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

Abstract

We consider the planar logarithmic Choquard equation $$-\Delta u+a(x)u+(\log |\cdot |\ast u^2)u=0,\text{in }{\mathbb{R}}^2$$ 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 $\Vert \cdot {\Vert }_X$ 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 $⟨\cdot ,\cdot {⟩}_u$, $u\in X$ with the $G$-equivariance property $⟨g\ast v,g\ast w{⟩}_{g\ast u}=⟨v,w{⟩}_u.$