论文标题
多种物种的非线性schrödinger系统$ {\ mathbb r}^n $中的多种非物种的非放置正解
Infinitely many nonradial positive solutions for multi-species nonlinear Schrödinger systems in ${\mathbb R}^N$
论文作者
论文摘要
在本文中，我们考虑了$ \ bbr^n $中的多物种非线性schrödinger系统： \ begin {equation*} \左\ {\ Aligned＆-Δu_j+v_j（x）u_j =μ_JU_J^3+\ sum_ {i = 1; i \ not = j}^dβ_{i，j} u_i^u_i^2u_j^2U_j \ quad \ quad \ quad \ quad \ quad \ quad \ quad \ quad \ quad \ bbr^in} ＆u_j（x）> 0 \ quad \ text {in} {\ mathbb r}^n， ＆u_j（x）\ to0 \ quad \ text {as} | x | \ to+\ infty，\ quad j = 1,2，\ cdots，d，d，\ endaligned \ right。 \ end {equation*} 其中$ n = 2,3 $，$μ_j> 0 $是常数，$β_{i，j} =β_{j，i} \ not = 0 $是耦合参数，$ d \ geq2 $和$ v_j（x）$是潜力。通过Ljapunov-schmidt的减少论点，我们在对潜在的一些轻度假设上构建了上述系统的许多非广泛的积极解决方案，$ v_j（x）$和耦合参数$ \ \ {β_{β_{i，j} \} \} \} $，{\ IT，{\ IT，{\ IT没有任何对称假设，对限制系统}。我们的结果，对Pistoia和Vaira \ Cite {pv22}的猜想给出了积极的答案，并将结果扩展在\ cite {pw13，pv22}中，在$ n = 2 $ and $ d = 2 $ and is camit中揭示了{\ it New new new new new new eToMenon} $ \ {β_{i，j} \} $。
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 $\{β_{i,j}\}$.
