On a Fourth-Order Boundary Value Problem at Resonance

Abstract

We investigate the spectrum structure of the eigenvalue problem u4x=λux,  x∈0,1;  u0=u1=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 -u4x+λ1ux+gx,ux=hx,  x∈0,1;  u0=u1=u′0=u′1=0, which models a statically elastic beam with both end-points being cantilevered or fixed, where λ1 is the first eigenvalue of the corresponding eigenvalue problem and nonlinearity g may be unbounded.