Quasilinear elliptic systems with nonstandard growth and weak monotonicity

We prove the existence of solutions for a quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{ll} -\text {div}\,\sigma (x,u,Du)&{}=f(x,u,Du)\quad \text {in}\varOmega ,\\ u&{}=0\quad \text {on}\partial \varOmega . \end{array} \right. \end{aligned}$$The results are obtained in Orlicz–Sobolev spaces by means of the Young measures.