@article{Boccardo2026, 
author = {Lucio Boccardo and Pasquale Imparato},
title = {The classical linear duality method in some semilinear and noncoercive Dirichlet problems},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {1},
pages = {48-54},
keywords = {Dirichlet problems, duality methods, semilinear equations, bounded solutions, elliptic systems},
url = {https://www.sciopen.com/article/10.3934/era.2026003},
doi = {10.3934/era.2026003},
abstract = {We study the following semilinear system:     (  ∗  )      {                                                u          ∈                                    W              0                              1                ,                2                                      (            Ω            )                    :                    −                                                                      d                  i                  v                                                              (          M          (          x          )                                D                    u          )          +          u          =          θ                    ψ                      |                    ψ                                    |                                                      p                ′                            −              2                                +          f          (          x          )          ;                                                          ψ          ∈                                    W              0                              1                ,                2                                      (            Ω            )                    :                    −                                                                      d                  i                  v                                                              (          M          (          x          )                                D                    ψ          )          +          ψ          =          u                      |                    u                                    |                                      p              −              2                                              and we prove the existence of bounded weak solutions in              W      0              1        ,        2              (    Ω    )  . Even if the system is nonlinear, we use a duality method.We dedicate this paper to Patrizia Pucci certain that she will help us for the study of (*) with nonlinear principal part.}
}