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:

Δu(x)+(ω+h2(x)x2+x+h(s)su2(s)ds)u(x)=u(x)p1u(x),- \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)=120rsu2(s)ds.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(1,3)p\in(1,3), the functional may be bounded from below or not, depending on ω\omega . Quite surprisingly, the threshold value for ω\omega 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