Existence of solutions for a quasilinear elliptic system with variable exponent

We consider the following quasilinear elliptic system in a Sobolev space with variable exponent:\[-\text{div}(a(|Du|)Du)=f,\]where a is a C^۱-function and f\in W^{-۱,p'(x)}(\Omega;\R^m). We use the theory of Young measures and weak monotonicity conditions to obtain the existence of solutions.

Keywords:

Quasilinear elliptic systems , Sobolev spaces with variable exponent , Weak solutions , Young measures