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

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, $$(-\mathrm{\Delta }{)}^{\alpha }u=\lambda u+(-\mathrm{\Delta }{)}^{\beta }\mathrm{\mid }u{\mathrm{\mid }}^{p-1}uin\mathrm{\Omega };(-\mathrm{\Delta }{)}^{j}u=0on\mathrm{\partial }\mathrm{\Omega },forj\in \mathbb{Z},0\le j<[\alpha ]$$ where $\mathrm{\Omega }$ 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, $$(-\mathrm{\Delta }{)}^{\alpha -\beta }u=\lambda (-\mathrm{\Delta }{)}^{-\beta }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u\text{quadin}\mathrm{\Omega };u=0\text{on}\mathrm{\partial }\mathrm{\Omega },$$ 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 $1<p\le 2_{\mu }^{\ast }-1$ where $\mu :=\alpha -\beta$ and $2_{\mu }^{\ast }=\frac{2N}{N-2\mu }$ is the critical exponent of the Sobolev embedding.