AI 中文总结
本文通过粘合/匹配论证，证明了具有不定算子的高阶非线性常微分方程与椭圆型偏微分方程存在可数解族，补充了L–S理论不适用情形下的解存在性结果。
AI 中文摘要
本文通过粘合/匹配论证，得到了来自反应扩散、薄膜和动力系统（DS）理论的、具有不定非强制算子的高阶非线性常微分方程与椭圆型偏微分方程解的可数子集。特别地，我们研究了 $\boldsymbol{\text{R}}$ 上的一些经典和拟线性退化常微分方程，其满足无穷远边界条件 $F(\boldsymbol{\text{\textbackslash}infty})=0$，且具有非奇异性非线性项，如：$F^{(4)}=-F+F^2$、$F^{(4)}=-F+F^2{\rm e}^{F-1}$、$(|F''|F'')''=-F + F^2$、$F^{(4)}=-|F|F+F^2$、$F^{(4)}=-F^3+F^4$、$F^{(6)}=F-F^2$ 等；以及具有奇异性和非光滑非线性项的方程，如：$F^{(4)}=-F+F^3$、$F^{(4)}=-F-(|F|F-F)''$、$F^{(4)}=- \frac{F}{\boldsymbol{\textbackslash}sqrt{|F|}}-(F^3-F)''$ 等。其中部分常微分方程为哈密顿型，已在动力系统（DS）理论中被详细研究。基于非线性算子、椭圆型偏微分方程和变分理论，$\boldsymbol{\text{R}}^N$ 上满足 $F(\boldsymbol{\text{\textbackslash}infty})=0$ 的相关多重调和椭圆型方程，如：$\boldsymbol{\text{\textbackslash}Delta}^2 F=-F+F^2$、$\boldsymbol{\text{\textbackslash}Delta}^2 F=-F -\boldsymbol{\text{\textbackslash}Delta}(F^2-F)$、$\boldsymbol{\text{\textbackslash}Delta}^3 F=F-F^2$ 等，也被证明存在可数的解族。对于这类与非奇异性泛函相关的方程，保证存在临界点序列的 Lusternik–Schnirel'man（L–S）亏格/范畴变分理论并不适用。这些常微分方程和椭圆型偏微分方程（如径向情形）被证明至少存在两个基本可数解族 $\boldsymbol{\text{\textbackslash}mathcal F}_{1,2}$，它们与两个周期轨道 $\boldsymbol{\text{\textbackslash}Gamma}_{\text{max/min}}$ 相关，具有正主导性。通过将周期轨道 $\boldsymbol{\text{\textbackslash}Gamma}$ 的任意有限样本通过指数衰减的尾部粘合在一起，得到的模式构成了 $\boldsymbol{\text{R}}^4$ 中任意复杂度的同宿轨的可数子集。
英文摘要
Countable subsets of solutions of higher-order nonlinear ODEs and elliptic PDEs with indefinite non-coercive operators from the reaction-diffusion, thin film and dynamical system (DS) theories are obtained via a gluing/matching argument. In particular we study some classic and quasilinear degenerate ODEs in $\mathbb{R}$, with boundary conditions at infinity $F(\infty)=0$, with non-odd nonlinearities such as $$ \begin{matrix} F^{(4)} =-F+F^2,\, \,\, F^{(4)}=-F+F^2{\rm e}^{F-1}, \, (|F''|F'')''=-F + F^2, \\ F^{(4)} =-|F|F+F^2, \,\,\,F^{(4)} =-F^3+F^4, \,\,\, F^{(6)}=F-F^2, \end{matrix} $$ etc, as well as equations with odd and non-smooth nonlinearities like $$F^{(4)}=-F+F^3, \,F^{(4)}=-F-(|F|F-F)'', \,\, F^{(4)}=- \frac F{\sqrt{|F|}}-(F^3-F)'',\,\mbox{etc.}$$ Some of these ODEs are Hamiltonian and were studied in detail in the DS theory. On the basis of nonlinear operators, elliptic PDEs and variational theory, related polyharmonic elliptic equations in $\mathbb{R}^N$, $F(\infty)=0$, such as $$Δ^2 F=-F+F^2, \quad Δ^2 F=-F -Δ(F^2-F), \quad Δ^3 F=F-F^2, \quad \mbox{etc.};$$ are also shown to admit countable families of solutions. For such equations with non-odd functionals associated Lusternik--Schnirel'man (L--S) genus/category variational theory guaranteeing existence of a sequence of critical points does not apply. These ODEs and elliptic PDEs (e.g., in the radial setting) are shown to admit at least two basic countable families ${\mathcal F}_{1,2}$ of positively dominant solutions connected with two periodic orbits $Γ_{\rm max/min}$. Patterns obtained by gluing together via exponentially decaying tails of arbitrary finite samples from $Γ$'s form a countable subset of homoclinics in $\mathbb{R}^4$ of an arbitrary complexity.
Comments58 pages
