Previous |  Up |  Next

Article

Title: Nonzero and positive solutions of a superlinear elliptic system (English)
Author: Zuluaga, Mario
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 37
Issue: 1
Year: 2001
Pages: 63-70
Summary lang: English
.
Category: math
.
Summary: In this paper we consider the existence of nonzero solutions of an undecoupling elliptic system with zero Dirichlet condition. We use Leray-Schauder Degree Theory and arguments of Measure Theory. We will show the existence of positive solutions and we give applications to biharmonic equations and the scalar case. (English)
Keyword: elliptic system
Keyword: Leray-Schauder degree
Keyword: maximum principle
MSC: 35B05
MSC: 35J45
MSC: 35J55
MSC: 35J60
MSC: 47H11
MSC: 47N20
idZBL: Zbl 1090.35525
idMR: MR1822763
.
Date available: 2008-06-06T22:28:21Z
Last updated: 2012-05-10
Stable URL: http://hdl.handle.net/10338.dmlcz/107787
.
Reference: [1] Adams R.: Sobolev Spaces.Academic Press, 1975. Zbl 0314.46030, MR 0450957
Reference: [2] Brown K. J.: Spatially inhomogeneous steady-state solutions for systems of equations describing interacting populations.J. of Math. Anal. and Appl. 95 (1983), 251–264. Zbl 0518.92017, MR 0710432
Reference: [3] Costa D. & Magalhães: A variational approach to subquadratic perturbations of elliptic systems.J. Differential Equations 111 (1994), No. 1, July 1, 103–122. MR 1280617
Reference: [4] De Figueiredo D. & Mitidieri E.: A maximum principle for an elliptic system and applications to semilinear problems.SIAM J. Math. Anal. 17 (1986), 836–849. MR 0846392
Reference: [5] Fleckinger J., Hernández J. & Thelin F. de: On maximum principle and existence of positive solutions for some cooperative elliptic systems.Differential and Integral Equations 8 (1995), 69–85. MR 1296110
Reference: [6] Krasnosel’skii M.: Topological Methods in the Theory of Nonlinear Integral Equations.Pergamon Press, 1964. MR 0159197
Reference: [7] Krasnosels’kii M. & Zabreico F.: Geometrical Methods of Nonlinear Analysis.Springer-Verlag, 1984. MR 0736839
Reference: [8] Lazer A. & Mckena P. J.: On steady-state solutions of a system of reaction-diffusion equations from biology.Nonlinear Anal. 6 (1982), 523–530. MR 0664014
Reference: [9] Lin F. H.: On the elliptic equation.$D_{i}\left[ a_{ij}D_{j}U\right] -k\left( x\right) U+k\left( x\right) U^{p}=0,$ Proc. Amer. Math. Soc. 95 (1985), 219–226. Zbl 0584.35031, MR 0801327
Reference: [10] Mitidieri E.: Nonexistence of positive solutions of semilinear elliptic systems in.$\mathbb{R}^{n},$ Differential Integral Equations (in press). Zbl 0848.35034, MR 1371702
Reference: [11] Mitidieri E.: A Rellich type identity and applications.Comm. Partial Differential Equations 18 (1993), 125–151. Zbl 0816.35027, MR 1211727
Reference: [12] Naito M.: A note on bounded positive entire solutions of semilinear elliptic equations.Hiroshima Math. J. 14 (1984), 211–214. Zbl 0555.35044, MR 0750398
Reference: [13] Ni W. M.: On the elliptic equation $\Delta u+K \left( x\right) u^{\frac{n+2}{n-2}}=0,$ its generalizations and applications in geometry.Indiana Univ. Math. J. 31 (1982), 493–529. MR 0662915
Reference: [14] Pucci P. & Serrin J.: A general variational identity.Indiana Univ. Math. J. 35 (1986), 681–703. MR 0855181
Reference: [15] Rothe F.: Global existence of branches of stationary solutions for a system of reaction-diffusion equations from biology.Nonlinear Anal. 5 (1981), 487–498. Zbl 0471.35031, MR 0613057
Reference: [16] Smoller J.: Shock Waves And Reaction-Diffusion Equations.Springer-Verlag, 1983. Zbl 0508.35002, MR 0688146
Reference: [17] Soto H. & Yarur C.: Some existence results of semilinear elliptic equations.Rendiconti di Matematica 15 (1995), 109–123. MR 1330182
Reference: [18] Yarur C.: Nonexistence of positive solutions for a class of semilinear elliptic systems.Electron. J. Differential Equations 1996 (1996), No. 08, 1–22. MR 1405040
Reference: [19] Zuluaga M.: On a nonlinear elliptic system: resonance and bifurcation cases.Comment. Math. Univ. Carolin. 40 (1999), No. 4, 701–711. Zbl 1064.35052, MR 1756546
Reference: [20] Zuluaga M.: A nonlinear undecoupling elliptic system at resonance.Russian J. Math. Phys. 6 (1999), No. 3, 353–362. Zbl 1059.35507, MR 1816949
Reference: [21] Zuluaga M.: Nonzero solutions of a nonlinear elliptic system at resonance.Nonlinear Anal. 31 (1998), No. 3/4, 445–454. Zbl 0921.35051, MR 1487555
.

Files

Files Size Format View
ArchMathRetro_037-2001-1_8.pdf 325.6Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo