Finding Gateaux-Saddles by a Local Minimax Method

Motivated by quasilinear elliptic PDEs in physical applications, Gateaux-saddles of a class of functionals J:H→{±∞}∪?, which are only Gateaux-differentiable at regular points, are considered. Since mathematical results and numerical methods for saddles of 𝒞¹ or locally Lipschitz continuous functionals in the literature are not applicable, the main objective of this article is to introduce a new mixed norm strong-weak topology approach such that a mathematical framework of a local minimax method is established to handle the singularity issue and to use the Gateaux-derivative of J for finding multiple Gateaux-saddles. Algorithm implementations on weak form and error control are presented. Numerical examples solving quasilinear elliptic problems from physical applications are successfully carried out to illustrate the method. Some interesting solution properties are to be numerically observed and open for analytical verification for the first time.