Semi-classical states for the Choquard equation

We study the nonlocal equation $-\varepsilon^2 Δu_\varepsilon + V u_\varepsilon = \varepsilon^{-α} \bigl(I_α\ast \lvert u_\varepsilon\rvert^p\bigr) \lvert u_\varepsilon \rvert^{p - 2} u_\varepsilon\quad\text{in \(\mathbf{R}^N\)},$ where $N \ge 1$, $α\in (0, N)$, $I_α(x) = A_α/\lvert x \rvert^{N - α}$ 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 $p \in [2, (N + α)/(N - 2)_+)$ then for all small $\varepsilon > 0$ the problem has a family of solutions concentrating to the local minimum of $V$ provided that: either $p > 1 + \max (α, \frac{α+ 2}{2})/(N - 2)_+$, or $p > 2$ and $\liminf_{\lvert x\rvert \to \infty} V (x) \lvert x \rvert^2 > 0$, or $p = 2$ and $\inf_{x \in \mathbb{R}^N} V (x) (1 + \lvert x \rvert^{N - α}) > 0$. 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.

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