On the behavior of solutions to the neumann problem in unbounded domains

The paper deals with the uniformly elliptic equation (a^ij (x)u_x)Xi = f(x) in an unbounded domain Ω ⊂ ℝⁿ and its solution u(x) that satisfies the homogeneous Neumann condition. The function f has a compact support. The domain Ω has the following structure: assume that {r_m} is an increasing sequence of positive numbers, h_m - r_m+1 - r_m, and the ratio h_m+1/h_m lies between positive constants C₁ and C₂. The intersection of Ω with the spherical layer between the spheres of radius r_m and r_m+1 with center at the origin satisfies a certain inequality of isoperimetric type. It is shown in this paper that the set of solutions splits up into three classes: (i) the solutions for which lim u(x) = ∞ and lim/|x| → ∞ U(x) = ∞ ; moreover, it is shown that these limits are attained with nearly the same speed (if C₁/C₂ = 1, then the speed is not less than the exponential one); (ii) the solutions for each of which a constant C exists such that lim/|x|→ ∞ u(x) = C and u(x) - C changes its sign for large |x|; here, the convergence to C is rapid (for C₁/C₂ = 1 this convergence is not slower than the exponential one); (iii) the solutions that do not change their sign for large |x| and increase or decrease to +∞ or ∞, respectively, with low speed (for C₁/C₂ = 1 with a linear speed) (one exception is possible here: a slow convergence to a constant). Bibliography: 7 titles.