## Differential and Integral Equations

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

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

#### 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

**Permanent link to this document**

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