J_\lambda$ is Fréchet differentiable at any point $u\in X_0$.

An eigenvalue problem for $-\mathcal L_K$ {#subsec:eigenvalue}
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Along the present paper we consider the following eigenvalue problem associated to the integro-differential operator $-\mathcal L_K$: $$\label{problemaautovalori}
\left\{\begin{array}{ll}
-\mathcal L_K u=\lambda\, u & \mbox{in } \Omega\\
u=0 & \mbox{in } {\mathbb R}^n\setminus \Omega\,.
\end{array}\right.$$ We denote by $\big\{ \lambda_k\big\}_{{k\in{\mathbb N}}}$ the sequence of the eigenvalues of the problem , with $$\label{lambdacrescente}
0<\lambda_1<\lambda_2\le \dots \le \lambda_k\le \lambda_{k+1}\le \dots$$ $${\mbox{$\lambda_k\to +\infty$ as $ k\to +\infty\,,$}}$$ and by $e_k$ the eigenfunction corresponding to $\lambda_k$. Moreover, we normalize $\big\{e_k\big\}_{{k\in{\mathbb N}}}$ in such a way that this sequence provides an orthonormal basis of $L^2(\Omega)$ and an orthogonal basis of $X_0$. For a complete study of the spectrum of the integro-differential operator $-\mathcal L_K$ we refer to [@sY Proposition 2.3], [@svlinking Proposition 9 and Appendix A] and [@servadeivaldinociBNLOW Proposition 4]. In particular, we recall that all eigenfunctions are Hölder continuous up to the boundary of $\Omega$.

Finally, we say that eigenvalue $\lambda_k$, $k\geq 2$, has multiplicity $m\in {\mathbb N}$ if $$\lambda_{k-1}<\lambda_k=\dots =\lambda_{k+m-1}<\lambda_{k+m}\,.$$ In this case the set of all the eigenvalues corresponding to $\lambda_k$ agrees with $$\mbox{span}\left\{e_k,\ldots,e_{k+m-1}\right\}\,.$$

In the following, for any $k\in {\mathbb N}$, we set $$\mathbb 