| 程永宽,沈尧天.含参数拟线性薛定谔方程的特征值问题*[J].数学年刊A辑,2023,44(2):113~120 |
| 含参数拟线性薛定谔方程的特征值问题* |
| The Eigenvalue Problem for a Class of Quasilinear Schr¨odinger Equations with a Parameter |
| Received:September 06, 2022 Revised:March 21, 2023 |
| DOI：10.16205/j.cnki.cama.2023.0009 |
| 中文关键词: 薛定谔方程, L∞ 估计, 特征值问题 |
| 英文关键词:Schr¨odinger equations, L∞ estimate, Eigenvalue problem |
| 基金项目:广东省基础与应用基础研究基金 (No.2020A1515010338) |
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| 中文摘要: |
| 本文考虑如下拟线性薛定谔方程：-△u + κu/2△u2 = λ|u|p?2u, x ∈ ?,这里 u ∈ H10(?), 2 < p < 2*, κ > 0, N > 3 且? 是有界区域.结合变分方法和摄动讨论, 作者证明了存在常数κ0 > 0, 使得对任何的κ ∈ (0, κ0)这类特征值问题有解 (λ, u). 特别地,如果限制 |u|pp = α, 作者发现对任何的 κ > 0 存在 α0 > 0,使得在 α < α0 时, 该特征值问题的解总是存在的.此外, 作者采用不同于 Morse迭代的方法构造出了常数κ0 和 α0 的精确表达式. |
| 英文摘要: |
| This paper considers a class of quasilinear Schr¨odinger equations of the form -△u + κu/2△u2 = λ|u|p?2u, x ∈ ?, where u ∈ H10(?), 2 < p < 2*, κ > 0, N > 3 and ? is a bounded domain. Combining variational approaches with perturbation arguments, the authors prove that there exists κ0 > 0 such that for any κ ∈ (0, κ0) this eigenvalue problem admits a solution (λ, u). More interestingly, if the eigenvalue problem is restricted to |u|pp = α, the authors observe that for any κ > 0, there exists α0 > 0 such that the solution of the eigenvalue problem exists under the situation of α < α0. Particularly, the authors construct the accurate expressions of κ0 and α0 and, different from the Morse estimate, the authors use another method to show the L∞ estimate. |
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