Bifurcation for some semilinear elliptic equations when the linearization has no eigenvalues

Wolfgang Rother

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 1, page 125-138
  • ISSN: 0010-2628

Abstract

top
We prove existence and bifurcation results for a semilinear eigenvalue problem in ℝ N ( N ≥ 2 ) , where the linearization — has no eigenvalues. In particular, we show that under rather weak assumptions on the coefficients λ = 0 is a bifurcation point for this problem in H 1 , H 2 and L p ( 2 ≤ p ≤ ∞ ) .

How to cite

top

Rother, Wolfgang. "Bifurcation for some semilinear elliptic equations when the linearization has no eigenvalues." Commentationes Mathematicae Universitatis Carolinae 34.1 (1993): 125-138. <http://eudml.org/doc/247520>.

@article{Rother1993,
abstract = {We prove existence and bifurcation results for a semilinear eigenvalue problem in $\mathbb \{R\}^N$$(N\ge 2)$, where the linearization — $\UnimplementedOperator $ has no eigenvalues. In particular, we show that under rather weak assumptions on the coefficients $\lambda =0$ is a bifurcation point for this problem in $H^1, H^2$ and $L^p$$(2\le p\le \infty )$.},
author = {Rother, Wolfgang},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {bifurcation point; variational method; eigenvalues; exponential decay; standing waves; existence; bifurcation},
language = {eng},
number = {1},
pages = {125-138},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Bifurcation for some semilinear elliptic equations when the linearization has no eigenvalues},
url = {http://eudml.org/doc/247520},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Rother, Wolfgang
TI - Bifurcation for some semilinear elliptic equations when the linearization has no eigenvalues
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 1
SP - 125
EP - 138
AB - We prove existence and bifurcation results for a semilinear eigenvalue problem in $\mathbb {R}^N$$(N\ge 2)$, where the linearization — $\UnimplementedOperator $ has no eigenvalues. In particular, we show that under rather weak assumptions on the coefficients $\lambda =0$ is a bifurcation point for this problem in $H^1, H^2$ and $L^p$$(2\le p\le \infty )$.
LA - eng
KW - bifurcation point; variational method; eigenvalues; exponential decay; standing waves; existence; bifurcation
UR - http://eudml.org/doc/247520
ER -

References

top
  1. Anderson D., Stability of time - dependent particle solutions in nonlinear field theories II, J. Math. Phys. 12 (1971), 945-952. (1971) 
  2. Berestycki H., Lions P.L., Nonlinear scalar field equations I: Existence of a ground state, Arch. Rat. Mech. Anal. 82 (1983), 313-345. (1983) Zbl0533.35029MR0695535
  3. Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, SpringerVerlag, Berlin, Heidelberg, New York, 1983. Zbl1042.35002MR0737190
  4. Hewitt E., Stromberg K., Real and Abstract Analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1975. Zbl0307.28001MR0367121
  5. Rother W., Bifurcation of nonlinear elliptic equations on ℝ N , Bull. London Math. Soc. 21 (1989), 567-572. (1989) MR1018205
  6. Rother W., Bifurcation of nonlinear elliptic equations on ℝ N with radially symmetric coefficients, Manuscripta Math. 65 (1989), 413-426. (1989) MR1019700
  7. Rother W., The existence of infinitely many solutions all bifurcating from λ = 0 , Proc. Royal Soc. Edinburgh 118A (1991), 295-303. (1991) Zbl0748.35029MR1121669
  8. Rother W., Nonlinear Scalar Field Equations, Differential and Integral Equations, to appear. Zbl0755.35082MR1167494
  9. Ruppen J.-H., The existence of infinitely bifurcation branches, Proc. Royal Soc. Edinburgh 101A (1985), 307-320. (1985) 
  10. Stampacchia G., Le probleème de Dirichlet pour les équations elliptique du second ordre à coefficients discontinues, Annls Inst. Fourier Univ. Grenoble 15 (1965), 189-257. (1965) MR0192177
  11. Stampacchia G., Équations elliptiques du second ordre à coefficients discontinues, Séminaire de Mathématiques Supérieurs, No. 16, Montreal, 1965. MR0251373
  12. Strauss W.A., Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55 (1977), 149-162. (1977) Zbl0356.35028MR0454365
  13. Stuart C.A., Bifurcation from the continuous spectrum in the L 2 - theory of elliptic equations on ℝ N , Recent Methods in Nonlinear Analysis and Applications, Proc. SAFA IV, Liguori, Napoli, 1981, pp. 231-300. MR0819032
  14. Stuart C.A., Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. (3) 45 (1982), 169-192. (1982) Zbl0505.35010MR0662670
  15. Stuart C.A., Bifurcation from the essential spectrum, Lecture Notes in Math. 1017 (1983), 575-596. (1983) Zbl0527.35010MR0726615
  16. Stuart C.A., Bifurcation in L p ( ℝ N ) for a semilinear elliptic equation, Proc. London Math. Soc. (3) 57 (1988), 511-541. (1988) MR0960098
  17. Stuart C.A., Bifurcation from the essential spectrum for some non-compact non-linearities, Math. Methods Appl. Sci. 11 (1989), 525-542. (1989) MR1001101
  18. Zhou H.-S., Zhu X.P., Bifurcation from the essential spectrum of superlinear elliptic equations, Appl. Analysis 28 (1988), 51-61. (1988) Zbl0621.35009MR0960586

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.