A priori bounds for solutions to fully anisotropic elliptic equations

We are concerned with a priori estimates, in rearrangement form, for weak solutions to fully anisotropic, nonlinear elliptic equations with lower-order terms whose prototype is \begin{equation*} \left\{ \begin{array} [c]{lll} -\hbox{\rm div} \; (a(x, u, \nabla u)) + b(u)=f(x) & \qquad\hbox{\rm in\ } \Omega \\ u=0 & \qquad\text{on}\;\partial\Omega. \end{array} \right. \end{equation*} Here, $\Omega$ is an open bounded set in $\mathbb{R}^{N}$, with $N\geq2$, $a(x, \eta, \xi)$ is a Carath\'{e}odory function fulfilling \begin{equation*} a(x,\eta,\xi)\cdot\xi\geq\Phi\left( \xi\right) \qquad \text{ for } \left( \eta,\xi\right) \in\mathbb{R}\times\mathbb{R}^{N}, \; \text{ for a. e. } x\in\Omega, \end{equation*} where $\Phi :\mathbb{R}^{N}\rightarrow\left[ 0,+\infty\right[ $ is an $N-$dimensional Young function, and $b:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous and strictly increasing function such that $b\left( 0\right)=0$. Finally, $f:\Omega \rightarrow\mathbb{R}$ is a nonnegative measurable function enjoying suitable integrability conditions.