Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Semiclassical states for Choquard type equations with critical growth:
  critical frequency case   In this paper we are interested in the existence of semiclassical states for
the Choquard type equation
  $$
  -\vr^2\Delta u +V(x)u =\Big(\int_{\R^N} \frac{G(u(y))}{|x-y|^\mu}dy\Big)g(u)
\quad \mbox{in $\R^N$},
  $$ where $0<\mu<N$, $N\geq3$, $\vr$ is a positive parameter and $G$ is the
primitive of $g$ which is of critical growth due to the
Hardy--Littlewood--Sobolev inequality. The potential function $V(x)$ is assumed
to be nonnegative with $V(x)=0$ in some region of $\R^N$, which means it is of
the critical frequency case. Firstly we study a Choquard equation with double
critical exponents and prove the existence and multiplicity of semiclassical
solutions by the Mountain-Pass Theorem and the genus theory. Secondly we
consider a class of critical Choquard equation without lower perturbation, by
establishing a global Compactness lemma for the nonlocal Choquard equation, we
prove the multiplicity of high energy semiclassical states by the
Lusternik--Schnirelman theory.

30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio