Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Local bifurcation from the second eigenvalue of the Laplacian in a square

Authors: Manuel del Pino, Jorge García-Melián and Monica Musso
Journal: Proc. Amer. Math. Soc. 131 (2003), 3499-3505
MSC (2000): Primary 35B32, 35J25
Published electronically: April 1, 2003
MathSciNet review: 1991761
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this work we study local bifurcation from the branch of trivial solutions for a class of semilinear elliptic equations, at the second eigenvalue $\lambda_2$ of a square. We find that the bifurcation set can be locally described as the union of exactly four bifurcation branches of nontrivial solutions which cross the bifurcation point $(\lambda_2,0)$. We also compute the Morse index of the solutions in the four branches.

References [Enhancements On Off] (What's this?)

  • 1. AMBROSETTI A. AND PRODI G., A primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1993). MR 94f:58016
  • 2.OHME R., Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme, Math. Z. 127 (1972), 105-126. MR 47:910
  • 3. CRANDALL M.G. AND RABINOWITZ P.H., Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340. MR 44:5836
  • 4. KRASNOSELSKI M.A., Topological methods in the theory of nonlinear integral equations, MacMillan, New York (1964). MR 28:2414
  • 5. MARINO A., La biforcazione nel caso variazionale, Confer. Sem. Mat. Univ. Bari (132) (1973), 14pp. MR 50:1068
  • 6. RABINOWITZ P.H., A bifurcation theorem for potential operators, J. Funct. Anal. 25 (4) (1977), 412-424. MR 57:3928

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35B32, 35J25

Retrieve articles in all journals with MSC (2000): 35B32, 35J25

Additional Information

Manuel del Pino
Affiliation: Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Jorge García-Melián
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, c/ Astrofísico Fco. Sánchez S/N – 38271 La Laguna, Spain

Monica Musso
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24 – 10129 Torino, Italy

Keywords: Local bifurcation, multiple branches, double eigenvalue
Received by editor(s): June 2, 2002
Published electronically: April 1, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society