Does the following minimum exist?

Question

Let $g \in L^{p}(\Omega ; \mathbb{R}^n)$ , $p>1$ and $\Omega\subset \mathbb{R}^n $ an open and bounded set with smooth boundary and consider the following functional:$$W_{0}^{1,p}\left(\Omega ; \mathbb{R} \right)\ni u\mapsto J\left(u\right):=\int_{\Omega}\left|\nabla u-g\right|^{p}\mathrm{d}x.$$ Does this functional has a minimum? And if yes, what is the Euler-Lagrange equation?