# A variational analysis of a gauged nonlinear Schrödinger equation

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 6, page 1463-1486
- ISSN: 1435-9855

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topPomponio, Alessio, and Ruiz, David. "A variational analysis of a gauged nonlinear Schrödinger equation." Journal of the European Mathematical Society 017.6 (2015): 1463-1486. <http://eudml.org/doc/277313>.

@article{Pomponio2015,

abstract = {This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $- \Delta u(x) + \left( \omega + \frac\{h^2(|x|)\}\{|x|^2\} + \int _\{|x|\}^\{+\infty \} \frac\{h(s)\}\{s\} u^2(s)\, ds \right) u(x) = |u(x)|^\{p-1\}u(x)$, where $h(r)= \frac\{1\}\{2\}\int _0^\{r\} s u^2(s) \, ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in (1,3)$, the functional may be bounded from below or not, depending on $\omega $. Quite surprisingly, the threshold value for $\omega $ is explicit. From this study we prove existence and non-existence of positive solutions.},

author = {Pomponio, Alessio, Ruiz, David},

journal = {Journal of the European Mathematical Society},

keywords = {gauged Schrödinger equations; Chern-Simons theory; variational methods; concentration compactness; gauged Schrödinger equations; Chern-Simons theory; concentration compactness},

language = {eng},

number = {6},

pages = {1463-1486},

publisher = {European Mathematical Society Publishing House},

title = {A variational analysis of a gauged nonlinear Schrödinger equation},

url = {http://eudml.org/doc/277313},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Pomponio, Alessio

AU - Ruiz, David

TI - A variational analysis of a gauged nonlinear Schrödinger equation

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 6

SP - 1463

EP - 1486

AB - This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $- \Delta u(x) + \left( \omega + \frac{h^2(|x|)}{|x|^2} + \int _{|x|}^{+\infty } \frac{h(s)}{s} u^2(s)\, ds \right) u(x) = |u(x)|^{p-1}u(x)$, where $h(r)= \frac{1}{2}\int _0^{r} s u^2(s) \, ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in (1,3)$, the functional may be bounded from below or not, depending on $\omega $. Quite surprisingly, the threshold value for $\omega $ is explicit. From this study we prove existence and non-existence of positive solutions.

LA - eng

KW - gauged Schrödinger equations; Chern-Simons theory; variational methods; concentration compactness; gauged Schrödinger equations; Chern-Simons theory; concentration compactness

UR - http://eudml.org/doc/277313

ER -

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