Partial differential equations
Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in ${ℝ}^{N}$
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 131-136.

We show that the elliptic problem $\Delta u+f\left(u\right)=0$ in ${ℝ}^{N}$, $N\ge 1$, with $f\in {C}^{1}\left(ℝ\right)$ and $f\left(0\right)=0$ 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 sign-changing. 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.150
Christos Sourdis 1

1 National and Kapodistrian University of Athens, Department of Mathematics, Athens, Greece.
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Christos Sourdis. Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\protect \mathbb{R}^N$. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 131-136. doi : 10.5802/crmath.150. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.150/

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