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A variational analysis of a gauged nonlinear Schrödinger equation

Alessio PomponioDavid Ruiz — 2015

Journal of the European Mathematical Society

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 + h 2 ( | x | ) | x | 2 + | x | + h ( s ) s u 2 ( s ) d s u ( x ) = | u ( x ) | p - 1 u ( x ) , where h ( r ) = 1 2 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 ( 1 , 3 ) , the functional may be bounded from below or not, depending on ω . Quite surprisingly, the threshold value for ω is explicit. From...

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