Consider the class of anisotropic elliptic problems in divergence form
\[ \begin{cases} -\text{div} (a(x,u,\nabla u)) = f(x) - \text{div} (g(x)) \quad &\text{in}\quad \Omega,\\ u =0 \quad&\text{on}\quad \partial\Omega, \end{cases}\tag{1} \]
on a bounded, open set \(\Omega \subset \mathbb{R}^N\), \(N\geq 2\), for suitable measurable functions \(f,g\), where the Carathéodory function \(a\) is assumed to satisfy the anisotropic growth condition \[ a(x,\eta,\xi) \geq \Phi (\xi) \qquad \text{for}\quad (\eta,\xi) \in \mathbb{R} \times \mathbb{R}^N, \] for an even convex function \(\Phi : \mathbb{R}^N \to [0,\infty)\) with \(\Phi(0)=0\) and \(\Phi(\xi)\to\infty\) as \(|\xi|\to\infty\) (i.e., a Young function). A model choice would be \(\Phi(\xi) = \sum_{i=1}^N \lambda_i |\xi_i|^{p_i}\); in particular, the growth in the different partial derivatives is allowed to be different, but \(\Phi\) need not be of polynomial type.
The goal of the paper is to prove, in the style of A. Cianchi [Commun. Partial Differ. Equations 32, No. 5, 693–717 (2007; Zbl 1219.35028)], a pointwise comparison theorem for weak solutions of (1) (more precisely, the symmetric decreasing rearrangement on the ball) in terms of the solution of a simpler symmetrized problem on the ball, as well as on integral expressions of the form \(\Phi(\nabla \,\cdot\,)\). This is applied to obtain regularity results for the problem (1) in the form of a priori bounds on (certain norms of) the solution. Applications are given for certain special choices of \(\Phi\) (such as the model choice mentioned above).
The principle novelty in the present paper is the treatment of the divergence form term on the right-hand side of (1).