In recent years, a great attention has been focused on studying fractional and nonlocal operators of elliptic type. This type of operator arises in a quite natural way in many different applications, such as the problem of thin obstacles, phase transition phenomena, population dynamics and game theory.
We consider the non local minimizing problem on $\mathbb{H}_0^s(\Omega) \subset L^{q_s}(\Omega)$, with $q_s=\frac{2n}{n-2s}$, $s\in ]0, 1[$ and $n\geq 3$:
\begin{equation}\label{premiereequation}
\inf_{\substack{u\in \mathbb{H}_0^s(\Omega) \\ ||u||_{L^{q_s}(\Omega)}=1}}\displaystyle\int_{\Omega}p(x) \displaystyle\int_{\R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dydx-\lambda \displaystyle\int_\Omega |u(x)|^2dx,
\end{equation}
where $\Omega$ is a bounded domain in $\R^n, \ p:\bar{\Omega} \rightarrow \R$ is a given positive weight presenting a global minimum $p_0 >0$ at $a \in \bar{\Omega}$ and $\lambda$ is a real constant. The objective of this work is to show that minimizers do exist for some $k, s, \lambda$ and $n$.
