Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 405-428
Central China Normal University, Hubei Key Laboratory of
Mathematical Sciences and
School of Mathematics and Statistics
Wuhan, 430079, P.R. China; ligb 'at' mail.ccnu.edu.cn
Central China Normal University, Hubei Key Laboratory of
Mathematical Sciences and
School of Mathematics and Statistics
Wuhan, 430079, P.R. China; luoxiaohf 'at' 163.com
Abstract. In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear Chern–Simons–Schrödinger equations in R2
-Δu - λu + (h2(|x|/|x|2 + ∫|x|+∞ h(s/s u2(s)ds)u = |u|p - 2u,
where
h(s) = 1/2 ∫0s ru2(r)dr.
To get such solutions we look for critical points of the energy functional
I(u) = 1/2 ∫R2 |∇u|2 + 1/2 ∫R2 |u|2/|x|2 (∫0|x| s/2 u2(s)ds)2 - 1/p ∫R2 |u|p
on the constraints
Sr(c) = {u ∈ Hr1(R2) : ||u||2L2(R2) = c}, c > 0.
When p = 4, we prove a sufficient condition for the nonexistence of constrain critical points of I on Sr(c) for certain c and get infinitely many minimizers of I on Sr(8π). For the value p ∈ (4,+∞) considered, the functional I is unbounded from below on Sr(c). By using the constrained minimization method on a suitable submanifold of Sr(c), we prove that for certain c > 0, I has a critical point on Sr(c). After that, we get an H1-bifurcation result of our problem. Moreover, by using a minimax procedure, we prove that there are infinitely many critical points of I restricted on Sr(c) for any c ∈ (0,4π/√(p - 3)).
2010 Mathematics Subject Classification: Primary 35J20, 35J60, 35J92.
Key words: Chern–Simons–Schrödinger, constrained minimization, bifurcation phenomenon, multiplicity.
Reference to this article: G. Li and X. Luo: Normalized solutions for the Chern–Simons–Schrödinger equation in R2. Ann. Acad. Sci. Fenn. Math. 42 (2017), 405-428.
https://doi.org/10.5186/aasfm.2017.4223
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