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

It is constructively proved that for class $A_{r,γ}=\{q\in L_{1,loc}(0,1): q\leq 0, \int_0^1 rq^γ\,dx\leqslant 1\}$, where $r\in C[0,1]$ is uniformly positive weight and $γ>1$, there exists a unique potential $\hat q\in A_{r,γ}$ such that minimal eigenvalue $λ_0(\hat q)$ of boundary problem $-y"+\hat qy=λy, y(0)=y(1)=0$ is equal to $M_{r,γ}=\sup_{q\in A_{r,γ}}λ_0(q)$. For case $γ=1$ we obtain that there exists a unique potential $\hat q\inΓ_{r,γ}$ with analogous property. Here $Γ_{r,γ}$ is a closure of $A_{r,γ}$ in the space $W_{2,loc}^{-1}(0,1)$ of generalized functions.

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