Multiple solutions for a $2$nth-order differential equation with Sturm-Liouville operator

Boundary value problems for higher-order nonlinear differential equations have been important in the flexibility mechanics and engineering physics. By using a variational method for a $2$nth-order differential equation with Sturm-liouville operator depending on a parameter $\lambda$, under some assumptions, we verify the existence of at least three solutions when $\lambda$ lies in two exactly determined open intervals, respectively. In our work, we shall look for local minima for the Euler functional correspondent to the problem. Meanwhile, as an application of our results, an example is given.