Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiple positive normalized solutions for nonlinear Schr\"odinger
  systems   We consider the existence of multiple positive solutions to the nonlinear
Schr\"odinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[
\left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1
+ \beta r_1 |u_1|^{r_1-2} u_1|u_2|^{r_2}, -\Delta u_2 &= \lambda_2 u_2 + \mu_2
|u_2|^{p_2 -2}u_2 + \beta r_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \end{aligned}
\right. \] under the constraint \[ \int_{\mathbb{R}^N}|u_1|^2 \, dx = a_1,\quad
\int_{\mathbb{R}^N}|u_2|^2 \, dx = a_2. \] Here $a_1, a_2 >0$ are prescribed,
$\mu_1, \mu_2, \beta>0$, and the frequencies $\lambda_1, \lambda_2$ are unknown
and will appear as Lagrange multipliers. Two cases are studied, the first when
$N \geq 1, 2 < p_1, p_2 < 2 + \frac 4N, r_1, r_2 > 1, 2 + \frac 4N < r_1 + r_2
< 2^*$, the second when $ N \geq 1, 2+ \frac 4N < p_1, p_2 < 2^*, r_1, r_2 > 1,
r_1 + r_2 < 2 + \frac 4N.$ In both cases, assuming that $\beta >0$ is
sufficiently small, we prove the existence of two positive solutions. The first
one is a local minimizer for which we establish the compactness of the
minimizing sequences and also discuss the orbital stability of the associated
standing waves. The second solution is obtained through a constrained mountain
pass and a constrained linking respectively.

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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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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: hep-th