A Ground State Solution to the Chern-Simons-Schrödinger System

Abstract

In this paper, we consider the Chern-Simons-Schrödinger system

$$\left\{{\matrix{{- \Delta u + \left[{{e^2}{{\left| {\bf{A}} \right|}^2} + \left({V\left(x \right) + 2e{A_0}} \right) + 2\left({1 + {{\kappa q} \over 2}} \right)N} \right]u + q{{\left| u \right|}^{p - 2}}u = 0,} \hfill \cr {- \Delta N + {\kappa ^2}{q^2}N + q\left({1 + {{\kappa q} \over 2}} \right)\,\,{u^2} = 0,} \hfill \cr {\kappa \left({{\partial _1}{A_2} - {\partial _2}{A_1}} \right) = - e{u^2},\,\,\,\,{\partial _1}{A_1} + {\partial _2}{A_2} = 0,} \hfill \cr {\kappa {\partial _1}{A_0} = {e^2}{A_2}{u^2},\,\,\,\kappa {\partial _2}{A_0} = - {e^2}{A_1}{u^2},} \hfill \cr}} \right.$$

where u ∈ H¹(ℝ²), p ∈ (2, 4), A_α: ℝ2 → ℝ are the components of the gauge potential (α = 0, 1, 2), N: ℝ2 → ℝ is a neutral scalar field, V(x) is a potential function, the parameters κ, q > 0 represent the Chern-Simons coupling constant and the Maxwell coupling constant, respectively, and e > 0 is the coupling constant. In this paper, the truncation function is used to deal with a neutral scalar field and a gauge field in the Chern-Simons-Schrödinger problem. The ground state solution of the problem (P) is obtained by using the variational method.