= Solution
Fix one <self-adjoint> choice of separated homogeneous <Dirichlet boundary conditions> or <Neumann boundary conditions> at each endpoint. The stated product condition then makes the boundary term $[awy y']_\alpha^\beta$ vanish. Multiplication of $Ly=\lambda wy$ by $y$ and <integration by parts> give
$$
\lambda\int_\alpha^\beta wy^2dx=\int_\alpha^\beta yLy\,dx=\int_\alpha^\beta[aw(y')^2-wcy^2]dx.
$$
Hence define the <quadratic functionals>
$$
\boxed{F[y]=\int_\alpha^\beta yLy\,dx=\int_\alpha^\beta[aw(y')^2-wcy^2]dx,\qquad G[y]=\int_\alpha^\beta wy^2dx,}
$$
and $\Lambda[y]=F[y]/G[y]$ for nonzero admissible $y$. The first expression for $F$ is linear in the operator $L$, and $G$ is linear in the weight $w$.
To prove the <minimum> statement, expand an admissible function in a complete weighted orthonormal system of <eigenfunctions> of the <Sturm-Liouville problem>, $y=\sum_n c_ny_n$, with ordered <eigenvalues> $\lambda_0\leq\lambda_1\leq\cdots$. Self-adjointness and <orthogonality of Sturm-Liouville eigenfunctions> give
$$
\Lambda[y]=\frac{\sum_n\lambda_n c_n^2}{\sum_n c_n^2}\geq\lambda_0.
$$
Equality is attained by a lowest <eigenfunction>, so
$$
\boxed{\lambda_{\min}=\min_{y\ne0}\Lambda[y].}
$$
This is the <Rayleigh-Ritz variational principle>. The endpoint product condition is shorthand for fixed choices of boundary type: allowing each trial function to choose different endpoint types would not define one fixed <self-adjoint> <eigenvalue> problem.
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