Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems

In this work, we consider the nonlinear eigenvalue problems u '' + ra(t) f(u) = 0, 0 < t < 1 u(0) = u(1) = 0 where a : [0, 1] -> [0, infinity) is continuous and a(t) not equivalent to 0 on any subinterval of [0, 1]; f : R -> R is continuous, and there exist two constants s(2) < 0 < s(1) such that f (s(1)) = f (s(2)) = f (0) = 0, f (s) > 0 for s epsilon (0, s(1)) U (s(1), + infinity), f (s) < 0 for S E (-infinity, s(2)) U (s(2), 0), the limits f(0) = lim(vertical bar s vertical bar -> 0) f(s)/s, f infinity = lim(vertical bar s vertical bar ->infinity) f(s)/s exist. Using global bifurcation techniques, we study s s the global behavior of the components of nodal solutions of the above asymptotically linear eigenvalue problems. (C) 2007 Elsevier Ltd. All rights reserved.