On a Fourth-Order Boundary Value Problem at Resonance. (11th June 2017) - Xu, Man; Ma, Ruyun
Abstract : We investigate the spectrum structure of the eigenvalue problem u 4 x = λ u x,     x ∈ 0, 1 ;     u 0 = u 1 = u ′ 0 = u ′ 1 = 0 . As for the application of the spectrum structure, we show the existence of solutions of the fourth-order boundary value problem at resonance - u 4 x + λ 1 u x + g x, u x = h x,     x ∈ 0, 1 ;     u 0 = u 1 = u ′ 0 = u ′ 1 = 0, which models a statically elastic beam with both end-points being cantilevered or fixed, where λ 1 is the first eigenvalue of the corresponding eigenvalue problem and nonlinearity g may be unbounded.