Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Abstract

Abstract. In this paper, we study the existence and multiplicity of solutions with a prescribed L 2 -norm for a class of nonlinear Chern-Simons-Schrödinger equations in RTo get such solutions we look for critical points of the energy functional, we prove a sufficient condition for the nonexistence of constrain critical points of I on S r (c) for certain c and get infinitely many minimizers of I on S r (8π). For the value p ∈ (4, +∞) considered, the functional I is unbounded from below on S r (c). By using the constrained minimization method on a suitable submanifold of S r (c), we prove that for certain c > 0, I has a critical point on S r (c). After that, we get an H 1 -bifurcation result of our problem. Moreover, by using a minimax procedure, we prove that there are infinitely many critical points of I restricted on S r (c) for any c ∈ 0,.

Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in $ \newcommand{\R}{\bf {\mathbb R}} \R^2$

“…In this aspect, we also would like to cite [43]. On the other hand, Li and Luo [28] also investigated the nonlocal equation (1.4) in the mass-critical case: p = 4 and mass-supercritical case: p > 4, for instance, the existence, H 1 (R 2 )-bifurcation and multiplicity of normalized solutions. The existence of stationary states with a vortex point has also been considered in [10,27].…”

Existence and multiplicity of normalized solutions for the nonlinear Chern–Simons–Schrödinger equations

“…Compared with [33], Li and Luo presented a certain constant c 0 = 4π √ p−3 , which improved the result for the case q > 4 in [33]. Moreover, the existence of normalized solutions for (1.3) was also considered in [16]. To this end, they used the approach introduced in [24] to construct a suitable submanifold of S r (c), which is defined by a condition which is a combination of the related Nehari functional and Pohozaev identity, i.e., (1.11) V (c) = {u ∈ S r (c) :…”

“…is strictly increasing on (−∞, 0) ∪ (0, ∞); (f 5 ) f is odd. Under the above conditions, it is well known (see [3,16]) that a solution of (1.4) with u 2 2 = c can be obtained as a constrained critical point of the functional…”

“…Moreover, they also analyzed the asymptotic behavior of u b n as b → 0 + . To the best knowledge of ours, little is known about the existence of normalized solutions of Chern-Simons-Schrödinger equations except for [8,16,33]. Set (1.10) e q (c) := inf…”

“…Furthermore, motivated by [1], the author also proved that for q > 4, (1.3) admits an unbounded sequence of couples of solutions (u n , λ n ) ⊂ H 1 r (R 2 ) × R − for c ∈ (0, c 0 ) sufficient small. In [16], Li and Luo considered problem (1.3) with q ≥ 4. For q = 4, they showed a sufficient condition for the nonexistence of constraint critical points of I q on S r (c) for certain c > 0 and obtained infinitely many minimizers of I q on S r (8π).…”