Majorants for eigenvalues of Sturm-Liouville problems with potentials lying in balls of weighted spaces

We study the problem concerning an exact a priori majorant for the least eigenvalue of the Sturm-Liouville problem
$$ -y''+qy=\lambda y,\qquad y(0)=y(1)=0 $$
with a condition of the form $\displaystyle\int_0^1 rq^\gamma\,dx\leqslant 1$ on the potential, where the weight $r\in C(0,1)$ is uniformly positive on the interval $(0,1)$. We give a constructive proof that this majorant is attainable for all $\gamma>1$ and, for a certain natural extension of the class of admissible potentials, also for $\gamma=1$.
Bibliography: 9 titles.