(Sign-changing solutions to a gauged nonlinear Schrödinger equation. Our results generalize the existing ones, see Li G. Moreover, we show that the energy of u λ is strictly larger than two times of the ground state energy and analyze the asymptotic behavior of u λ as λ ↘ 0 +. Under some suitable assumptions on f ∈ C ( R ), we apply the constraint minimization argument to establish a least energy sign-changing solution u λ with precisely two nodal domains. We study the existence and asymptotic behavior of least energy sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth − Δ u + ω u + λ h 2 ( | x | ) | x | 2 + ∫ | x | ∞ h ( s ) s u 2 ( s ) d s u = f ( u ) i n R 2, u ∈ H r 1 ( R 2 ), where ω, λ > 0 are constants and h ( s ) = ∫ 0 s r 2 u 2 ( r ) d r. We study a class of gauged nonlinear Schrödinger equations in the plane Moreover, orbital instability of ground states is explored. Finally, in the mass supercritical case, we prove the existence of ground state and infinitely many radially symmetric solutions. Secondly, in the mass critical case, we investigate the existence and nonexistence of normalized solutions. In addition, we discuss the radial symmetry and uniqueness of minimizer to the minimization problem. As a by-product of the compactness of any minimizing sequence, orbital stability of the set of minimizers to the minimization problem is achieved.

Firstly, in the mass subcritical case, we establish the compactness of any minimizing sequence to the associated global minimization problem. The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint. In this paper, we study normalized solutions to the Chern-Simons-Schrödinger system, which is a gauge-covariant nonlinear Schrödinger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory.

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