  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.
