Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems

Abstract

We consider the boundary value problem Lu(x) = p(x)u(x) + g(x, u (0) (x),...,u (2m-1 )(x))u(x), x∈(0,π), (*) where (i) L is a 2mth order, self-adjoint, disconjugate ordinary differential operator on [0, π], together with separated boundary conditions at 0 and π; (ii) p is continuous and p ≥ 0 on [0, π], while p ≢ 0 on any interval in [0, π]; (iii) g: [0, π] × ℝ 2m →ℝ is continuous and there exist increasing functions ζ u , ζ l : [0, ∞) → [0, ∞) such that with lim t→∞ ζ l (t) = ∞ (the non-linear term in (*) is superlinear as u(x) → ∞). We obtain a global bifurcation result for a related bifurcation problem. We then use this to obtain infinitely many solutions of (*) having specified nodal properties. © 2002 Elsevier Science (USA). All rights reserved.