arXiv 2210.033302022-10-07math.APmath-phmath.MP

多种群非线性薛定谔方程组在${\mathbb R}^N$中的无穷多个非径向正解

Infinitely many nonradial positive solutions for multi-species nonlinear Schrödinger systems in ${\mathbb R}^N$

Tuoxin Li, Juncheng Wei, Yuanze Wu

备注：: 49 pages; any comment welcome

摘要：

本文考虑多种群非线性薛定谔系统$\bbr^N$：\begin{equation*} \left\{\aligned&-Δu_j+V_j(x)u_j=μ_ju_j^3+\sum_{i=1;i\not=j}^dβ_{i,j} u_i^2u_j\quad\text{in }\bbr^N, &u_j(x)>0\quad\text{in } {\mathbb R}^N, &u_j(x)\to0\quad\text{as }|x|\to+\infty,\quad j=1,2,\cdots,d,\endaligned\right. \end{equation*}其中$N=2,3$，$μ_j>0$为常数，$β_{i,j}=β_{j,i}\not=0$为耦合参数，$d\geq2$和$V_j(x)$为势.利用Ljapunov-Schmidt约化论证，在对势$V_j(x)$和耦合参数$\{β_{i,j}\}$的一些温和假设下，构造了上述系统的无穷多个非径向正解，{ \it对上述系统的极限情形没有任何对称性假设}.我们的结果肯定地回答了Pistoia和Vaira的\cite{PV22}猜想，推广了\cite{PW13,PV22}的结果，揭示了$N=2$和$d=2$的{ \it新现象}，并且对于耦合参数$\{β_{i,j}\}$是{ \it几乎最优的}.

英文摘要：

In this paper, we consider the multi-species nonlinear Schrödinger systems in $\bbr^N$: \begin{equation*} \left\{\aligned&-Δu_j+V_j(x)u_j=μ_ju_j^3+\sum_{i=1;i\not=j}^dβ_{i,j} u_i^2u_j\quad\text{in }\bbr^N, &u_j(x)>0\quad\text{in } {\mathbb R}^N, &u_j(x)\to0\quad\text{as }|x|\to+\infty,\quad j=1,2,\cdots,d,\endaligned\right. \end{equation*} where $N=2,3$, $μ_j>0$ are constants, $β_{i,j}=β_{j,i}\not=0$ are coupling parameters, $d\geq2$ and $V_j(x)$ are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials $V_j(x)$ and coupling parameters $\{β_{i,j}\}$, {\it without any symmetric assumptions on the limit case of the above system}. Our result, giving a positive answer to the conjecture in Pistoia and Vaira \cite{PV22} and extending the results in \cite{PW13,PV22}, reveals {\it new phenomenon} in the case of $N=2$ and $d=2$ and is {\it almost optimal} for the coupling parameters $\{β_{i,j}\}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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