Solution Branches for Non-linear Equilibrium Problems—Bifurcation and Domain Perturbations

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Abstract
Consider the following simple, but typical, example of a non-linear equilibrium (differential equation) problem: Δu = −λf(u) for (x, y) є D, u = 0 for (x, y) є D. Usually the eigenvalues of the (small u) linearized problem are simple, and each simple eigen value generates two solution branches for the full problem. However, the full problem nearly always has many other solution branches, and this paper describes how to find these other branches and why they arise.

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