Nonlinear fractional elliptic problem with singular term at the boundary

TL;DR

This paper investigates the existence and non-existence of solutions for a nonlinear fractional elliptic boundary value problem with a boundary singular term, depending on parameters s and q.

Contribution

It provides new results on the conditions for existence and non-existence of solutions to a fractional elliptic problem with boundary singularities, extending previous work.

Findings

Existence of solutions for certain ranges of q and s.

Non-existence results when parameters fall outside these ranges.

Characterization of solution behavior near the boundary.

Abstract

Let $\Omega\subset \mathbb{R}^N$ be a bounded regular domain, $0<s<1$ and $N>2s$. We consider $$ (P)\left\{ \begin{array}{rcll} (-\Delta)^s u &= & \frac{u^{q}}{d^{2s}} & \text{ in }\Omega , \\ u &> & 0 & \text{in }\Omega , \\ u & = & 0 & \text{ in }\mathbb{R}^N\setminus\Omega ,% \end{array}% \right. $$ where $0<q\le 2^*_s-1$, $0<s<1$ and $d(x) = dist(x,\partial\Omega)$. {The main goal } of this paper is to analyze existence and non existence of solution to problem $(P)$ according to the value of $s$ and $q$.