On a singular minimizing problem

We investigate the non-homogeneous modular Dirichlet problem Δ _p (·)u(x) = f (x) (where Δ _p (·)u(x) = div(|?u|^p(x-2)?u(x)) from the functional analytic point of view and we prove the stability of the solutions \({\left( {{u_{{p_i}}}} \right)_i}\) of the equation \({\Delta _{{p_i}\left( \cdot \right)}}{u_{{p_i}\left( \cdot \right)}} = f\) as p _i (·) → q(·) via Gamma-convergence of sequence of appropriate functionals.