Summary: We prove, using a variational method in the functional framework of Orlicz-Sobolev spaces, the existence and uniqueness of the weak solution for the problem \[ \begin{aligned} & \sum\limits_{|\alpha|\leq m-1} (-1)^{|\alpha|}D^{\alpha}\left[a_\alpha\left(D^\alpha(Ku)\right)\right]=f,\\ & D^\alpha u=0\quad \text{on}\quad \partial \Omega, \quad |\alpha|\leq m-1, \end{aligned} \] where \(\Omega\subset {\mathbb R}^n\) is a bounded domain with a sufficiently regular boundary \(\partial\Omega\), and \(Ku=\sum\limits_{i=1}^n\frac{\partial u}{\partial x_i}\).