Semilinear Neumann boundary value problems on a rectangle

We consider a semilinear elliptic equation Deltau + lambdaf(u) = 0, x is an element of Omega, partial derivativeu/partial derivativen = 0, x is an element of partial derivativeOmega, where Omega is a rectangle (0, a) x (0, b) in R-2. For balanced and unbalanced f, we obtain partial descriptions of global bifurcation diagrams in (lambda, u) space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when lambda --> infinity. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.