Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems. (2010) - Xu Xu, Jia Jia; Han Han, XiaoLing XiaoLing
Abstract : We consider the fourth-order two-point boundary value problem u ′′′′ + M u = λ h ( t ) f ( u ), 0 < t < 1, u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = u ′ ( 1 ) = 0, where λ ∈ ℝ is a parameter, M ∈ ( - π 4, π 4 / 64 ) is given constant, h ∈ C ( [ 0, 1 ], [ 0, ∞ ) ) with h ( t ) ≢ 0 on any subinterval of [ 0, 1 ], f ∈ C ( ℝ, ℝ ) satisfies f ( u ) u > 0 for all u ≠ 0, and lim   u → - ∞ f ( u ) / u = 0, lim   u → + ∞ f ( u ) / u = f + ∞, lim   u → 0 f ( u ) / u = f 0 for some f + ∞, f 0 ∈ ( 0, + ∞ ) . By using disconjugate operator theory and bifurcation techniques, we establish existence and multiplicity results of nodal solutions for the above problem.