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

## TL;DR

This paper investigates the existence of positive solutions for a class of higher order fractional elliptic equations involving inverse fractional operators, establishing conditions under which solutions exist or do not exist depending on parameters and critical exponents.

## Contribution

It introduces new existence and nonexistence results for positive solutions to fractional elliptic problems with inverse operators, extending the understanding of such equations.

## Key findings

- Existence of solutions for subcritical exponents
- Nonexistence beyond the critical exponent
- Dependence of solutions on the parameter λ

## 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, $$(-\Delta)^{\alpha} u=\lambda u+ (-\Delta)^{\beta}|u|^{p-1}u \quad \mbox{in}\quad \Omega;\qquad (-\Delta)^{j}u=0\quad \mbox{on}\quad \partial\Omega,\quad \mbox{for}\quad j\in\mathbb{Z},\: 0\leq j< [\alpha]$$ 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.