Semi-classical states for the Choquard equation

Abstract: We study the nonlocal equation $$-\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon = \varepsilon^{-\alpha} \bigl(I_\alpha \ast \lvert u_\varepsilon\rvert^p\bigr) \lvert u_\varepsilon \rvert^{p - 2} u_\varepsilon\quad\text{in (\mathbf{R}^N)}, $$ where $N \ge 1$, $\alpha \in (0, N)$, $I_\alpha (x) = A_\alpha/\lvert x \rvert^{N - \alpha}$ is the Riesz potential and $\varepsilon > 0$ is a small parameter. We show that if the external potential $V \in C (\mathbb{R}^N; [0, \infty))$ has a local minimum and $p \in [2, (N + \alpha)/(N - 2)+)$ then for all small $\varepsilon > 0$ the problem has a family of solutions concentrating to the local minimum of $V$ provided that: either $p > 1 + \max (\alpha, \frac{\alpha + 2}{2})/(N - 2)+$, or $p > 2$ and $\liminf_{\lvert x\rvert \to \infty} V (x) \lvert x \rvert^2 > 0$, or $p = 2$ and $\inf_{x \in \mathbb{R}^N} V (x) (1 + \lvert x \rvert^{N - \alpha}) > 0$. Our assumptions on the decay of $V$ and admissible range of $p\ge 2$ are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.