Vol. 62, No. 1, 1976

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Entire solutions of linear elliptic equations with Laplacian principal part

Gerald Norman Hile

Vol. 62 (1976), No. 1, 127–140
Abstract

Consider the equations in Rn, n 2,

Δφ = f + b⋅∇φ
(*)

Δ φ = b⋅∇ φ
(I)

where f and b are locally Hölder continuous, and as |x|→∞, f(x) = O(|x|τ), b(x) = O(|x|σ), σ,τ > 1. It is shown that if 0 ρ < σ 1, there is a one-to-one correspondence between entire C2 solutions of () whose gradients grow no faster than O(|x|ρ), and harmonic polynomials with gradients of the same growth. For (I) therefore solutions whose gradients grow no faster than O(|x|ρ) form a finite dimensional vector space. These results for (I) give analogues to the concept of “generating pairs” in pseudo-analytic function theory.

Mathematical Subject Classification 2000
Primary: 35J15
Secondary: 30A64
Milestones
Received: 16 July 1975
Published: 1 January 1976
Authors
Gerald Norman Hile