A Complete Bifurcation Scenario for the 2-d Nonlinear Laplacian with Neumann Boundary Conditions on the Unit Square
Eugene L. Allgower, Klaus Böhmer, Zhen Mei · Birkhäuser Basel eBooks · 1991
In this paper we describe the complete scenario of solutions bifurcating from the trivial solution and their stability for a model problem in bifurcation analysis the two-dimensional Laplacian with Neumann boundary conditions on the unit square. We show that at a corank ρ bifurcation point, there are exactly (3 ρ −1)/2 different solution branches bifurcating front the trivial solution curve. Our results are easily extended to other boundary conditions, e.g. Dirichlet or Dirichlet and Neumann along different sides, etc. which can be embedded in the periodic boundary conditions.