The global boundedness of minimizers is proved for functionals \(\int_ \Omega f(x,Du)dx\) (\(\Omega\) bounded open subset of \(R^ n\)), where \(f(x,z)\) is a Carathéodory function, defined for \(x\in \Omega\), \(z\in R^ n\), satisfying anisotropic growth conditions: \[ f(x,z)\geq m\sum^ n_{i=1}|\xi_ i|^{q_ i} (m>0, q_ i>1),\quad f(x,0)\in L^ r(\Omega)\quad (r>1) \] and \[ {\bar q^*\over\bar q}\left(1-{1\over r}\right)>1\quad\text{with }{1\over \bar q}={1\over n}\sum^ n_{i=1}{1\over q_ i},\quad\bar q^*={n\bar q\over n-\bar q}\quad (\bar qq_ i'\), \(q_ i'/=q_ i/(q_ i-1)\) and we prove the global boundedness of solutions if \((\bar q^*/\bar q)\min_{1\leq i\leq n} (1-(q_ i'/r_ i))>1\).