Fractional Eigenvalues

We study the non-local eigenvalue problem $$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$for large values of $$p$$ and derive the limit equation as $$p\rightarrow \infty $$. Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.