Multi-bump solutions for critical Schrödinger equations with electromagnetic fields and logarithmic nonlinearity.

In this paper, we are interested in the existence and multiplicity of multi-bump solutions for critical Schrödinger equations with electromagnetic fields and logarithmic nonlinearity of the following type: − (∇ + i A (x)) 2 u + (λ Z (x) + (x)) u = u log | u | 2 + | u | 2 ∗ − 2 u ,   u ∈ H 1 (ℝ N , ℂ) , where N ≥ 3 , the magnetic potential A ∈ L loc 2 (ℝ N , ℝ N) , ∈ (1 , + ∞) , the parameter λ ≥ 1 and Z (x) , (x) : ℝ N → ℝ are the non-negative continuous functions. Applying variational methods, we obtain that the above equations have at least 2 k − 1 multi-bump solutions as λ ≥ 1 is sufficiently large. To some extent, we extend and complement the results of [C. O. Alves and C. Ji, Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well, Sci. China Math. 65 (2022) 1577–1598; J. Wang and Z. Yin, Multi-bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity, Math. Nachr. 298 (2025) 328–355] from subcritical case to critical case. [ABSTRACT FROM AUTHOR]