove nonexistence of positive C2 solutions for systems of semilinear elliptic inequalities, for polyharmonic semilinear inequalities in cones and, under better conditions on the nonlinearity, for bounded positive solutions of elliptic semilinear equations in half spaces. Using a blow-up argument, these results allow us to prove a-priori bounds for a class of semilinear elliptic systems of equations in bounded domains.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 6 , 1998 , pp. 1217 - 1247
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Agmon, S., Douglis, A. and Nirenberg, L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure Appl. Math. 12 (1959), 623–727.CrossRefGoogle Scholar
2Berestycki, H., Dolcetta, I. Capuzzo and Nirenberg, L.. Problèmes elliptiques indéfinis et théoremès de Liouville non-linéaires. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 945–50.Google Scholar
3Berestycki, H., Dolcetta, I. Capuzzo and Nirenberg, L.. Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4 (1995), 59–78.CrossRefGoogle Scholar
4Berestycki, H., Nirenberg, L. and Varadhan, S. R. S.. The principal eigenvalue and maximum principle for second order elliptic operator in general domains. Comm. Pure Appl. Math. 47 (1994), 47–92.Google Scholar
5Caristi, G. and Mitidieri, E.. Non existence of positive solutions for quasilinear equations. Adv. Differential Equations 2 (1997), 319–59.CrossRefGoogle Scholar
6Clément, Ph., de Figueiredo, D. G. and Mitidieri, E.. A priori estimates for positive solutions of semilinear elliptic systems via Hardy–Sobolev inequalities. In Nonlinear Partial Differential Equations, Pitman Research Notes in Mathematics 343, 3–91 (Harlow: Longman, 1996).Google Scholar
7Clément, Ph., Manasevich, R. and Mitidieri, E.. Positive solutions for a quasilinear system via blow up. Comm. Partial Differential Equations 18 (1993), 2071–106.CrossRefGoogle Scholar
8Dancer, E. N.. Some notes on the method of moving planes. Bull. Austral. Math. Soc. 46 (1992), 425–34.Google Scholar
9de Figueiredo, D. G. and Felmer, P.. A Liouville-type theorem for elliptic systems. Ann. Scuola Norm. Cl. Sci.(4) (1992), 387–97.Google Scholar
10Gidas, B. and Spruck, J.. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 35 (1981), 525–98.Google Scholar
11Gidas, B. and Spruck, J.. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (1981), 883–901.CrossRefGoogle Scholar
12Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1983).Google Scholar
13Mitidieri, E.. Non existence of positive solutions for semilinear elliptic systems in ℝn. Differential Integral Equations 9 (1996), 465–79.Google Scholar
14Nicolescu, M.. Les fonctions polyharmoniques (Paris: Herman, 1936).Google Scholar
15Serrin, J. and Zou, H.. Non-existence of positive solutions of Lane–Emden systems. Differential Integral Equations 9 (1996), 635–53.Google Scholar
16Souto, M. A. S.. A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems. Differential Integral Equations 8 (1995), 1245–58.Google Scholar
17Sweers, G.. Personal communication (1996).Google Scholar
18Treves, F.. Basic Linear Partial Differential Equations (New York: Academic Press, 1975).Google Scholar