| Abstract: |
| In this talk we consider a Schr\odinger-Choquard equation of the type
$$
-\Delta u +V(x)u=(I_\alpha\ast |u|^p)|u|^{p-2}u+\lambda u \quad x \in \mathbb R^N
$$
where $V$ is a periodic and non costant potential, $I_\alpha$ denotes the Riesz potential and $N\geq 3$. In this case, the spectrum of the self-adjoint operator $-\Delta + V$ in $L^2(\mathbb R^N)$ is purely continuous and may contain gaps.
An interesting physical and mathematical issue is establishing the existence of branches of solutions converging towards the trivial solution as $\lambda$ approaches some {\emph{bifurcation point}} of the spectrum.
An intriguing situation is the so called {\emph{gap-bifurcation}}, occurring at boundary points of the spectral gaps:
we review the main known results and open problems in the local case (in presence of a local perturbation $f(u)$) and we address the non local case. |
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