Bifurcations for a problem with jumping nonlinearities

Lucie Kárná; Milan Kučera

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 3, page 481-496
  • ISSN: 0862-7959

Abstract

top
A bifurcation problem for the equation Δ u + λ u - α u + + β u - + g ( λ , u ) = 0 in a bounded domain in N with mixed boundary conditions, given nonnegative functions α , β L and a small perturbation g is considered. The existence of a global bifurcation between two given simple eigenvalues λ ( 1 ) , λ ( 2 ) of the Laplacian is proved under some assumptions about the supports of the functions α , β . These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to λ ( 1 ) , λ ( 2 ) .

How to cite

top

Kárná, Lucie, and Kučera, Milan. "Bifurcations for a problem with jumping nonlinearities." Mathematica Bohemica 127.3 (2002): 481-496. <http://eudml.org/doc/249045>.

@article{Kárná2002,
abstract = {A bifurcation problem for the equation \[ \Delta u+\lambda u-\alpha u^++\beta u^-+g(\lambda ,u)=0 \] in a bounded domain in $^N$ with mixed boundary conditions, given nonnegative functions $\alpha ,\beta \in L_\infty $ and a small perturbation $g$ is considered. The existence of a global bifurcation between two given simple eigenvalues $\lambda ^\{(1)\},\lambda ^\{(2)\}$ of the Laplacian is proved under some assumptions about the supports of the functions $\alpha ,\beta $. These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to $\lambda ^\{(1)\}, \lambda ^\{(2)\}$.},
author = {Kárná, Lucie, Kučera, Milan},
journal = {Mathematica Bohemica},
keywords = {nonlinearizable elliptic equations; jumping nonlinearities; global bifurcation; half-eigenvalue; nonlinearizable elliptic equations; jumping nonlinearities; global bifurcation; half-eigenvalue},
language = {eng},
number = {3},
pages = {481-496},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bifurcations for a problem with jumping nonlinearities},
url = {http://eudml.org/doc/249045},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Kárná, Lucie
AU - Kučera, Milan
TI - Bifurcations for a problem with jumping nonlinearities
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 3
SP - 481
EP - 496
AB - A bifurcation problem for the equation \[ \Delta u+\lambda u-\alpha u^++\beta u^-+g(\lambda ,u)=0 \] in a bounded domain in $^N$ with mixed boundary conditions, given nonnegative functions $\alpha ,\beta \in L_\infty $ and a small perturbation $g$ is considered. The existence of a global bifurcation between two given simple eigenvalues $\lambda ^{(1)},\lambda ^{(2)}$ of the Laplacian is proved under some assumptions about the supports of the functions $\alpha ,\beta $. These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to $\lambda ^{(1)}, \lambda ^{(2)}$.
LA - eng
KW - nonlinearizable elliptic equations; jumping nonlinearities; global bifurcation; half-eigenvalue; nonlinearizable elliptic equations; jumping nonlinearities; global bifurcation; half-eigenvalue
UR - http://eudml.org/doc/249045
ER -

References

top
  1. Sur certains problemes elliptiques asymetriques avec poids indefinis, C. R. Acad. Sci., Paris, Ser. I, Math. 332 (2001), 215–218. (2001) MR1817364
  2. On some nonlinear Sturm-Liouville problems, J. Differ. Equations 26 (1977), 375–390. (1977) Zbl0331.34020MR0481230
  3. A Prüfer approach to half-linear Sturm-Liouville problems, Proc. Edinburgh Math. Soc. 41 (1998), 573–583. (1998) MR1697591
  4. On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh, Sect. A 76 (1977), 283–300. (1977) MR0499709
  5. Bifurcation of solutions to reaction-diffusion systems with jumping nonlinearities, Applied Nonlinear Analysis, A. Sequeira, H. Beirao da Veiga, J. H. Videman (eds.), Kluwer Academic/Plenum Publishers, 1999, pp. 79–96. (1999) MR1727442
  6. Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions, Fields Institute Comm. 25 (2000), 239–256. (2000) MR1759546
  7. Boundary value problems with jumping nonlinearities, Čas. Pěst. Mat. 101 (1976), 69–87. (1976) MR0447688
  8. Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983. (1983) MR0737190
  9. Bifurcation points of variational inequalities, Czechoslovak Math. J. 32 (1982), 208–226. (1982) MR0654057
  10. Reaction-diffusion systems: Stabilizing effect of conditions described by quasivariational inequalities, Czechoslovak Math. J. 47 (1997), 469–486. (1997) MR1461426
  11. Global Bifurcation in Variational Inequalities, Springer, New York, 1997. (1997) MR1438548
  12. Topics in Nonlinear Functional Analysis, Courant Institut, New York, 1974. (1974) Zbl0286.47037MR0488102
  13. Spectral analysis of variational inequalities, Comment. Math. Univ. Carolin. 27 (1986), 605–629. (1986) MR0873631
  14. Solvability and multiplicity results of variational inequalities, Comment. Math. Univ. Carolin. 30 (1989), 281–302. (1989) MR1014128
  15. Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1987), 487–513. (1987) MR0301587
  16. The Fučík spectrum of general Sturm-Liouville problems, J. Differ. Equations 161 (2000), 87–109. (2000) Zbl0976.34024MR1740358

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.