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

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,p), (*) where (i) L is a 2mth order, self-adjoint, disconjugate ordinary differential operator on [0, p], together with separated boundary conditions at 0 and p; (ii) p is continuous and p = 0 on [0, p], while p ? 0 on any interval in [0, p]; (iii) g: [0, p] × R^2m ?R is continuous and there exist increasing functions ?_u, ?_l : [0, 8) ? [0, 8) such that with lim_t?8 ?_l(t) = 8 (the non-linear term in (*) is superlinear as u(x) ? 8). 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.