@article{Zhang2026, 
author = {Le Zhang and Xiao Long and Shuibo Huang and Yonglin Xu},
title = {Existence of solutions to fractional elliptic problem with nonlocal gradient term and lower order term},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {9365-9379},
keywords = {fractional Laplace problem, nonlocal gradient term, lower-order term},
url = {https://www.sciopen.com/article/10.3934/math.2026388},
doi = {10.3934/math.2026388},
abstract = {Let     Ω  ⊂            R        N          (          3      ⩽      N      <                        2          s          (          2          −          θ          )          −          4          t                θ              )  be a smooth bounded domain. For     0  <  t  <            2      s      (      2      −      θ      )      −      3      θ        4    <  s  <  1  ,we consider the following fractional partial differential equation:                                 {                                                                                                                (                      −                      Δ                                              )                        s                                            u                      (                      x                      )                                                                                    =                      B                                                                        |                                                                                                                                                      D                                                                                                                            t                                                                                                                    (                                                                                                u                                                                                                                                                    2                                        −                                        θ                                                                            2                                                                                                                                      (                                x                                )                                                            )                                                                                |                                                2                                            +                      λ                      f                      (                      x                      )                      ,                                                                                  x                      ∈                      Ω                      ,                                                                                                  u                      (                      x                      )                                                                                    >                      0                      ,                                                                                  x                      ∈                      Ω                      ,                                                                                                  u                      (                      x                      )                                                                                    =                      0                      ,                                                                                  x                      ∈                                                                        R                                                N                                            ∖                      Ω                      ,                                                                                                                        where    B  >  0 is a constant,    θ  ∈      (          0      ,                        4          s                          2          s          +          3                      )  ,    0  <  f  ∈      L    m    (  Ω  ), and    λ  >  0 is a real parameter. In addition,                      D                    t            (                  u                                            2              −              θ                        2                              (      x      )        )   denotes a nonlocal gradient term. Problems involving local gradient terms and lower-order terms have been extensively investigated in the existing literature. Motivated by this, we focus on the corresponding problem with nonlocal gradient terms and lower-order terms in the present paper. Specifically, our aim is to analyze the influence of the lower-order term on the existence of solutions to the fractional Laplace problem. For    0  <  λ  ⩽      λ    ∗  , we prove the existence of solutions to this problem when    m  >      N    s  .}
}