Bifurcation into spectral gaps for strongly indefinite Choquard equations

Abstract

We consider the semilinear elliptic equations $$\{\begin{array}{ll}& -\Delta u+V(x)u=\left(I_{\alpha }\ast |u{|}^p\right)|u{|}^{p-2}u+\lambda uforx\in {\mathbb{R}}^N,\\ & u(x)\to 0\; as\; |x|\to \infty ,\end{array}$$ where $I_{\alpha }$ is a Riesz potential, $p\in (\frac{N+\alpha }{N},\frac{N+\alpha }{N-2})$, $N\ge 3$, 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 $H_{loc}^2({\mathbb{R}}^N)$. Final remarks are given about the eventual occurrence of a bifurcation from infinity in $\lambda =a$.