To the Issue of Justification of the Absolutely Continuous Spectrum Eigenfunctions Asymptotics in the Problem of Three One-Dimensional Short-Range Quantum Particles with Repulsion

Limit profiles for singularly perturbed Choquard equations with local repulsion

We study Choquard type equation of the form $$-Δu +\varepsilon u-(I_α*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$ where $N\geq3$, $I_α$ is the Riesz potential with $α\in(0,N)$, $p>1$, $q>2$ and $\varepsilon\ge 0$. Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qual…