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arXiv · 1902.02028

A note on deformation argument for $L^2$ constraint problem

Norihisa Ikoma·Kazunaga Tanaka
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Abstract

We study the existence of $L^2$ normalized solutions for nonlinear Schr\"odinger equations and systems. Under new Palais-Smale type conditions we develop new deformation arguments for the constraint functional on $S_m=\{ u; \, \int_{\mathbf{R}^N} | u |^2=m\}$ or $S_{m_1} \times S_{m_2}$. As applications, we give other proofs to the results of [\cite[J:20], \cite[BdV:6], \cite[BS1:7]]. As to the results of [\cite[J:20], \cite[BdV:6]], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [\cite[BS1:7]], via our deformation result we can show the existence of vector solution without using constraint related to the Pohozaev identity.

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Norihisa Ikoma, Kazunaga Tanaka. 2019-02-06. A note on deformation argument for $L^2$ constraint problem. https://doi.org/10.57262/ade/1571731543

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