On a minimization problem for a functional generated by the Sturm–Liouville problem with integral condition on the potential

In this article we consider the minimization problem of the functional $R[Q,y]=\frac{\int_{0}^{1}y'^2dx- \int_{0}^{1}Q(x)y^2dx}{\int_{0}^{1}y^2dx}$ generated by a Sturm–Liouville problem with Dirichlet boundary conditions and with an integral condition on the potential. Estimation of the infimum of functional in some class of functions $y$ and $Q(x)$ is reduced to estimation of a nonlinear functional non depending on the potential $Q(x)$. This leads to related parameterized nonlinear boundary value problem. Upper and lower estimates for $\inf_{y\in H_{0}^{1}(0,1)}R[Q,y]$ are obtained for different values of parameter.