A weighted eigenvalue problem for mixed local and nonlocal operators with potential

We study an indefinite weighted eigenvalue problem for an operator of mixed-type (that includes both the classical p-Laplacian and the fractional p-Laplacian) in a bounded open subset RN \, (N2) with Lipschitz boundary, which is given by align* -ₚ u + (-ₚ) ˢu+V (x) |u|^p-2u&= g (x) |u|^p-2u~in~, u&=0~in~RN, align* where >0 is a parameter, exponents 0 0 a. e. in. Using the variational tools together with a weak comparison and strong maximum principles, we investigate the existence and uniqueness of principal eigenvalue and discuss its qualitative properties. Moreover, with the help of Ljusternik-Schnirelman category theory, it is proved that there exists a nondecreasing sequence of positive eigenvalues which goes to infinity. Further, we show that the set of all positive eigenvalues is closed, and eigenfunctions associated with every positive eigenvalue are bounded.