On a criterion for discrete or continuous spectrum of pp-Laplace eigenvalue problems with singular sign-changing weights

In this paper we study the existence of spectrum for one-dimensional pp-Laplace eigenvalue problems with singular weights subject to Dirichlet boundary condition. For certain class of singular sign-changing weights, we prove the existence of discrete spectrum. Proofs are based on the C1[0,1]C1[0,1]-regularity of solutions and construction of the first eigenvalue in a variational set-up. We also present an example of a weight for which the spectrum is continuous and any corresponding eigenfunction does not belong to C1[0,1]C1[0,1].