$\mathbb{R}^N$中一类反应扩散方程定域定态的不稳定性和无序性
Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in $\mathbb{R}^N$
摘要：
我们证明了椭圆问题$Δu+f(u)=0$在$\mathbb{R}^N$，$N\geq 1$，与$f\in C^1(\mathbb{R})$和$f(0)=0$没有非平凡的稳定的解决方案，衰减到零，在无穷远，只要$f$是非增加的原点附近。作为推论，我们可以证明任何两个在无穷远处衰减到零的非平凡解必须相交，只要其中至少有一个是变号的。这个属性以前只知道在两个解决方案是积极的情况下，一个不同的方法。我们还讨论了相应的反应扩散方程的单调异宿解的存在性。
英文摘要：
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.
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