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.