  Recently, a great attention has been focused on the study of fractional and
nonlocal operators of elliptic type, both for the pure mathematical research
and in view of concrete real-world applications. We consider the following
nonlocal 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}
\int_{\mathbb{R}^n}p(x)
\bigg(\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dy\bigg)dx-\lambda
\int_\Omega |u(x)|^q dx, \qquad (1.1) \end{equation} where $\Omega$ is a
bounded domain in $\mathbb{R}^n, p :\mathbb{R}^n \rightarrow \mathbb{R}$ is a
given positive weight presenting a global minimum $p_0 >0$ at $a \in \Omega$
and $\lambda$ is a real constant. In this work we show that for $q=2$ the
infimum of (1.1) over the set $\{u\in \mathbb{H}_0^s(\Omega),
||u||_{L^{q_s}(\Omega)}=1\}$ does exist for some $k, s, \lambda$ and $n$ and
for $q\geq 2$ we study non-ground state solutions using the mountain pass
Theorem.
