Fractional eigenvalues. (English) Zbl 1292.35193

The authors investigate the nonlocal eigenvalue problem \[ 2\int_{R^{n}}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|y-x|^{\alpha p}}dy+\lambda|u(x)|^{p-2}u(x)=0 \] in a bounded domain \(\Omega\subset \mathbb R^{n},\) \(p>2,\) \(n<\alpha p<n+p,\) as Euler-Lagrange equation, arising from the minimization of the fractional Rayleigh quotient and especially the cases of large \(p\) and the limit case \(p\rightarrow \infty.\) The solutions are treated in the viscosity sense. It is proved that positive viscosity solutions are unique up to a normalization and the first eigenvalue is isolated. For sign changing solutions strange phenomena are detected, caused by the influence of points far away appearing in the domain of integration for the non-local operator.
For sign changing solutions of the \(\infty\)-eigenvalue equation \(\max\{\mathcal{L}_{\infty}\) \({ u(x)},\) \( \mathcal{L}_{\infty}^{-}\) \(u(x)+\) \(\lambda u(x)\}=0\) it is needed to introduce the open sets \(\{u<0\}\) and the nodal line \(\{u=0\}.\) Strange phenomena occur in the case of the \(\infty\)-eigenvalue equation, such as nodal domains – which are the connected components of the open sets \(\{u>0\}\) and \(\{u<0\}\) – not having the same first \(\infty\)-eigenvalue, yet they all come from the same higher \(\infty\)-eigenfunction; the restriction of a higher eigenfunction to one of its nodal domains is not an \(\infty\)-eigenfunction for the nodal domain. The last phenomenon arises even in one-dimensional examples.


35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35R11 Fractional partial differential equations
35J60 Nonlinear elliptic equations
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