Nonlinear elastic beam problems with the parameter near resonance

Abstract

AbstractIn this paper, we consider the nonlinear fourth order boundary value problem of the form$$ \begin{array}\text \left\{ \begin{aligned} &u^{(4)}(x)-\lambda u(x)=f(x, u(x))-h(x), \ \ x\in (0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0,\\ \end{aligned}\right. \end{array} $$which models a statically elastic beam with both end-points cantilevered or fixed. We show the existence of at least one or two solutions depending on the sign of λ−λ1, where λ1 is the first eigenvalue of the corresponding linear eigenvalue problem and λ is a parameter. The proof of the main result is based upon the method of lower and upper solutions and global bifurcation techniques.