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

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@article{CRMATH_2021__359_2_131_0, author = {Sourdis, Christos}, title = {Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\protect \mathbb{R}^N$}, journal = {Comptes Rendus. Math\'ematique}, pages = {131--136}, publisher = {Acad\'emie des sciences, Paris}, volume = {359}, number = {2}, year = {2021}, doi = {10.5802/crmath.150}, language = {en}, url = {http://www.numdam.org/articles/10.5802/crmath.150/} }

TY - JOUR AU - Sourdis, Christos TI - Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\protect \mathbb{R}^N$ JO - Comptes Rendus. Mathématique PY - 2021 SP - 131 EP - 136 VL - 359 IS - 2 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.150/ DO - 10.5802/crmath.150 LA - en ID - CRMATH_2021__359_2_131_0 ER -

%0 Journal Article %A Sourdis, Christos %T Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\protect \mathbb{R}^N$ %J Comptes Rendus. Mathématique %D 2021 %P 131-136 %V 359 %N 2 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.150/ %R 10.5802/crmath.150 %G en %F CRMATH_2021__359_2_131_0

Sourdis, Christos. 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. http://www.numdam.org/articles/10.5802/crmath.150/

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