## Differential and Integral Equations

### Uniqueness and nonuniqueness of positive solutions for a semilinear elliptic equation in $\mathbb{R}^n$

Achilles Tertikas

#### 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.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 4 (1995), 829-848.

Dates
First available in Project Euclid: 20 May 2013

https://projecteuclid.org/euclid.die/1369055613

Mathematical Reviews number (MathSciNet)
MR1306594

Zentralblatt MATH identifier
0823.35052

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

#### Citation

Tertikas, Achilles. Uniqueness and nonuniqueness of positive solutions for a semilinear elliptic equation in $\mathbb{R}^n$. Differential Integral Equations 8 (1995), no. 4, 829--848. https://projecteuclid.org/euclid.die/1369055613