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

Abstract

We study the existence of $L^2$ normalized solutions for nonlinear Schrödinger 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.

36 pages

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