Buckling of a tapered elastica

The nonlinear eigenvalue problem (1)-(3) below is a model for the buckling of a tapered elastic rod. The coefficient A is an element of C([0, 1]) is s uch that A(s) > 0 for s > 0 and there exist p greater than or equal to 0 an d L is an element of (0, infinity) such that lim(s-->o) A(s)/s(p) = L. For 0 less than or equal to p < 2, there is bifurcation only at values of it in a discrete subset of (0, infinity) whereas for p = 2 every point in the in terval [L/4, infinity) is a bifurcation point. Furthermore, at p = 2, one o bserves a dramatic change in the shape of the equilibrium configurations. L et u(u) not equivalent to 0 be a configuration which minimizes the energy. For 0 less than or equal to p < 2, lim(s-->o) u(u) (s) is an element of (-p i, pi), whereas for p greater than or equal to 2, lim(s-->o) u(u)(s) = +/-p i. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsev ier SAS.