Existence under lack of convexity

Abstract

It is well-established that there are two main methods to show existence of critical points for coercive functionals: either one focuses on proving the existence of global minimizers through the direct method and weak lower semicontinuity; or else, one relies on the Palais-Smale condition and goes directly to showing the existence of such critical, non-minimizer points. Both essentially require the convexity of the principal part of the underlying functional. In this contribution, we would like to explore the possibility of being dispensed with such convexity condition in such a way that neither of the two methods is applicable. We will show that, in most regular, coercive cases, associated Euler-Lagrange equations of optimality admit (weak) solutions in spite of non-convexity. Our results are even valid for vector problems of PDEs. Important restrictions to be taken into account in particular situations like positivity for scalar equations, or injectivity for systems, would require further insight.