Hِlder regularity for the gradients of solutions of the strong -Laplacian

Under some assumptions on the continuous function p ( x ) > 1 , we obtain the local C 1 , α regularity of solutions of the strong p ( x ) -Laplacian − div ( | ∇ u | p ( x ) − 2 ∇ u ) + | ∇ u | p ( x ) − 2 log ( | ∇ u | ) ∇ u ⋅ ∇ p = 0 by using the integral estimates in Campanato spaces.