In this paper we study the boundary value problem for a class of high order nonlinear elliptic equations in the bounded domain $G_0$ with smooth boundary $\Gamma_0$ in the space $\mathbf{R}^n$ where the domain $G_0$ is considered as a limit (in some sense) of the family of domain $\{G_t\}$ which depend smoothly on a parameter $t\in [0; T]$ in S. G. Krein's sense when $t$ tends to 0.