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We show that the elliptic problem $\Delta u+f(u)=0$ in $\mathbb{R}^N$, $N\geq1$, with $f\in C^1(\mathbb{R})$ and $f(0)=0$ does not have nontrivial stablesolutions that decay to zero at infinity, provided that $f$ is nonincreasingnear the origin. As a corollary, we can show that any two nontrivial solutionsthat decay to zero at infinity must intersect each other, provided that atleast one of them is signchanging. This property was previously known only inthe case where both solutions are positive with a different approach. We alsodiscuss implications of our main result on the existence of monotoneheteroclinic solutions to the corresponding reaction-diffusion equation.

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