Asymptotics of variational eigenvalues for a general nonlocal $p$-Laplacian with varying horizon

From the recent developing of nonlocal gradients with finite horizon $\delta >0$ based on general kernels, we introduce a new nonlocal $p$-Laplacian and study the eigenvalue problem associated with it. Furthermore, by virtue of $\mathrm{\Gamma }$-convergence arguments, we establish stability results of the solutions for varying horizon in the extreme cases $\delta \to 0^{+}$ and $\delta \to \mathrm{\infty }$, recovering the solutions for the local eigenvalue problem associated with the $p$-Laplacian, and the ones associated with the $H^{s,p}$-Laplacian, respectively.