@article{Li2022, 
author = {Xiangrui Li and Shuibo Huang and Meirong Wu and Canyun Huang},
title = {Existence of solutions to elliptic equation with mixed local and nonlocal operators},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {7},
pages = {13313-13324},
keywords = {non-smooth variational approach, mixed local and nonlocal operators},
url = {https://www.sciopen.com/article/10.3934/math.2022735},
doi = {10.3934/math.2022735},
abstract = {In this paper, making use of a new non-smooth variational approach established by Moameni[13,14,15,16], we establish the existence of solutions to the following mixed local and nonlocal elliptic problem         {                            −          Δ          u          +          (          −          Δ                      )            s                    u          =          μ          g          (          x          ,          u          )          +          b          (          x          )          ,                          x          ∈          Ω          ,                                                                                                                                                                                                                                                        u          ≥          0          ,                                                                            x          ∈          Ω          ,                                                                                                                                                                                                                                                        u          =          0          ,                                                                            x          ∈                                    R                                      N                                ∖          Ω          ,                        where    Ω  ⊂            R              N       is a bounded smooth domain,    (  −  Δ      )          s       is the restricted fractional Laplacian,    μ  >  0,    0  <  s  <  1,    N  >  2  s,    g satisfies some growth condition and    b  (  x  )  ∈      L    m    (  Ω  ) for    m  ≥  2. The interesting feature of our work is that we show that the nonlocal operator has an important influence in the existence of solutions to the above equation since    g has new growth condition.}
}