S-Shaped Connected Component for Nonlinear Fourth-Order Problem of Elastic Beam Equation. (24th July 2017) - Wang, Jinxiang; Ma, Ruyun; Wen, Jin
Abstract : We investigate the existence of S -shaped connected component in the set of positive solutions of the fourth-order boundary value problem: u ′ ′ ′ ′ x = λ h x f u x, x ∈ ( 0, 1 ), u ( 0 ) = u ( 1 ) = u ′ ′ 0 = u ′ ′ 1 = 0, where λ > 0 is a parameter, h ∈ C [ 0, 1 ], and f ∈ C [ 0, ∞ ) with f 0 ≔ lim s → 0 ⁡ ( f ( s ) / s ) = ∞ . We develop a bifurcation approach to deal with this extreme situation by constructing a sequence of functions f [ n ] satisfying f [ n ] → f and ( f [ n ] ) 0 → ∞ . By studying the auxiliary problems, we get a sequence of unbounded connected components C [ n ], and, then, we find an unbounded connected component C in the set of positive solutions of the fourth-order boundary value problem which satisfies 0, 0 ∈ C ⊂ lim ⁡ sup ⁡ C [ n ] and is S -shaped.