*Accepted Paper*

**Inserted:** 19 sep 2014

**Last Updated:** 2 mar 2016

**Journal:** Adv. Calc. Var.

**Pages:** 37

**Year:** 2015

**Abstract:**

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian}
in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem
\[
\inf \{\lambda_2(\Omega)\,:\,

\Omega

=c\}.
\]
We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.

**Keywords:**
spectral optimization, quasilinear nonlocal operators, nonlocal eigenvalue problems, Caccioppoli estimates

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