A variational analysis of a gauged nonlinear Schrödinger equation

Pomponio, 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 -