On the eigenvalues and Fučík 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 \begin{equation*} $\mathcal{T}$(u) :=- Δ_p u + (- Δ_p)^s u - μ\frac{|u|^{p-2}u}{|x|^{p θ}}, \end{equation*} where $0<s<1<p<N$, $\theta \in [s,1]$, and $\mu \in (0,{\mu }_0(\theta ))$. First, we establish a mixed interpolated Hardy inequality and then show the existence of eigenvalues and their properties. We also investigate the Fučík spectrum, the existence of the first nontrivial curve in the Fučík 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.