A note on deformation argument for $L^2$ normalized solutions of nonlinear Schrödinger equations and systems
A note on deformation argument for $L^2$ normalized solutions of nonlinear Schrödinger equations and systems
复制标题
DOI：
10.57262/ade/1571731543
复制
发表时间：
2019-02
影响因子：
1.4
通讯作者：
N. Ikoma;Kazunaga Tanaka
中科院分区：
文献类型：
--
作者：
N. Ikoma;Kazunaga Tanaka
我们研究非线性 Schr\"odinger 方程和系统的 $L^2$ 归一化解的存在性。在新的 Palais-Smale 类型条件下，我们为 $S_m=\{ u; \, \int_{\mathbf{R}^N} | u |^2=m\}$ 或 $S_{m_1} \times 上的约束泛函开发新的变形参数 S_{m_2}$。作为应用，我们对 [\cite[J:20], \cite[BdV:6], \cite[BS1:7]] 的结果给出其他证明。对于[\cite[J:20], \cite[BdV:6]]的结果，我们的变形结果使我们能够将属理论直接应用到相应的泛函上以获得无穷多个 解决方案。至于结果 [\cite[BS1:7]]，通过我们的变形结果，我们可以在不使用与 Pohozaev 恒等式相关的约束的情况下证明向量解的存在性。
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.