Orlicz-Sobolev versus H\"older local minimizer for nonlinear Robin problems

In this paper, we establish a regularity results for weak solutions of Robin problems driven by the well-known Orlicz $g$-Laplacian operator. Precisely, by using a suitable variation of the Moser iteration technique, we prove that every weak solution of our problem is bounded. Moreover, we combine this result with the Lieberman regularity theorem, to show that every $C^1(\overline{\Omega})$-local minimizer is also a $W^{1,G}(\Omega)$-local minimizer for the corresponding energy functional of Robin-Orlicz problem.