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MONOTONICITY OF POLYNOMIAL GROWING SOLUTIONS FOR UNIFORMLY ELLIPTIC FRACTIONAL EQUATIONS
  • PENGYAN WANG,
  • ZHIHAO WANG,
  • YUNXIA MA
PENGYAN WANG
Xinyang Normal University

Corresponding Author:[email protected]

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ZHIHAO WANG
Xinyang Normal University
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YUNXIA MA
Xinyang Normal University
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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.
27 Oct 2024Submitted to Mathematical Methods in the Applied Sciences
29 Oct 2024Submission Checks Completed
29 Oct 2024Assigned to Editor
01 Nov 2024Review(s) Completed, Editorial Evaluation Pending
01 Nov 2024Reviewer(s) Assigned