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)^m u^(2m) (x) = ?g(u(x))u(x), x ? (-1, 1), u⁽ⁱ⁾ (-1) = u⁽ⁱ⁾ (1) = 0, i = 0,..., m - 1, where ? ? R, and the function g: R ? R is C¹ and satisfies g(0) > 0, ±g' (?) > 0, ±? > 0, together with some further conditions as ? ? 8. 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 ? lying in various intervals in R. © 2003 Elsevier Inc. All rights reserved.