Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth
Abstract
We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} \Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\ \int_{\mathbb{R}^N} u^2 \, dx = \rho > 0, \end{cases} where $\rho$ is prescribed and $(\lambda, u) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints is demonstrated, which allows to provide general growth assumptions imposed on $g$. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at $0$.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2002.08344
 arXiv:
 arXiv:2002.08344
 Bibcode:
 2020arXiv200208344B
 Keywords:

 Mathematics  Analysis of PDEs;
 35J20;
 35J60;
 35Q55
 EPrint:
 to appear in Journal of Functional Analysis