Global $c^{1,α}$ regularity and fine behavior at critical points for quasilinear elliptic equations with Orlicz growth and unbounded drift

Abstract

We investigate the global regularity up to the boundary and the fine pointwise behavior along the critical set for bounded weak solutions to general quasilinear elliptic Dirichlet problems in Orlicz-Sobolev spaces. Our structural framework encompasses operators with non-standard Lieberman-type growth alongside lower-order terms featuring natural gradient growth, an unbounded source term in $L^q$, and an unbounded spatial drift weight in $L^s$. To overcome the analytical obstructions posed by the unbounded drift across the anisotropic growth gap, we first establish global Hölder continuity up to the boundary via Moser iteration. We then construct a novel smooth regularization scheme for general homogeneous constant-coefficient operators, merging the approximation frameworks of Lieberman and Cianchi-Maz'ya, to derive global $C^{1,α}$ regularity up to the boundary through a variational perturbation approach. Building upon this global theory, we employ a geometric compactness and blow-up argument to prove that at any interior or boundary critical point where the gradient vanishes, the solution achieves an improved pointwise $C^{1,α}$ regularity. Remarkably, at boundary critical points, this quantitative improvement decouples from the lower-order terms and completely bypasses both the boundary datum regularity and the domain smoothness. Finally, we discuss the sharpness of the integrability threshold required for the unbounded drift weight in the subcritical regime.