Boundary value problems for linear quasi-elliptic $A + \sigma$-type operators with variable coefficients are studied in the unbounded region of $\mathbf{R}^{n}$, definited by $x_{k} > 0$, $k = 1,\cdots,n$; $\sigma$ means a perturbation whose behaviour is assigned at infinity and in the angular points of the domain. It is proved that the operator related to the problem has closed range and finite dimensional null space. The study is developed within a new class of dissimetric Sobolev weighted spaces.