A Class of Functionals on the Sequence Space $s$ Satisfying the Palais-Smale Condition
Kaveh Eftekharinasab
TL;DR
This work develops a variational framework for a class of functionals on the sequence space $s$ of rapidly decreasing sequences, called $\mathcal{F}_s$-functionals, defined as sums of quadratic and convex terms with quadratic growth. The authors establish that such functionals satisfy the Palais--Smale condition and admit a unique global minimizer, and prove that the PS-condition is preserved under linear homeomorphisms, enabling pullbacks to isomorphic Fréchet spaces. By transferring the variational problem across isomorphisms (e.g., to $\mathcal{S}(\mathbb{R})$, $\mathscr{D}[a,b]$, $C^{\infty}_{2\pi}(\mathbb{R})$, and $C^{\infty}[a,b]$) via explicit basis representations (Hermite, Fourier, Chebyshev), the paper provides a toolkit to analyze nonlinear operator problems through diagonalization in $s$. The framework is demonstrated with concrete nonlinear spectral problems and a semilinear PDE, proving the existence, uniqueness, and regularity of solutions as global minimizers in $s$, which in turn yield smooth, rapidly decreasing solutions in the original function spaces.
Abstract
We introduce a class of functionals on the space of rapidly decreasing sequences $s$, called $\mathcal{F}_s$-functionals, defined as decomposable sums of quadratic and convex terms with quadratic growth. We prove that such functionals satisfy the Palais-Smale condition and admit a unique global minimum. Furthermore, we show that the Palais-Smale condition is preserved under linear homeomorphisms. This allows us to construct corresponding functionals satisfying the Palais-Smale condition on Fréchet spaces isomorphic to $s$. We then show how this framework provides a tool for the proof of existence and uniqueness of solutions for specific operator problems, where coupled infinite-dimensional systems are transformed into diagonalized problems in the space $s$.