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

Abstract 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 $$C^{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> regularity up to the boundary.