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arXiv · 2410.15138

Solitary waves for the power degenerate NLS -- existence and stability

Vishnu Iyer·Atanas G. Stefanov
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Abstract

We consider a semilinear Schr\"odinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. 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.

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Vishnu Iyer, Atanas G. Stefanov. 2024-10-19. Solitary waves for the power degenerate NLS -- existence and stability. https://arxiv.org/abs/2410.15138

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