In this paper we analyse possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a non-local eigenvalue problem of fractional type that approximates, when taking a suitable limit, the classical Steklov eigenvalue problem.

Using variational methods and depending on a parameter $\unicode[STIX]{x1D706}$ we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$:

$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}=0$

$\unicode[STIX]{x1D6FE}=1$

In this paper, starting from classical non-convex and nonlocal3D-variational model of the electric polarization in a ferroelectricmaterial, via an asymptotic process we obtain a rigorous2D-variational model for a thin film. Depending on the initial boundaryconditions, the limit problem can be either nonlocal or local.

We first prove an abstract result for a class of nonlocalproblems using fixed point method. We apply this result toequations revelant from plasma physic problems. These equationscontain terms like monotone or relative rearrangement of functions.So, we start the approximation study by using finite element todiscretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.