A variational analysis of a gauged nonlinear Schrödinger equation

Alessio Pomponio; David Ruiz

doi:10.4171/jems/535

JournalsjemsVol. 17, No. 6pp. 1463–1486

A variational analysis of a gauged nonlinear Schrödinger equation

Alessio Pomponio
Politecnico di Bari, Italy
David Ruiz
Universidad de Granada, Spain

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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:

- Δ u (x) + (ω + \frac{h ^{2} ( ∣ x ∣ )}{∣ x ∣ ^{2}} + \int_{∣ x ∣}^{+ \infty} \frac{h ( s )}{s} u^{2} (s) d s) u (x) = ∣ u (x) ∣^{p - 1} u (x),

where

h (r) = \frac{1}{2} \int_{0}^{r} s u^{2} (s) d s .

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 $ω$ . Quite surprisingly, the threshold value for $ω$ is explicit. From this study we prove existence and non-existence of positive solutions.

Cite this article

Alessio Pomponio, David Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation. J. Eur. Math. Soc. 17 (2015), no. 6, pp. 1463–1486

DOI 10.4171/JEMS/535