Parametric quasilinear elliptic equations under Neumann boundary condition with full gradient dependence

We study a class of parametric quasilinear elliptic problems under homogeneous Neumann boundary condition. The equation is driven by the p -Laplacian with an additive coercive term, and the reaction combines a variable-exponent source with a Carathéodory perturbation that depends nonlinearly on the gradient. We develop a nonvariational sub-supersolution approach based on an auxiliary truncated problem and the surjectivity theorem for pseudomonotone operators. This method ensures the existence of a positive weak solution, its location between explicit positive constant sub- supersolutions, and global C¹ C 1 regularity up to the boundary.