Work

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

Year: 2026

Type: article

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].

Source: SIAM Journal on Mathematical Analysis

DOI: 10.1137/25m1748846

Authors: O. Cabrera, S. Cingolani, T. Weth

Institutions: Goethe University Frankfurt, University of Bari Aldo Moro

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Deutscher Akademischer Austauschdienst, Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni

Award: G2433072319