Critical point theory on convex subsets with applications in differential equations and analysis

We shall establish a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space. This variational principle has many applications in partial differential equations while unifying and generalizing several results in nonlinear analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality. As a consequence, several substantial new results are emerged. We shall also provide concrete applications in local and nonlocal partial differential equations, including the symmetry properties of the Allen-Cahn equation on bounded domains, for which the standard methodologies have major limitations to be applied. Crown Copyright (C) 2020 Published by Elsevier Masson SAS. All rights reserved.