Solution curves and exact multiplicity results for 2mth order boundary value problems

For any integer m 2, we consider the 2mth order boundary value problem (−1)mu(2m)(x) = λg u(x) u(x), x ∈ (−1, 1), u(i)(−1) = u(i)(1) = 0, i= 0, . . . , m − 1, where λ ∈ R, and the function g :R→R is C1 and satisfies g(0) > 0, ±g (ξ ) > 0, ±ξ >0, together with some further conditions as |ξ|→∞. We obtain curves of nontrivial solutions of this problem, bifurcating from u = 0 at the eigenvalues of the linearised problem, and obtain the exact number of solutions of the problem for λ lying in various intervals in R.  2003 Elsevier Inc. All rights reserved