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

Alexander Quaas; Aliang Xia

doi:10.3934/dcds.2017113

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2017, Volume 37, Issue 5: 2653-2668. Doi: 10.3934/dcds.2017113

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

Alexander Quaas^1, and
Aliang Xia^2,

1.
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla: V-110, Avda. España 1680, Valparaíso, Chile
2.
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

Received: October 31, 2015

Revised: December 31, 2016

Published: May 2017

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Abstract

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation:

$ \begin{eqnarray*}(-Δ)^α u=λ a(x)u-b(x)u^p&\ \ \ {\rm in}\,\,\mathbb{R}^N, \end{eqnarray*}$

where $ α∈(0, 1) $, $ N≥ 2 $, $λ >0$, $a$ and $b$ are positive smooth function in $\mathbb{R}^N$ satisfying

$a\left( x \right) \to {a^\infty } > 0\;\;\;\;{\rm{and}}\;\;\;b\left( x \right) \to {b^\infty } > 0\;\;\;\;{\rm{as}}\;\;\;{\rm{|}}\mathit{x}{\rm{|}} \to \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.

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Mathematics Subject Classification: 35J60, 47G20.

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References

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