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Nonnegative Solutions for Weakly Nonlinear Elliptic Equations

Published online by Cambridge University Press:  20 November 2018

Walter Allegretto
Walter Allegretto*
Affiliation:
University of Alberta, Edmonton, Alberta
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Let x = (x1, … xn ) denote a point of Euclidean n space En and set Di = ∂/∂xi for i = 1, … n. Let Ω denote an exterior domain in En with smooth boundary and consider in Ω the formal elliptic problem:

1

We first consider the problem of finding nonnegative generalized solutions of (1) when τ ≧ 0, τ ≢ 0, and r(x) ≡ 0. Under more stringent conditions on the coefficients and for suitable r(x), we then show the existence of a locally bounded solution. Next, we show that, under stronger assumptions, our main criterion is also necessary. The final arguments are devoted to the consideration of illustrative examples.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 1 , 01 February 1984 , pp. 71 - 83
Copyright
Copyright © Canadian Mathematical Society 1984

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