Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some remarks on a minimization problem associated to a fourth order
  nonlinear Schr\"odinger equation   Let $\gamma > 0\,$, $\beta > 0\,$, $\alpha > 0$ and $0 < \sigma N < 4$. In
the present paper, we study, for $c > 0$ given, the constrained minimization
problem \begin{equation*} \label{MinL2fixed} m(c):=\inf_{u\in S (c) }E(u),
\end{equation*} where \begin{equation*} E
(u):=\frac{\gamma}{2}\int_{\mathbb{R}^N}|\Delta u|^2\, dx
-\frac{\beta}{2}\int_{\mathbb{R}^N}|\nabla u|^2\,
dx-\frac{\alpha}{2\sigma+2}\int_{\mathbb{R}^N}|u|^{2\sigma+2}\, dx,
\end{equation*} and \begin{equation*} S(c):=\left\{u\in
H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^{2}\, dx=c\right\}. \end{equation*}
The aim of our study is twofold. On one hand, this minimization problem is
related to the existence and orbital stability of standing waves for the mixed
dispersion nonlinear biharmonic Schr\"odinger equation \begin{equation*} i
\partial_t \psi -\gamma \Delta^2 \psi - \beta \Delta \psi + \alpha
|\psi|^{2\sigma} \psi =0, \quad \psi (0, x)=\psi_0 (x),\quad (t, x) \in
\mathbb{R} \times \mathbb{R}^N. \end{equation*} On the other hand, in most of
the applications of the Concentration-Compactness principle of P.-L. Lions, the
difficult part is to deal with the possible dichotomy of the minimizing
sequences. The problem under consideration provides an example for which, to
rule out the dichotomy is rather standard while, to rule out the vanishing,
here for $c > 0$ small, is challenging. We also provide, in the limit $c \to
0$, a precise description of the behaviour of the minima. Finally, some
extensions and open problems are proposed.

7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph