On majorants of eigenvalues of Sturm-Liouville problems with potentials from balls of weighted spaces

It is constructively proved that for class $A_{r,gamma}={qin L_{1,loc}(0,1): qleq 0, int_0^1 rq^gamma,dxleqslant 1}$, where $rin C[0,1]$ is uniformly positive weight and $gamma>1$, there exists a unique potential $hat qin A_{r,gamma}$ such that minimal eigenvalue $lambda_0(hat q)$ of boundary problem $$-y+hat qy=lambda y, y(0)=y(1)=0 $$ is equal to $M_{r,gamma}=sup_{qin A_{r,gamma}}lambda_0(q)$. For case $gamma=1$ we obtain that there exists a unique potential $hat qinGamma_{r,gamma}$ with analogous property. Here $Gamma_{r,gamma}$ is a closure of $A_{r,gamma}$ in the space $W_{2,loc}^{-1}(0,1)$ of generalized functions.