NONLINEAR ELLIPTIC EIGENVALUE PROBLEM - Budden, P; Norbury, J (1979)
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-u 3. The trivial solution, where u=0, exists for all λ. Non-trivial solutions bifurcate from the trivial solution only when λ=λ n(δ) (where λ( n, n = 1, 2, are the eigenvalues of the linearized problem in which f(u) is replaced by u). If λ n(0) is not a simple eigenvalue (for exampkle ), λ 2 then, for δ and λ-λ n(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)|∇u 2dx 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. © 1979 Academic Press Inc. (London) Ltd.