On countable subsets of solutions of nonlinear higher-order ODEs and elliptic PDEs with indefinite operators

Abstract

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 $$\Delta^2 F=-F+F^2, \quad \Delta^2 F=-F -\Delta (F^2-F), \quad \Delta^3 F=F-F^2, \quad \mbox{etc.};$$ 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 $\Gamma_{\rm max/min}$. Patterns obtained by gluing together via exponentially decaying tails of arbitrary finite samples from $\Gamma$'s form a countable subset of homoclinics in $\mathbb{R}^4$ of an arbitrary complexity.