On the Neumann problem for higher-order divergent elliptic equations in an unbounded domain,close to a cylinder

The Neumann problem for a higher-order divergent elliptic equation, defined in an unbounded domain, close to a cylinder, is investigated. It is proved that each solution, having a slowly increasing energy integral, tends at infinity to a certain polynomial and, in the case of an exponential decrease of the righthand side of the equation, the convergence rate is also exponential. Existence and uniqueness are obtained in classes of functions with bounded or unbounded energy integral. Formulas, expressing the coefficients of the limit polynomial in terms of the right-hand side of the equation and of the Dirichlet data at the base of an unbounded domain, close to a semiinfinite cylinder, are derived.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 191–217, 1992.