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

In this paper, we study the existence and uniqueness of positive
solutions for the following nonlinear fractional elliptic equation:
(−∆)αu = λa(x)u − b(x)u
p
in R
N ,
where α ∈ (0, 1), N ≥ 2, λ > 0, a and b are positive smooth function
in R
N satisfying
a(x) → a
∞ > 0 and b(x) → b
∞ > 0 as |x| → ∞.
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.