Anisotropic classes of uniqueness of the solution of the Dirichlet problem for quasi-elliptic equations

We select a class of uniqueness of the solutions of the quasi-elliptic equation with the Dirichlet condition on the boundary of an unbounded domain $\Omega\subset\mathbb R^{n+1}$ and show that for domains with irregular behaviour of the boundary this class can be wider than that established in [10] for second-order elliptic equations. For the Laplace equation we construct an example of non-uniqueness of solution of the Dirichlet problem that shows that the class of uniqueness found in this paper cannot be essentially extended.