Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in ℝ N

We show that the elliptic problem ∆u + f (u) = 0 in RN , N ≥ 1, with f ∈C 1(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 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. Manuscript received 26th August 2020, revised 13th November 2020, accepted 15th November 2020.