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This paper is devoted to the study of the existence of positive solutions fora problem related to a higher order fractional differential equation involvinga 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 fractionalelliptic 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 solutionsdepending on the parameter $\lambda>0$, up to the critical value of theexponent $p$, 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 Sobolevembedding.

Positive solutions for semilinear fractional elliptic problems involving an inverse fractional operator