@article{Li2022, 
author = {Yanfeng Li and Haicheng Liu},
title = {A multiplicity result for double phase problem in the whole space},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {9},
pages = {17475-17485},
keywords = {double phase operator, Musielak-Orlicz-Sobolev space, critical point},
url = {https://www.sciopen.com/article/10.3934/math.2022963},
doi = {10.3934/math.2022963},
abstract = {In the present paper, we discuss the solutions of the following double phase problem     −      d    i    v    (      |    ∇  u            |                                        p          −          2                      ∇  u  +  μ  (  x  )      |    ∇  u            |                                        q          −          2                      ∇  u  )  +      |    u            |                                        p          −          2                      u  +  μ  (  x  )      |    u            |                                        q          −          2                      u  =  f  (  x  ,  u  )  ,    x  ∈            R        N    ,where    N  ≥  2,    1  <  p  <  q  <  N and    0  ≤  μ  ∈      C                                    0          ,          α                      (            R        N    )  ,    α  ∈  (  0  ,  1  ]. Based on the theory of the double phase Sobolev spaces        W                                    1          ,          H                      (            R        N    ), we prove the existence of at least two non-trivial weak solutions.}
}