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

Abstract

We consider the planar logarithmic Choquard equation $$- \Delta u + a(x)u + (\log|\cdot| \ast u^2)u = 0,\qquad \text{in } \mathbb{R}^2$$ 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 $\Phi$ 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 $\langle g \ast v , g \ast w \rangle_{g \ast u} = \langle v , w \rangle_u.$