## Abstract

The paper deals with the uniformly elliptic equation (a^{ij} (x)u_{x})Xi = f(x) in an unbounded domain Ω ⊂ ℝ^{n} 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 C_{1} and C_{2}. 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 C_{1}/C_{2} = 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 C_{1}/C_{2} = 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 C_{1}/C_{2} = 1 with a linear speed) (one exception is possible here: a slow convergence to a constant). Bibliography: 7 titles.

Original language | English |
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Pages (from-to) | 2373-2384 |

Number of pages | 12 |

Journal | Journal of Mathematical Sciences |

Volume | 85 |

Issue number | 6 |

DOIs | |

State | Published - 1997 |