Vol. 44, No. 2, 1973

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ISSN: 0030-8730
Bounded entire solutions of elliptic equations

Avner Friedman

Vol. 44 (1973), No. 2, 497–507
Abstract

Let

     ∑n       -∂2u--  ∑n      ∂u-
Lu =     aij(x)∂xi∂xj +   bi(x)∂xi.
i,j=1              i=1
(1.1)

Consider the equation

Lu(x) = f(x).
(1.2)

It is shown, under some general conditions on the coefficients of L, that if f(x) is locally Hölder continuous and

f(x) = O (|x|− 2− μ) as |x| → ∞ (μ > 0)
(1.3)

then there exists a bounded solution of (1.2) in Rn when n 3. If n = 2 then bounded entire solutions may not exist, but there exists a nonnegative solution of (1.2) in R2 which is bounded above by 0(log |x|). An application of these results to the Cauchy problem is given in the final section of the paper.

Mathematical Subject Classification 2000
Primary: 35J15
Milestones
Received: 15 October 1971
Published: 1 February 1973
Authors
Avner Friedman