Nodal solutions for the Choquard equation

We consider the general Choquard equations $$-\mathrm{\Delta }u+u=(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$$ where $I_{\alpha }$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N

• \alpha}{N}, \frac{N + \alpha}{N - 2})$andminimalactionnodalsolutionsfor$p \in (2,\frac{N + \alpha}{N - 2})$. We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais–Smale sequences. The nonlinear Schr"odinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.