Continuity of Derivatives of a Convex Solution to a Perturbed One-Laplace Equation by p-Laplacian

Abstract We consider a one-Laplace equation perturbed by p -Laplacian with $$1<p<\infty $$ 1 < p ∞ . prove that weak solution is continuously differentiable $$(C^1)$$ ( C ) if it convex. Note similar result fails to hold for the unperturbed equation. The main difficulty show $$C^{1}$$ -regularity of at boundary facet where gradient vanishes. For this purpose we blow-up and its limit constant function establishing Liouville-type result, which proved showing strong maximum principle. Our argument rather elementary, since assume A few generalizations are also discussed.