Normalized solutions to the Chern-Simons-Schrödinger system under the nonlinear combined effect

• Published in: Science China. Mathematics Vol. 66; no. 9; pp. 2057 - 2080
• Main Authors: Yao, Shuai, Chen, Haibo, Sun, Juntao
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.09.2023; Springer Nature B.V
• Subjects: Applications of Mathematics; Asymptotic properties; Critical point; Lagrange multiplier; Mathematics; Mathematics and Statistics; Nonlinearity; Schrodinger equation; Chern-Simons-Schrödinger system; 35J20; 35B38; 35J61; variational method; 35B35; normalized solution; 35Q40; constraint manifold
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-021-2021-8

We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f ( u ) = ∣ u ∣ p −2 u + μ ∣ u ∣ q −2 u in ℝ 2 ,where μ ∈ {± 1} and 2 < p , q < ∞. The solutions correspond to critical points of the underlying energy functional subject to the L 2 -norm constraint, namely, ∫ ℝ 2 | u | 2 d x = c for c > 0 given. Of particular interest is the competing and double L 2 -supercritical case, i.e., μ = −1 and min{ p , q } > 4. We prove several existence and multiplicity results depending on the size of the exponents p and q . It is worth emphasizing that some of them are also new even in the study of the Schrödinger equations. In addition, the asymptotic behaviors of the solutions and the associated Lagrange multipliers λ as c → 0 are described.