A min-max principle for non-differentiable functions with a weak compactness condition
The authors extend the min-max principle of \textit{N. Ghoussoub} [Proc. Am. Math. Soc. 117, No. 2, 439--447 (1993; Zbl 0791.49028)] to local Lipschitz continuous functions satisfying a weak (PS) condition, which induces both the non-smooth (PS)\(^+\) or (PS)\(^-\) condition [Definition 2 of \textit{K.-C. Chang}, J. Math. Anal. Appl. 80, 102--129 (1981; Zbl 0487.49027)] and non-smooth Cerami condition. The main result i.e., Theorem 3.1 is proved by adapting Ghoussoub's approach to the non-smooth framework and exploiting Ekeland's variational principle with a suitable metric of geodesic type, and argument of non-smooth analysis. The authors also give two direct consequences of Theorem 3.1, i.e., Theorems 3.2 and 3.3. Theorem 3.2 provides a more refined version of several recent critical point results and Theorem 3.3 might reveal useful to get non-smooth variants of structure or multiplicity results for \(C^1\) functions.
scientific article; zbMATH DE number 5546292
scientific article