Some multiplicity results for an elastic beam equation at resonance (Q1261524)

This paper deals with multiplicity results for nonlinear elastic beam equation of the type \(-d^ 4 u/dx^ 4+\pi^ 4 u+g(x,u)=e(x)\) \(0<x<1\), \(u(0)=u(1)=u''(0)=u''(1)=0\), where \(g: [0,1]\times{\mathbf R}\to{\mathbf R}\) satisfies Carathéodory condition and \(e\in L^ 2[0,1]\). By combining the Lyapunov-Schmidt procedure with the technique of connected set, we establish several multiplicity results under suitable condition.