@article{Yi2023, 
author = {Xing Yi},
title = {Nonhomogeneous nonlinear integral equations on bounded domains},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {9},
pages = {22207-22224},
keywords = {integral equation, Hardy-Littlewood-Sobolev inequality, blowing-up and rescaling argument, Ekeland variational principle},
url = {https://www.sciopen.com/article/10.3934/math.20231132},
doi = {10.3934/math.20231132},
abstract = {This paper investigates the existence of positive solutions for a nonhomogeneous nonlinear integral equation of the form         u          p      −      1        (  x  )  =      ∫          Ω                  u      (      y      )                      |            x      −      y                        |                          n          −          α                      d  y  +      ∫          Ω                  f      (      y      )                      |            x      −      y                        |                          n          −          α                      d  y  ,    x  ∈            Ω      ¯      where              2      n              n      +      α        ≤  p  <  2  ,    1  <  α  <  n,    n  >  2  ,     Ω is a bounded domain in              R              n      . We show that under suitable assumptions on    f  , the integral equation admits a positive solution in        L                            2          n                          n          +          α                          (    Ω    )  . Our method combines the Ekeland variational principle, a blow-up argument and a rescaling argument which allows us to overcome the difficulties arising from the lack of Brezis-Lieb lemma in        L                            2          n                          n          +          α                      (  Ω  ).}
}