H\"{o}lder continuity for nonlinear elliptic problem in Musielak-Orlicz-Sobolev space

Under appropriate assumptions on the $N(\mathrm{\Omega })$-fucntion, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev space to prove the H"{o}lder continuity of fully nonlinear elliptic problems. As the applications, the H"{o}lder continuity of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the H"{o}lder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.