  In this paper we discuss the existence and non-existence of weak solutions to
parametric fractional equations involving the square root of the Laplacian
$A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$)
and with zero Dirichlet boundary conditions. Namely, our simple model is the
following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda
f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega.
\end{array}\right. \end{equation*} The existence of at least two non-trivial
$L^{\infty}$-bounded weak solutions is established for large value of the
parameter $\lambda$ requiring that the nonlinear term $f$ is continuous,
superlinear at zero and sublinear at infinity. Our approach is based on
variational arguments and a suitable variant of the Caffarelli-Silvestre
extension method.
