Nonzero and positive solutions of a superlinear elliptic system

Mario Zuluaga Uribe

Archivum Mathematicum (2001)

  • Volume: 037, Issue: 1, page 63-70
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

Zuluaga Uribe, Mario. "Nonzero and positive solutions of a superlinear elliptic system." Archivum Mathematicum 037.1 (2001): 63-70. <http://eudml.org/doc/248745>.

@article{ZuluagaUribe2001,
abstract = {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.},
author = {Zuluaga Uribe, Mario},
journal = {Archivum Mathematicum},
keywords = {elliptic system; Leray-Schauder degree; maximum principle; elliptic system; Leray-Schauder degree; maximum principle},
language = {eng},
number = {1},
pages = {63-70},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Nonzero and positive solutions of a superlinear elliptic system},
url = {http://eudml.org/doc/248745},
volume = {037},
year = {2001},
}

TY - JOUR
AU - Zuluaga Uribe, Mario
TI - Nonzero and positive solutions of a superlinear elliptic system
JO - Archivum Mathematicum
PY - 2001
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 037
IS - 1
SP - 63
EP - 70
AB - 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.
LA - eng
KW - elliptic system; Leray-Schauder degree; maximum principle; elliptic system; Leray-Schauder degree; maximum principle
UR - http://eudml.org/doc/248745
ER -

References

top
  1. Adams R., Sobolev Spaces, Academic Press, 1975. (1975) Zbl0314.46030MR0450957
  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. (1983) Zbl0518.92017MR0710432
  3. Costa D. & Magalhães, A variational approach to subquadratic perturbations of elliptic systems, J. Differential Equations 111 (1994), No. 1, July 1, 103–122. (1994) MR1280617
  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. (1986) MR0846392
  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. (1995) MR1296110
  6. Krasnosel’skii M., Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, 1964. (1964) MR0159197
  7. Krasnosels’kii M. & Zabreico F., Geometrical Methods of Nonlinear Analysis, Springer-Verlag, 1984. (1984) MR0736839
  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. (1982) MR0664014
  9. Lin F. H., On the elliptic equation , Proc. Amer. Math. Soc. 95 (1985), 219–226. (1985) Zbl0584.35031MR0801327
  10. Mitidieri E., Nonexistence of positive solutions of semilinear elliptic systems in , Differential Integral Equations (in press). Zbl0848.35034MR1371702
  11. Mitidieri E., A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), 125–151. (1993) Zbl0816.35027MR1211727
  12. Naito M., A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), 211–214. (1984) Zbl0555.35044MR0750398
  13. Ni W. M., On the elliptic equation Δ u + K x u n + 2 n - 2 = 0 , its generalizations and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493–529. (1982) MR0662915
  14. Pucci P. & Serrin J., A general variational identity, Indiana Univ. Math. J. 35 (1986), 681–703. (1986) MR0855181
  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. (1981) Zbl0471.35031MR0613057
  16. Smoller J., Shock Waves And Reaction-Diffusion Equations, Springer-Verlag, 1983. (1983) Zbl0508.35002MR0688146
  17. Soto H. & Yarur C., Some existence results of semilinear elliptic equations, Rendiconti di Matematica 15 (1995), 109–123. (1995) MR1330182
  18. Yarur C., Nonexistence of positive solutions for a class of semilinear elliptic systems, Electron. J. Differential Equations 1996 (1996), No. 08, 1–22. (1996) MR1405040
  19. Zuluaga M., On a nonlinear elliptic system: resonance and bifurcation cases, Comment. Math. Univ. Carolin. 40 (1999), No. 4, 701–711. (1999) Zbl1064.35052MR1756546
  20. Zuluaga M., A nonlinear undecoupling elliptic system at resonance, Russian J. Math. Phys. 6 (1999), No. 3, 353–362. (1999) Zbl1059.35507MR1816949
  21. Zuluaga M., Nonzero solutions of a nonlinear elliptic system at resonance, Nonlinear Anal. 31 (1998), No. 3/4, 445–454. (1998) Zbl0921.35051MR1487555

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.