On a notion of category depending on a functional, Part I: Theory and application to critical point theory

We introduce the notion of category depending on a fixed functional ff defined on a topological space. This notion permits us to obtain a better lower bound to the number of critical points of ff than the bound obtained with the Lusternik–Schnirelman category. We also introduce the notion of truncated category depending on the functional ff which permits us to obtain a lower bound to the number of critical points of unbounded functionals. Finally, we extend our result after having introduced the notions of relative category and limit relative category depending on the functional ff.