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 Ω⊂ RN \,(N≥2) with { Lipschitz boundary} ∂ Ω, which is given by {align*} -Δ_p u + (-Δ_p)^su+V(x)|u|ᵖ⁻²u&=λ g(x)|u|ᵖ⁻²u~in~Ω, u&=0~in~R^NΩ, {align*} where λ >0 is a parameter, exponents $0<s<1<p<N$, and V, g∈ Lq(Ω) for q∈ (N/sp, ∞) with V≥ 0, g > 0 a.e. in Ω. 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.