STUDY OF THE BUCKLING OF A TAPERED ROD WITH THE GENUS OF A SET.

This paper, which can be considered a continual, ion of the papers [C. A. Stuart, J. Math. Pures Appl., 80 (2001), pp. 281-337] and [C. A. Stuart, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), pp. 729 7(14], is concerned with the study of the buckling of a tapered rod. This physical phenomenon leads to tile nonlinear eigenvalue problem {A(s)u'(s)}' + μsin u(s) = 0 for all s ∈ (0, 1), u(1) = lim A(s)u'(s) = 0, s → 0 ∫ [sup1 sub0] A(s)'(s)²ds < ∞ where A(s) ∈ C([0, 1]) is such t, hat, A(s) > 0 for all s > 0 and lim[s sub&rarr0] A(s)/s[supp] = L for some constants p ≥ 0 and L ∈ (0, ∞). We study the set of all solutions of tile problem and, in particular, find tile points μ ∈ R[sub+] such that bifurcation occurs at, (μ, 0). As was shown by Stuart in [J. Math. Pures Appl., 80 (2001), pp. 281 387], there is a number A(A) ≥ 0 such that, for μ ≤ A(A), μ ≡ 0 is the only solution of the problem, and it minimizes the energy in the space of all admissible configurations. For μ > A(A), the energy is minimized by a nontrivial solution. For 0 ≤ p < 2, bifurcation occurs at a discrete set of eigenvalues μ[subi], i ∈ N* = {1,2,...}, which satisfy μ1 A(A), μ[subi] < μ[subi]+1 for all i ∈ N* and lim[subi → ∞] μ[subi] = ∞. At p 2, changes occur. For 0 ≤ p le; 2, A(A) > 0, whereas A(A) = 0 for p > 2. For p 2, there is a number A[sube;(A) ∈ [A(A),∞) such that bifurcation occurs at every value μ ∈ [A[sube](A), ∞). In this paper, we show tile following points: • For p = 2, if A(A) < A[sube](A), bifurcation from the solution u ≡ 0 also occurs at a finite or countable set, of eigenvalues μ[subi] I ⊂ N*,... [ABSTRACT FROM AUTHOR]