  This paper is devoted to the study of the existence of positive solutions for
a problem related to a higher order fractional differential equation involving
a nonlinear term depending on a fractional differential operator,
$$(-\Delta)^{\alpha} u=\lambda u+ (-\Delta)^{\beta}|u|^{p-1}u \quad
\mbox{in}\quad \Omega;\qquad (-\Delta)^{j}u=0\quad \mbox{on}\quad
\partial\Omega,\quad \mbox{for}\quad j\in\mathbb{Z},\: 0\leq j< [\alpha]$$
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $0<\beta<1$,
$\beta<\alpha<\beta+1$ and $\lambda>0$. In particular, we study the fractional
elliptic problem, $$ (-\Delta)^{\alpha-\beta} u= \lambda(-\Delta)^{-\beta}u+
|u|^{p-1}u \quad\mbox{in} \quad \Omega;\qquad u=0 \quad \hbox{on} \quad
\partial\Omega,$$ and we prove existence or nonexistence of positive solutions
depending on the parameter $\lambda>0$, up to the critical value of the
exponent $p$, i.e., for $1<p\leq 2_{\mu}^*-1$ where $\mu:=\alpha-\beta$ and
$2_{\mu}^*=\frac{2N}{N-2\mu}$ is the critical exponent of the Sobolev
embedding.
