Bifurcation into spectral gaps for strongly indefinite Choquard equations

We consider the semilinear elliptic equations $\left\{ \begin{array}{ll} &-Δu+V(x)u=\left(I_α\ast |u|^p\right)|u|^{p-2}u+λu\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right.$ where $I_α$ is a Riesz potential, $p\in(\frac{N+α}N,\frac{N+α}{N-2})$, $N\geq3$, and $V$ is continuous periodic. We assume that $0$ lies in the spectral gap $(a,b)$ of $-Δ+ V$. We prove the existence of infinitely many geometrically distinct solutions in $H^1(\mathbb R^N)$ for each $λ\in(a, b)$, which bifurcate from $b$ if $\frac{N+α}N< p < 1 +\frac{2+α}{N}$. Moreover, $b$ is the unique gap-bifurcation point (from zero) in $[a,b]$. When $λ=a$, 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 $λ=a$.

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