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

Abstract

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, $$(-Δ)^α u=λu+ (-Δ)^β|u|^{p-1}u \quad \mbox{in}\quad Ω;\qquad (-Δ)^{j}u=0\quad \mbox{on}\quad \partialΩ,\quad \mbox{for}\quad j\in\mathbb{Z},\: 0\leq j< [α]$$ 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, $$ (-Δ)^{α-β} u= λ(-Δ)^{-β}u+ |u|^{p-1}u \quad\mbox{in} \quad Ω;\qquad u=0 \quad \hbox{on} \quad \partialΩ,$$ 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.