2024/10/19 by Vishnu Mahadeva Iyer, Iyer, Vishnu, Atanas Stefanov +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2410.15138
openalex publication_date 2024/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a semilinear Schrödinger equation, driven by the power degenerate second order differential operator ∇⋅ (|x|2a ∇), a∈ (0,1). We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg's inequality. Depending on the parameter a and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open -- such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness.