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A new framework for Ljusternik-Schnirelmann theory and its application to planar Choquard equations

Omar Cabrera, Silvia Cingolani, Tobias Weth

TL;DR

The paper introduces a novel G-equivariant Ljusternik-Schnirelman variational framework tailored to strongly indefinite planar logarithmic Choquard equations. By constructing a $G$-invariant family of norms via a generalized barycenter and establishing a $G$-equivariant Cerami-type (NCG) condition, the authors recover compactness up to translations and obtain an unbounded sequence of critical points. They apply this to the indefinite Choquard energy, proving infinitely many high-energy solutions under $\mathbb{Z}^2$ translation symmetry and extending to radial and nonradial (sign-changing) multiplicity results through varied symmetry groups. The work blends abstract LS theory with a careful functional-analytic setup for the logarithmic nonlocal term, offering a unified variational approach that also recovers known results in positive-potential regimes. The results have significant implications for nonlocal PDEs with indefinite potentials and translation invariance, providing a robust toolkit for multiplicity in strongly indefinite settings.$

Abstract

We consider the planar logarithmic Choquard equation $$- Δu + a(x)u + (\log|\cdot| \ast u^2)u = 0,\qquad \text{in } \mathbb{R}^2$$ in the strongly indefinite and possibly degenerate setting where no sign condition is imposed on the linear potential $a \in L^\infty(\mathbb{R}^2)$. In particular, we shall prove the existence of a sequence of high energy solutions to this problem in the case where $a$ is invariant under $\mathbb{Z}^2$-translations. The result extends to a more general $G$-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 $Φ$ associated with the logarithmic Choquard equation is only defined on a subspace $X \subset H^1(\mathbb{R}^2)$ with the property that $\|\cdot\|_X$ is not translation invariant. The new approach is based on a new $G$-equivariant version of the Cerami condition and on deformation arguments adapted to a family of suitably constructed scalar products $\langle \cdot, \cdot \rangle_u$, $u \in X$ with the $G$-equivariance property $\langle g \ast v , g \ast w \rangle_{g \ast u} = \langle v , w \rangle_u.$

A new framework for Ljusternik-Schnirelmann theory and its application to planar Choquard equations

TL;DR

The paper introduces a novel G-equivariant Ljusternik-Schnirelman variational framework tailored to strongly indefinite planar logarithmic Choquard equations. By constructing a -invariant family of norms via a generalized barycenter and establishing a -equivariant Cerami-type (NCG) condition, the authors recover compactness up to translations and obtain an unbounded sequence of critical points. They apply this to the indefinite Choquard energy, proving infinitely many high-energy solutions under translation symmetry and extending to radial and nonradial (sign-changing) multiplicity results through varied symmetry groups. The work blends abstract LS theory with a careful functional-analytic setup for the logarithmic nonlocal term, offering a unified variational approach that also recovers known results in positive-potential regimes. The results have significant implications for nonlocal PDEs with indefinite potentials and translation invariance, providing a robust toolkit for multiplicity in strongly indefinite settings.$

Abstract

We consider the planar logarithmic Choquard equation in the strongly indefinite and possibly degenerate setting where no sign condition is imposed on the linear potential . In particular, we shall prove the existence of a sequence of high energy solutions to this problem in the case where is invariant under -translations. The result extends to a more general -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 associated with the logarithmic Choquard equation is only defined on a subspace with the property that is not translation invariant. The new approach is based on a new -equivariant version of the Cerami condition and on deformation arguments adapted to a family of suitably constructed scalar products , with the -equivariance property

Paper Structure

This paper contains 15 sections, 35 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. A new version of G-equivariant LS theory
  3. The main result in an abstract setting
  4. Some Lipschitz properties
  5. A Deformation Lemma under the $(NCG)$-condition
  6. Completion of the proof of Theorem \ref{['c-k-to-infty']}
  7. A variant case
  8. A $G$-equivariant functional-analytic setting for the logarithmic Choquard equation
  9. A Lipschitz continuous generalized barycenter map
  10. An equivariant metric
  11. Action of subgroups of $E(2)$
  12. High energy solutions of the indefinite logarithmic Choquard equation
  13. Completeness of energy intervals
  14. Verification of the $(NCG)$-condition
  15. Completion of the proof of Theorem \ref{['thm:multiplicityN-']}

Key Result

Theorem 1.1

Let $a \in L^\infty(\mathbb{R}^2)$ be $\mathbb{Z}^2$-invariant. Then the equation eq:choquard has a sequence $(\pm u_n)_n$ of nontrivial solution pairs with $\Phi(\pm u_n) \to \infty$ as $n \to \infty$. Moreover, if $\mathop{\mathrm{ess\,inf}}\limits_{\mathbb{R}^2} a > 0$, then

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.1
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • ...and 60 more