Global Behavior of the Components for the Second Order m-Point Boundary Value Problems. (24th December 2007) - An, Yulian; Ma, Ruyun
Abstract : We consider the nonlinear eigenvalue problemsu ″ + r f ( u ) = 0, 0 < t < 1, u ( 0 ) = 0, u ( 1 ) = ∑ i = 1 m − 2 α i u ( η i ), wherem ≥ 3, η i ∈ ( 0, 1 ), andα i > 0 fori = 1, …, m − 2, with∑ i = 1 m − 2 α i < 1 ;r ∈ ℝ ;f ∈ C 1 ( ℝ, ℝ ) . There exist two constantss 2 < 0 < s 1 such thatf ( s 1 ) = f ( s 2 ) = f ( 0 ) = 0 andf 0 : = lim u → 0 ( f ( u ) / u ) ∈ ( 0, ∞ ), f ∞ : = lim | u | → ∞ ( f ( u ) / u ) ∈ ( 0, ∞ ) . Using the global bifurcation techniques, we study the global behavior of the components of nodal solutions of the above problems.