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