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

TL;DR

This paper proves that certain reaction-diffusion equations in ^N do not admit nontrivial stable solutions decaying at infinity if the nonlinearity is nonincreasing near zero, impacting the understanding of solution stability and intersection properties.

Contribution

It establishes the nonexistence of stable, decaying solutions under specific conditions and explores solution intersection and implications for heteroclinic solutions.

Findings

Stable solutions decay to zero are absent under the given conditions.

Any two decaying solutions must intersect if at least one changes sign.

Results influence the understanding of monotone heteroclinic solutions.

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.