Minimization of the functional of the gradient by Baire's theorem

We give sufficient conditions for the existence of solutions of the minimum problem $$ {cal P}_{u_0}: qquad hbox{Minimize}quad int_Omega g(Du(x))dx, quad uin u_0 + W_0^{1,p}(Omega,{Bbb R}), $$ based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum $u_0in W_0^{1,p}(Omega,{Bbb R})$.