Bifurcation into spectral gaps for strongly indefinite Choquard equations

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 $\lambda=a$.