On the eigenvalues and Fu{c}\'{}k spectrum of p-Laplace local and nonlocal operator with mixed interpolated Hardy term

In this article, we are concerned with the eigenvalue problem driven by the mixed local and nonlocal p-Laplacian operator having the interpolated Hardy term {equation*} T(u) :=- Δ_p u + (- Δ_p)^s u - μ {|u|ᵖ⁻²u}{|x|p θ}, {equation*} where $0<s<1<p<N$, θ ∈ [s,1], and μ ∈ (0,μ₀(θ)). First, we establish a mixed interpolated Hardy inequality and then show the existence of eigenvalues and their properties. We also investigate the Fu{c}\'{}k spectrum, the existence of the first nontrivial curve in the Fu{c}\'{}k spectrum, and prove some of its properties. Moreover, we study the shape optimization of the domain with respect to the first two eigenvalues, the regularity of the eigenfunctions, the Faber-Krahn inequality, and a variational characterization of the second eigenvalue.