Multiple normalized solutions of Chern–Simons–Schrödinger system

Abstract

In this paper, we consider the following equationfor p > 2 and λ > 0, which appeared in Byeon et al. (J Funct Anal 263(6):1575-1608, 2012) to find the standing wave solutions of the ChernSimons-Schrödinger system. By using the minimax theorem, we get the multiplicity results for the L 2 -normalized solutions to the equation, and thus there are multiple L 2 -normalized solutions of the Chern-SimonsSchrödinger system. Mathematics Subject Classification. 35Q55, 35A15, 35B30.

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

“…When q > 4, using a minimax procedure motivated by [1], the authors proved the multiplicity of normalized solutions for (1.3) for c ∈ (0, 4π √ p−3 ). 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].…”

“…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…”

“…In [8], Byeon et al proved that problem (1.10) admits a positive minimizer provided that c > 0 is sufficiently small whenever q ∈ (3, 4) or c > 0 is arbitrary whenever q ∈ (2,3]. If c and q satisfy the above assumptions, then Yuan [33] obtained infinitely many distinct pairs of solutions (u n , λ n ) ⊂ H 1 r (R 2 ) × R − of (1.3) for each n ∈ N + via the argument of Krasnoselski genus (see [26]). 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.…”

Standing waves for the Chern–Simons–Schrödinger equation with critical exponential growth

“…In [18], the authors researched the Chern-Simons-Schrödinger system without Ambrosetti-Rabinowitz condition. The other related research for system (1.1), we may refer to [13,16,20]. However, to our knowledge, the Chern-Simons-Schrödinger system with critical exponential growth was not considered until now, that is, f behaves like exp(α|u| 2 ) as |u| → ∞.…”

The existence of nontrivial solutions to Chern-Simons-Schrödinger systems

“…The existence and non-existence standing wave solutions have been shown under the assumptions that f (u) = λ|u| p−1 u, λ > 0 and p > 2 by variational methods in [4], see also [9] and [10], [5]. A series of their existence results of solitary waves has been established in [6], [13], [16], [17] and [23]. We studied the existence, non-existence, and multiplicity of standing waves to the nonlinear CSS systems with an external potential V (x) without the Ambrosetti-Rabinowitz condition in [20], and the concentration of solutions in [21].…”