The branching theory of solutions of certain nonlinearelliptic partial differential equations is developed, when the nonlinearterm is perturbed from unforced to forced. We findfamilies of branching points and the associated nonisolated solutionswhich emanate from a bifurcation point of the unforced problem.Nontrivial solution branches are constructed which contain the nonisolated solutions, and the branching is exhibited. An iterationprocedure is used to establish the existence of these solutions, anda formal perturbation theory is shown to give asymptotically validresults. The stability of the solutions is examined and certainsolution branches are shown to consist of minimal positive solutions.Other solution branches which do not contain branching points arealso found in a neighborhood of the bifurcation point.
The qualitative features of branching points and theirassociated nonisolated solutions are used to obtain useful informationabout buckling of columns and arches. Global stability characteristicsfor the buckled equilibrium states of imperfect columns andarches are discussed. Asymptotic expansions for the imperfectionsensitive buckling load of a column on a nonlinearly elastic foundationare found and rigorously justified.
