论文标题
在$ \ mathbb {r}^n $中，对局部稳态的不稳定性和非定位稳态对反应扩散方程式
Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\mathbb{R}^N$
论文作者
论文摘要
我们表明，椭圆问题$ΔU+f（u）= 0 $ in $ \ mathbb {r}^n $，$ n \ geq 1 $，c^1 in c^1（\ mathbb {r}）$和$ f（r}）$ and $ f（0）= 0 $ in Infinity in Infinity in Infinity in Innin IS n in Incor，该$ f（0）= 0 $。作为推论，我们可以证明，在无穷大时腐烂到零的任何两种非平凡溶液都必须相交，但前提是其中至少一个是签名的。以前仅在两种解决方案均以不同的方法为正的情况下才知道该属性。我们还讨论了我们的主要结果对相应反应扩散方程的单调杂斜溶液的存在的含义。
We show that the elliptic problem $Δu+f(u)=0$ 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.
