Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 429-449

EXISTENCE AND MULTIPLICITY OF NORMALIZED SOLUTIONS FOR THE NONLINEAR CHERN–SIMONS–SCHRÖDINGER EQUATIONS

Haibo Chen and Weihong Xie

Central South University, School of Mathematics and Statistics
Changsha, Hunan 410083, P.R. China; math_chb 'at' 163.com

Central South University, School of Mathematics and Statistics
Changsha, Hunan 410083, P.R. China; xieweihong0218 'at' 163.com

Abstract. In this paper, we prove the existence and multiplicity results of solutions with prescribed L²-norm for a class of nonlinear Chern–Simons–Schrödinger equations in R²

–Δu – λu + κ(h²(|x|)/|x|² + ∫_|x|^∞h(s)/s u²(s) ds)u = f(u),

where λ ∈ R, κ > 0, f ∈ C(R,R) and

h(s) = 1/2 ∫₀^sru²(r) dr.

To obtain such solutions, we look into critical points of the energy functional

E_κ(u) = 1/2 ∫_R2|∇u|² + κ/2 ∫_R2|u|²/|x|² (∫₀^|x|r/2 u²(r) dr)² – ∫_R2F(u)

constrained on the L²-spheres S_r(c) = {u ∈ H_r¹(R²) : ‖u‖₂² = c}. Here, c > 0 and F(s) := ∫₀^sf(t) dt. Under some mild assumptions on f, we show that critical points of E_κ unbounded from below on S_r(c) exist for certain c > 0. In addition, we establish the existence of infinitely many critical points {u_n^κ} of E_κ on S_r(c) provided that f is odd. Finally, we regard κ as a parameter and and present a convergence property of u_n^κ as κ ↘ 0. These results improve and generalize the existing ones in the literature.

2010 Mathematics Subject Classification: Primary 35J20, 35J60.

Key words: Chern–Simons–Schrödinger equations, normalized solutions, multiplicity results, asymptotic behavior.

Reference to this article: H. Chen and W. Xie: Existence and multiplicity of normalized solutions for the nonlinear Chern–Simons–Schrödinger equations. Ann. Acad. Sci. Fenn. Math. 45 (2020), 429-449.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4518