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1995 Uniqueness and nonuniqueness of positive solutions for a semilinear elliptic equation in $\mathbb{R}^n$
Achilles Tertikas
Differential Integral Equations 8(4): 829-848 (1995).

Abstract

This article is concerned with semilinear elliptic equations of the form $$ \Delta u+\lambda g(x)f(u)=0\quad\text{in }\Bbb R^n. $$ In particular, it addresses the questions of nonexistence, uniqueness and nonuniqueness of positive solutions $u$ such that $$ \begin{align} &\lim_{|x|\to\infty}u(x)=0\quad\text{when }n\ge3,\qquad \text{no condition at infinity when }n=1,2. \end{align} $$ A typical example for $f$ is the function $f(u)=u-u^{1+p}$, $p>0$, arising in population genetics models; moreover, the function $g$ may change sign. Our results show the importance of the behavior of $g$ at infinity. We derive the nonexistence and uniqueness results by establishing a series of identities involving solutions of the problem, as well as solutions of a linear problem. For nonuniqueness, we use sub and super solution techniques in connection with monotonicity arguments.

Citation

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Achilles Tertikas. "Uniqueness and nonuniqueness of positive solutions for a semilinear elliptic equation in $\mathbb{R}^n$." Differential Integral Equations 8 (4) 829 - 848, 1995.

Information

Published: 1995
First available in Project Euclid: 20 May 2013

zbMATH: 0823.35052
MathSciNet: MR1306594

Subjects:
Primary: 35J60
Secondary: 35B05

Rights: Copyright © 1995 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.8 • No. 4 • 1995
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