@article{Brandolini2023, 
author = {Barbara Brandolini and Florica C. Cîrstea},
title = {Anisotropic elliptic equations with gradient-dependent lower order terms and        L    1   data},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {4},
pages = {1-33},
keywords = {nonlinear anisotropic elliptic equations, Leray–Lions operators, pseudo-monotone operators, lower order terms, summable data},
url = {https://www.sciopen.com/article/10.3934/mine.2023073},
doi = {10.3934/mine.2023073},
abstract = {We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as        A    u  +  Φ  (  x  ,  u  ,  ∇  u  )  =      B    u  +  f in    Ω, where    Ω is a bounded open subset of              R        N   and    f  ∈      L    1    (  Ω  ) is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator        A  , the prototype of which is        A    u  =  −      ∑          j      =      1        N        ∂    j    (      |        ∂    j    u            |                      p        j            −      2            ∂    j    u  ) with        p    j    >  1 for all    1  ≤  j  ≤  N and        ∑          j      =      1        N    (  1      /        p    j    )  >  1. As a novelty in this paper, our lower order terms involve a new class of operators        B   such that        A    −      B   is bounded, coercive and pseudo-monotone from        W    0          1      ,              p        →              (  Ω  ) into its dual, as well as a gradient-dependent nonlinearity    Φ with an "anisotropic natural growth" in the gradient and a good sign condition.}
}