@article{Wang2023, 
author = {Li Wang and Jun Wang and Daoguo Zhou},
title = {Concentration of solutions for double-phase problems with a general nonlinearity},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {13593-13622},
keywords = {double-phase problems, penalization method, variational methods},
url = {https://www.sciopen.com/article/10.3934/math.2023690},
doi = {10.3934/math.2023690},
abstract = {In this paper, we study the following problems with a general nonlinearity:         {                                              −                      Δ            p                    u          −                      Δ            q                    u          +          V          (          ε          x          )          (                      |                    u                                    |                                      p              −              2                                u          +                      |                    u                                    |                                      q              −              2                                u          )          =          f          (          u          )          ,                                      i            n                                                        R                        N                    ,                                              u          ∈                      W                          1              ,              p                                (                                    R                        N                    )          ∩                      W                          1              ,              q                                (                                    R                        N                    )          ,                                      i            n                                                        R                        N                    ,                        where    ε  >  0 is a small parameter,    2  ≤  p  <  q  <  N, the potential    V is a positive continuous function having a local minimum.       f  :      R    →      R   is a        C    1   subcritical nonlinearity. Under some proper assumptions of    V and    f  , we obtain the concentration of positive solutions with the local minimum of    V by applying the penalization method for above equation. We must note that the monotonicity of              f      (      s      )              s              p        −        1             and the so-called Ambrosetti-Rabinowitz condition are not required.}
}