Nonlinear elastic beam problems with the parameter near resonance
The article is devoted to the existence of solutions for the following fourth-order Dirichlet boundary value problem: \[ \begin{gathered} u^{(4)}(x)-\lambda u(x)=f(x,u(x))-h(x),\quad 0< x<1,\\ u(0)=u(1)=u'(0)=u'(1)=0, \end{gathered} \] where $\lambda$ is a parameter, $h\in C[0,1]$, and $f\in C([0,1]\times\mathbb{R},\mathbb{R})$. The nonlinearity $f$ is assumed to satisfy a weak monotonocity condition together with an asymptotic sublinear growth assumption. The authors first recall a spectrum result on the linear eigenvalue problem: \[ \begin{gathered} u^{(4)}(x)=\lambda u(x),\quad 0< x<1,\\ u(0)=u(1)=u'(0)=u'(1)=0, \end{gathered} \] a result which comes from their work [ibid. 16, 1176--1186 (2018; Zbl 07045499)]. Then the authors prove the existence of strict upper and lower solutions in case $h=0$. Finally The Rabinowitz global bifurcation technique at infinity is used to establish existence of connected component (second solution). An illustrating example with sublinear $f$ is supplied.
scientific article; zbMATH DE number 7045499
scientific article