A Non-Linear Elliptic Eigenvalue Problem

The solutions (λ, u(x, y)) of ∇2u = −f(u) in the rectangleD(δ) ≡ {(x, y)|0 < x Y < t, 0 < y < 1 + δ}which have u=0 on the boundary of D(δ) are considered in detail for the cases f(u)=sinh u and f(u)=u-u3. The trivial solution, where u=0, exists for all λ. Non-trivial solutions bifurcate from the trivial solution only when λ=λn(δ) (where λ(n, n = 1, 2, are the eigenvalues of the linearized problem in which f(u) is replaced by u). If λn(0) is not a simple eigenvalue (for exampkle ), λ2 then, for δ and λ−λn(δ) both small, secondary bifurcation occurs from the branches of non trivial solutions. Approximations to u(x, y) are found (i) near both the primary and secondary bifurcation points, where E ≡ ∫D(δ)|∇u2dx dy is small and (ii) when E ↑ ∞. Finite-difference solutions (computed using Newton's method), are found, and these match smoothly on to the appropriate analytical approximations as E ↑ 0 and as E ↑ ∞. Both the analystical and numerical techniques can be applied to wider classes of functions f(x, y; u) and boundary conditions.