Conditional rearrangement comparison for elliptic equations with Orlicz growth and gradient-dependent lower-order terms

In this paper, we study nonlinear Dirichlet problems with Orlicz growth and gradient-dependent lower-order terms, − div ( a ( x , u , ∇ u ) ) = H ( x , u , ∇ u ) . The principal part is governed by an N -function A , while the gradient-dependent lower-order term is controlled by a second N -function B . Assuming that B ∘ A − 1 is concave and that F ( t ) : = t B ( t ) A ( t ) is nondecreasing, we derive a Talenti-type differential inequality for the decreasing rearrangement of admissible rearrangement solutions. The gradient contribution is represented by the nonsingular kernel K ( s ) = θ C s 1 − 1 / n F ( C s 1 − 1 / n ( − ( u ∗ ) ′ ( s ) ) ) , thereby avoiding the potentially singular quotient B ( t ) / A ( t ) at t = 0 . We then construct a radial comparison model on the symmetrized ball and reduce it to a one-dimensional nonlinear flux equation. Provided that the radial model is solvable and the associated Volterra regularity condition holds, we obtain a rearrangement comparison principle for admissible rearrangement solutions.