Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Semi-classical states for the Choquard equation   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.

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The categories of this paper are: nucl-th
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin