Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\mathbb{R}^N$

Abstract: We show that the elliptic problem $\Delta u+f(u)=0$ in $\mathbb{R}^N$, $N\geq 1$, with $f\in C^1(\mathbb{R})$ and $f(0)=0$ does not have nontrivial stable solutions that decay to zero at infinity, provided that $f$ is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is signchanging. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of monotone heteroclinic solutions to the corresponding reaction-diffusion equation.