The aim of this thesis is to prove the existence of a weak solution for semilinear fractional elliptic equations given by \begin{eqnarray*}\left\{ \begin{array}{ll} (-\Delta)^s u=|u|^{p-2} u+ f(x),& \quad x\in \Omega,\\ u=0, & \quad x \in \R^n \backslash \Omega, %0, & \quad x \in \partial \Oemga, \end{array} \right. \end{eqnarray*} where $(-\Delta)^s$ denotes the fractional Laplace operator with $s \in (0,1],$ $n > 2s,$ $\Omega$ is an open bounded domain in $\R^n$ with $C^2$-boundary and $f \in L^2(\Omega).$ We are interested in extending this result for $s \in (0, 1]$ to $p$ greater than the critical Sobolev exponent where the compact embedding fails to hold. We shall make use of a new variational principle presented in \cite{Mo1} that allows one to deal with problems well beyond the compact structure.