Some multiplicity results for an elastic beam equation at resonance. (English) Zbl 0776.73037

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.


74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34B15 Nonlinear boundary value problems for ordinary differential equations
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