Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in R^N involving fractional Laplacian

TL;DR

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional elliptic equations in imensional space, establishing conditions for solutions and analyzing their asymptotic behaviors.

Contribution

It provides new results on the existence, uniqueness, and asymptotic properties of solutions to fractional elliptic equations with logistic-type nonlinearities.

Findings

Proved existence and uniqueness of positive solutions.

Analyzed asymptotic behavior of solutions at infinity.

Developed comparison principles for fractional elliptic equations.

Abstract

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-\Delta)^\alpha u=\lambda a(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ \alpha\in(0,1) $, $ N\ge 2 $, $\lambda >0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.