A global branch approach to normalized solutions for the Schr\"odinger equation

Louis Jeanjean; Jianjun Zhang; Xuexiu Zhong

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arXiv · 2112.05869

A global branch approach to normalized solutions for the Schr\"odinger equation

Louis Jeanjean·Jianjun Zhang·Xuexiu Zhong

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Abstract

We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schr\"odinger equation of the form \begin{equation*} -\Delta u+\lambda u=g(u), \quad u \in H^1(\mathbb{R}^N), \, N \geq 1. \end{equation*} Our approach permits to handle in a unified way nonlinearities $g(s)$ which are either mass subcritical, mass critical or mass supercritical. Among its main ingredients is the study of the asymptotic behaviors of the positive solutions as $\lambda\rightarrow 0^+$ or $\lambda\rightarrow +\infty$ and the existence of an unbounded continuum of solutions in $(0, + \infty) \times H^1(\mathbb{R}^N)$.

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Louis Jeanjean, Jianjun Zhang, Xuexiu Zhong. 2021-12-10. A global branch approach to normalized solutions for the Schr\"odinger equation. https://arxiv.org/abs/2112.05869

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