On the first eigenvalue of the Sturm–Liouville problem with a weighted integral condition on the potential

In this paper, we consider the Sturm–Liouville problem on the segment $[0,1]$ with the Dirichlet boundary conditions and a weighted integral condition on the potential, which allows the potential to have different orders of singularities at the endpoints of the segment $[0,1]$. We obtain an additional integral condition for the potential under which the first eigenvalue of the problem exists. For values of the parameters of the weighted integral condition that provide the existence of potentials satisfying both integral conditions, we examine estimates of the first eigenvalue of the problem.