Bifurcation into spectral gaps for strongly indefinite Choquard equations

TL;DR

This paper proves the existence of multiple solutions for a class of indefinite Choquard equations with spectral gap bifurcations, revealing complex solution structures depending on parameters and spectral properties.

Contribution

It establishes the first results on bifurcation and multiplicity of solutions for strongly indefinite Choquard equations with spectral gap considerations.

Findings

Infinitely many solutions bifurcate from the spectral gap boundary.

Solutions exist for all parameters within the spectral gap.

Unique bifurcation point at the upper spectral edge.

Abstract

We consider the semilinear elliptic equations $$ \left\{ \begin{array}{ll} &-\Delta u+V(x)u=\left(I_\alpha\ast |u|^p\right)|u|^{p-2}u+\lambda u\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right. $$ where $I_\alpha$ is a Riesz potential, $p\in(\frac{N+\alpha}N,\frac{N+\alpha}{N-2})$, $N\geq3$, and $V $ is continuous periodic. We assume that $0$ lies in the spectral gap $(a,b)$ of $-\Delta + V$. We prove the existence of infinitely many geometrically distinct solutions in $H^1(\mathbb R^N)$ for each $\lambda\in(a, b)$, which bifurcate from $b$ if $\frac{N+\alpha}N< p < 1 +\frac{2+\alpha}{N}$. Moreover, $b$ is the unique gap-bifurcation point (from zero) in $[a,b]$. When $\lambda=a$, we find infinitely many geometrically distinct solutions in $H^2_{loc}(\mathbb R^N)$. Final remarks are given about the eventual occurrence of a bifurcation from infinity in $\lambda=a$.