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,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) is continuous and there exist increasing functions ζu, ζl : [0,∞)→[0,∞) such that with limt→∞ ζl(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.