  We prove the existence of a weak solution for a general class of Dirichlet
anisotropic elliptic problems such as $\mathcal Au+\Phi(x,u,\nabla
u)=\mathfrak{B}u+f$ in $\Omega$, where $\Omega$ is a bounded open subset of
$\mathbb R^N$ and $f\in L^1(\Omega)$ is arbitrary. The principal part is a
divergence-form nonlinear anisotropic operator $\mathcal A$, the prototype of
which is $\mathcal A u=-\sum_{j=1}^N \partial_j(|\partial_j
u|^{p_j-2}\partial_j u)$ with $p_j>1$ for all $1\leq j\leq N$ and $\sum_{j=1}^N
(1/p_j)>1$. As a novelty in this paper, our lower order terms involve a new
class of operators $\mathfrak B$ such that $\mathcal{A}-\mathfrak{B}$ is
bounded, coercive and pseudo-monotone from $W_0^{1,\overrightarrow{p}}(\Omega)$
into its dual, as well as a gradient-dependent nonlinearity $\Phi$ with an
"anisotropic natural growth" in the gradient and a good sign condition.
