We study the existence of an integral representation for the functional $$ L^p_\mu(\Omega;\mathbb{R}^m)\ni u\mapsto\inf\left\{\liminf_{n\to+\infty}\int_\Omega f(\nabla u_n)d\mu:C^\infty\big(\overline{\Omega};\mathbb{R}^m\big)\ni u_n\stackrel{L^p_\mu}{\to} u\right\} $$ when $\mu$ is a positive Radon measure on $\mathbb{R}^N$, $\Omega\subset\mathbb{R}^N$ is a bounded open set, and $f:\mathbb{M}^{m\times N}\to[0,+\infty[$ is a continuous function not necessarily convex with growth conditions of order $p>1$.