Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2023-10-31 , DOI: 10.1007/s41980-023-00823-2 Ziheng Zhang , Jianlun Liu , Qingle Guan
This paper is concerned with the existence and multiplicity of normalized solutions to the following biharmonic Schrödinger equation:
$$\begin{aligned} \left\{ \begin{array}{ll} {\Delta }^2u-h(\varepsilon x) |u|^{p-2}u=\lambda u\quad \text{ in }\ {\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N} u^2 {\textrm{d}}x = c, \\ \end{array} \right. \end{aligned}$$where \(\varepsilon , c>0,\) \(N\ge 1,\) \(2<p<2+\frac{8}{N},\) \(\lambda \in {\mathbb {R}}\) is a Lagrangian multiplier and \(h:{\mathbb {R}}^N\rightarrow {{\mathbb {R}}}\) is a continuous function. Under a class of reasonable assumptions on h, we obtain the existence of ground-state normalized solutions. Meanwhile, we also prove that the number of normalized solutions is at least the number of global maximus points of h when \(\varepsilon \) is small enough. Some recent results are generalized and improved.
中文翻译：
亚临界增长双调和薛定谔方程归一化解的存在性和重数
本文关注以下双调和薛定谔方程的归一化解的存在性和多重性：
$$\begin{对齐} \left\{ \begin{array}{ll} {\Delta }^2u-h(\varepsilon x) |u|^{p-2}u=\lambda u\quad \text { 在 }\ {\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N} u^2 {\textrm{d}}x = c, \\ \end{array } \正确的。\end{对齐}$$其中\(\varepsilon , c>0,\) \(N\ge 1,\) \(2<p<2+\frac{8}{N},\) \(\lambda \in {\mathbb { R}}\)是拉格朗日乘子，\(h:{\mathbb {R}}^N\rightarrow {{\mathbb {R}}}\)是连续函数。在h的一类合理假设下，我们得到基态归一化解的存在性。同时，我们还证明了当\(\varepsilon \)足够小时，归一化解的数量至少为h的全局极大点的数量。最近的一些结果得到了概括和改进。
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