The existence solution for a fourth-order equation by using variational methods and critical point theory

In this paper by using variational methods and critical point theory, we obtain some sufficient conditions to guarantee that the following problem has infinitely many classical solutions. \begin{equation}\label{1.1} \left\{\begin{array}{ll} u^{(4)}(x)= \lambda h(x)k(u(x)) \ \ \ \mathrm{in}~~~ [0,1],\u(0)=u (0)=0,\u (1)=0,\qquad u (1)=\mu g(u(1)), \end{array}\right. \end{equation} where $h\in L^1([0,1])$ such that $h(x)\geq 0$ a.e. $x\in [0,1]$, $h\not\equiv 0,$ and $k:\mathbb{R}\rightarrow\mathbb{R}$ be a nonnegative continuous function.