2022/10/23 by Peng Luo, Luo, Peng, Kefan Pan +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2210.12604
openalex publication_date 2022/10/23 · openalex created_date 2022/10/31 · openalex updated_date 2026/07/28
We are concerned with the Moser-Trudinger problem \begincases -Δu=λueu2~~amp;in~Ω,
ugt;0 ~~ amp;in~Ω,
u=0~~amp;on~∂ Ω, \endcases where Ω⊂ ℝ2 is a smooth bounded domain and λ>0 is sufficiently small. Qualitative analysis for Moser-Trudinger nonlinearities has been studied in recent decades, however there is still a lot of clarity about this issue, even for a low energy. The reason is that this problem is a critical exponent for dimension two and will lose compactness. Here by using a variety of local Pohozaev identities, we qualitatively analyze the positive solutions of Moser-Trudinger problem with a low energy, which contains the Morse index, non-degeneracy, asymptotic behavior, uniqueness and symmetry of solutions. Since the fundamental solution of -Δ in Ω⊂ ℝ2 is in logarithmic form and the corresponding bubble is exponential growth, more precise asymptotic behavior of the solutions is needed, which is of independent interest. Moreover, to obtain our results, some ODE's theory will be used to a prior estimate of the solutions and some elliptic theory in dimension two will play a crucial role.