A New Framework for Ljusternik–Schnirelmann Theory and Its Application to Planar Choquard Equations

Abstract. We consider the planar logarithmic Choquard equation [Formula: see text] in the strongly indefinite and possibly degenerate setting where no sign condition is imposed on the linear potential [Formula: see text]. In particular, we shall prove the existence of a sequence of high energy solutions to this problem in the case where [Formula: see text] is invariant under [Formula: see text]-translations. The result extends to a more general [Formula: see text]-equivariant setting, for which we develop a new variational approach which allows us to find critical points of Ljusternik–Schnirelmann type. In particular, our method resolves the problem that the energy functional [Formula: see text] associated with the logarithmic Choquard equation is only defined on a subspace [Formula: see text] with the property that [Formula: see text] is not translation-invariant. The new approach is based on a new [Formula: see text]-equivariant version of the Cerami condition and on deformation arguments adapted to a family of suitably constructed scalar products [Formula: see text], [Formula: see text] with the [Formula: see text]-equivariance property [Formula: see text].