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In this paper we obtain the interior Hölder regularity of the gradient for the elliptic-Laplacian equation with the logarithmic functionunder some proper assumptions on the Hölder continuous functionsand  A . This extends previous results obtained by Giannetti and Passarelli di Napoli [Regularity results for a new class of functionals with non-standard growth conditions. J. Differ Equ. 2013;254(3):1280–1305], where  A  is the identity matrix and hence does not depend on  x .

Hölder estimates for the elliptic p ( x )-Laplacian equation with the logarithmic function