On the existence of solutions to one-dimensional fourth-order equations (Q2034137)

In this paper, the authors consider the following fourth-order boundary-value problem: \begin{align*} & u^{(4)}h(x,u')-u''=[\lambda f(x,u)+g(u)]h(x,u')\ in\ ]0, 1[\\ & u(0)=u(1)=0=u''(0)=u''(1), \tag{1}\end{align*} where \(\lambda\) is a positive parameter, \(f:[0, 1]\times\mathbb{R}\longrightarrow\mathbb{R}\) is an \(L^{1}-\)Caratheodory function, \(g:\mathbb{R}\longrightarrow\mathbb{R}\) is a Lipschitz continuous function and \(h:[0, 1]\times\mathbb{R}\longrightarrow [0,\infty[\) is a bounded and continuous function. Using variational methods and Ricceri's variational principle, they prove under suitable conditions on \(\lambda,f,g,h\) that problem (1) admits at least one nontrivial weak solution.