A Class of Functionals on the Sequence Space $s$ Satisfying the Palais-Smale Condition

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 Fr\'echet 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 $s$.