About correctness of the Dirichlet problem for a multivariate elliptic system with varying coefficients

The property of a system of partial differential equations with variable coefficients to belong to one or another homotopic type depends on the domain point at which this system is considered. The degeneration manifolds split the original region into parts. The study of the influence of such degeneration on the solvability character of the boundary value problems is important [1]. We consider the system of $n$ partial second order differential equations
$$ -\Lambda(x)\Delta u_j+\mu\frac{\partial}{\partial x_j} \sum_{i=1}^{n}\frac{\partial u_i}{\partial x_i}=0,\quad j=1,\ldots,n, $$
with a real function $\Lambda(x)$, $x=(x_1,\ldots,x_n)$. We obtain the conditions, under which the modified Dirichlet problem for this system is solvable up to an arbitrary harmonic function of $n-1$ variables.