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

A. I. Ibragimov, E. M. Landis

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The paper deals with the uniformly elliptic equation (aij (x)ux)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 {rm} is an increasing sequence of positive numbers, hm - rm+1 - rm, and the ratio hm+1/hm lies between positive constants C1 and C2. The intersection of Ω with the spherical layer between the spheres of radius rm and rm+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.

Original languageEnglish
Pages (from-to)2373-2384
Number of pages12
JournalJournal of Mathematical Sciences
Issue number6
StatePublished - 1997


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