A symmetry-breaking problem in the annulus

Abstract

We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ▵u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ▵z + [math001](u)z, z isin; C²$⁰([math001]), is non-trivial) and existence of nonradial solutions for the semi-linear equation. These nonradial (asymmetric) solutions are obtained via a bifurcation procedure from the radial (symmetric) ones. This phenomena is called symmetry-breaking. The bifurcation results are proved by a Conley index argument

Keywords:

• Symmetry-breaking
• annulus
• elliptic

Keywords:

• 34A10
• 35B05
• 35B30
• 35B32
• 35J25
• 35J60
• 35P99

¹Supported in part by AFOSR-88-0025 amd NSF-DMS-86-0035

²Current address: Department of Mathematics, North Carolina State University, Box 8205, Raleigh NC 27695-8205

¹Supported in part by AFOSR-88-0025 amd NSF-DMS-86-0035

²Current address: Department of Mathematics, North Carolina State University, Box 8205, Raleigh NC 27695-8205

Notes

¹Supported in part by AFOSR-88-0025 amd NSF-DMS-86-0035

²Current address: Department of Mathematics, North Carolina State University, Box 8205, Raleigh NC 27695-8205

Additional information

Notes on contributors

Xabier Garaizar

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