MONOTONICITY OF POLYNOMIAL GROWING SOLUTIONS FOR UNIFORMLY ELLIPTIC
FRACTIONAL EQUATIONS
Abstract
In this paper we consider the following equation involving uniformly
elliptic nonlocal operator in the upper half space { A 2 s u ( x )= f (
u ( x )) , x ∈ R + n , u ( x ) > 0 , x ∈ R + n , u ( x )= 0
, x ∉ R + n . We first develop a narrow region principle for
antisymmetric functions in unbounded domains, in which we assume that
u has polynomial growth instead of the usual decay condition
u→0 at infinity, which is the improvement of the partial result
of Wu-Qu-Zhang-Zhang[Math Meth Appl Sci. 2023;46:3721-3740]. Then we
obtain the monotonicity of positive solutions in the upper half space by
a direct method of moving planes, which extends the monotonic result of
Li[Adv. Nonlinear Stud., 24 (2024), 451-462]. Based on the
monotonicity results, we prove the existence of positive bounded
solutions for uniformly elliptic nonlocal equation on the whole space.
We believe that these methods will also be helpful for equations
involving other non local operators.