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, Volume 65, Issue 4, pp 413–426 | Cite as

Bifurcation for a semilinear elliptic equation on ℛN with radially symmetric coefficients

  • Wolfgang Rother


We consider the semilinear eigenvalue problem\( - \Delta u - q(x)\left| u \right|^\sigma u = \lambda \cdot u\) on ℛN (N ≥ 2) (N≥2) and investigate the question under which conditions on the radially symmetric function q, λ=0 is a bifurcation point for this equation in H1, In H2 and in Lp for 2≤p≤+∞.


Eigenvalue Problem Number Theory Elliptic Equation Algebraic Geometry Bifurcation Point 
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  1. [1]
    HELLWIG,G.: Partial Differential Equations. Stuttgart: B. G. Teubner (1977)Google Scholar
  2. [2]
    ROTHER,W.: Bifurcation of nonlinear elliptic equations on ℛN. Bull. London Math. Soc. (to appear)Google Scholar
  3. [3]
    RUPPEN,H.-J.: The existence of infinitely many bifurcation branches. Proc. Roy. Soc. Edinburgh101A, 307–320 (1985)Google Scholar
  4. [4]
    STRAUSS,W.A.: Existence of Solitary Waves in Higher Dimensions. Commun. Math. Phys.55, 149–162 (1977)Google Scholar
  5. [5]
    STUART,C.A.: Bifurcation from the continuous spectrum in the L2-theory of elliptic equations on ℛN. Recent Methods in Nonlinear Analysis and Applications. Proc. of SAFA IV, Liguori, Napoli (1981)Google Scholar
  6. [6]
    STUART,C.A.: Bifurcation for Dirichlet problems without eigenvalues. Proc. London Math. Soc. (3),45, 169–192 (1982)Google Scholar
  7. [7]
    STUART,C.A.: Bifurcation from the essential spectrum. Lect. Notes in Math.,1017, 575–596 (1983)Google Scholar
  8. [8]
    STUART,C.A.: A variational approach to bifurcation in Lp on an unbounded symmetrical domain. Math. Ann.,263, 51–59 (1983)Google Scholar
  9. [9]
    STUART,C.A.: Bifurcation in LpN for a semilinear elliptic equation. Proc. London Math. Soc.,57, 511–541 (1988)Google Scholar
  10. [10]
    STUART, C.A.: Bifurcation from the essential spectrum for some non-compact non-linearities. Math. Meth. in Appl. Sci.(to appear)Google Scholar
  11. [11]
    ZHOU,H.-S., ZHU,X.P.: Bifurcation from the essential spectrum of superlinear elliptic equations. Appl. Analysis,28 51–61 (1988)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Wolfgang Rother
    • 1
  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthBund. Rep. Deutschland

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