@article{Deng2023, 
author = {Shengbing Deng and Qiaoran Wu},
title = {Existence of normalized solutions for the Schrödinger equation},
year = {2023},
journal = {Communications in Analysis and Mechanics},
volume = {15},
number = {3},
pages = {575-585},
keywords = {normalized solutions, Schrödinger equation, Sobolev critical nonlinearities, approximation method, mountain-pass type solution},
url = {https://www.sciopen.com/article/10.3934/cam.2023028},
doi = {10.3934/cam.2023028},
abstract = {In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities.                                       {                                                    −                Δ                u                =                λ                u                +                μ                |                u                                  |                                      q                    −                    2                                                  u                +                |                u                                  |                                      p                    −                    2                                                  u                                                              in                                                                                      R                                    N                                ,                                                                                      ∫                                                                                    R                                            N                                                                      |                u                                  |                  2                                d                x                =                                  a                  2                                ,                                                                  where    N  ⩾  3,    2  <  q  <  2  +      4    N  ,    p  =      2    ∗    =            2      N              N      −      2      ,    a  ,  μ  >  0 and    λ  ∈      R   is a Lagrange multiplier. Since the existence result for    2  +      4    N    <  p  <      2    ∗   has been proved, using an approximation method, that is let    p  →      2    ∗  , we obtain that there exists a mountain-pass type solution for    p  =      2    ∗  .}
}