On a fourth-order boundary value problem at resonance (Q2360755)

In this paper, the authors consider the following fourth-order boundary value problem at resonance \[ \begin{cases} &-u^{(4)}(x)+\lambda_1 u(x)+g(x,u(x))=h(x),\;x\in (0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0,\end{cases} \] which models a statically elastic beam with both end-points being fixed, where \(\lambda_1\) is the first eigenvalue of the corresponding eigenvalue problem \[ \begin{cases} &-u^{(4)}(x)+\lambda_1 u(x)=0,\;x\in (0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0. \end{cases} \] Firstly, they show that the eigenvalues of the above eigenvalue problem are simple and form a sequence \(0<\lambda_1<\lambda_2<\lambda_3<\cdots \rightarrow \infty\). Then, by using Leray-Schauder continuation method, they obtain the existence of solutions with unbounded nonlinearity \(g\).