Half-linear Sturm-Liouville problem with weights

We prove a necessary and sufficient conditions for discreteness of the set of all eigenvalues (with the usual Sturm--Liouville properties) of half--linear eigenvalue problem with locally integrable weights. Our conditions appear to be equivalent to the compact embedding of certain weighted Sobolev and Lebesgue spaces. Every eigenvalue allows the variational characterization of Ljusternik--Schnirelmann type.

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