Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mass Concentration and Local Uniqueness of Ground States for
  $L^2$-subcritical Nonlinear Schr\"{o}dinger Equations   We consider ground states of $L^2$-subcritical nonlinear Schr\"{o}dinger
equation (1.1), which can be described equivalently by minimizers of the
following constraint minimization problem
  $$ e(\rho):=\inf\{E_{\rho}(u):u\in \mathcal{H}(\mathbb{R}^d),\|u\|_2^2=1\}.$$
The energy functional $E_{\rho}(u)$ is defined by $$
E_{\rho}(u):=\frac{1}{2}\int_{\mathbb{R}^d}|\nabla u|^2dx
+\frac{1}{2}\int_{\mathbb{R}^d}V(x)|u|^2dx-\frac{\rho^{p-1}}{p+1}\int_{\mathbb{R}^d}|u|^{p+1}dx,$$
where $d\geq1$, $\rho>0$, $p\in\big(1, 1+\frac{4}{d}\big)$ and $0\leq
V(x)\to\infty$ as $|x| \to\infty$. We present a detailed analysis on the
concentration behavior of ground states as $\rho\to\infty$, which extends the
concentration results shown in [22]. Moreover, the uniqueness of nonnegative
ground states is also proved when $\rho$ is large enough.

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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin