论文标题
$ \ mathbb {r}^d $的半线性椭圆系统的局部径向解决方案的建设性证明
Constructive proofs for localized radial solutions of semilinear elliptic systems on $\mathbb{R}^d$
论文作者
论文摘要
椭圆问题的基态解决方案已在部分微分方程的理论中进行了广泛的分析，因为它们代表了许多模型方程中的基本空间模式。虽然标量方程以及某些特定类别的椭圆系统的结果是全面的，但在非线性椭圆方程的通用系统中，这些局部解决方案知之甚少。在本文中，我们提出了一种通用方法，可以建设性地证明在$ \ mathbb {r}^d $上存在椭圆系统的局部径向对称解。这种解决方案基本上是由非自治的普通微分方程的系统描述的。我们使用动力学系统理论和计算机辅助的证明技术研究了这些系统，将适当选择的Lyapunov-Perron运算符与Newton-Kantorovich型定理相结合。我们通过在$ \ mathbb {r}^3 $上证明了该方法的特定局部径向解决方案，从而证明了这种方法的力量，在$ \ mathbb {r}^2 $上的swift-hohenberg方程，在$ \ mathbb {r}^2 $上，以及$ \ mathbbbbbbbbbbbbbbbbbbbbbbb {r}^2 $}^2 $。这些结果表明，通过建设性证明，可以触犯各种椭圆系统中的基态解决方案。
Ground state solutions of elliptic problems have been analyzed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localized solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localized radially symmetric solutions of elliptic systems on $\mathbb{R}^d$. Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov-Perron operator with a Newton-Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localized radial solutions of the cubic Klein-Gordon equation on $\mathbb{R}^3$, the Swift-Hohenberg equation on $\mathbb{R}^2$, and a three-component FitzHugh-Nagumo system on $\mathbb{R}^2$. These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.
