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

摘要

We consider the boundary value problem Lu(x) = p(x)u(x) + g(x, u((0))(x), ... , u((2m-1))(x))u(x), xis an element of(0, pi), where (i) L is a 2mth order, self-adjoint, disconjugate ordinary differential operator on [0, pi], together with separated boundary conditions at 0 and pi; (ii) p is continuous and p greater than or equal to, 0 on [0, pi], while p not equivalent to 0 on any interval in [0, pi]; (iii) g: [0, pi] x R-2m --> R is continuous and there exist increasing functions xi(u), xi(1) : [0, infinity) --> [0, infinity) such that xi(u)(xi)greater than or equal tog(x, xi)greater than or equal toxi(1)(xigreater than or equal to0, lim g(x, xi) = 0, xi-->0 with lim1-->infinity xi(1)(t) = infinity (the non-linear term in (*) is superlinear as u(x) --> infinity). 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. (C) 2002 Elsevier Science (USA). All rights reserved. [References: 9]