arXiv 1908.110902019-08-30math.AP

具有竞争与合作混合项的薛定谔方程组最小能量正解的存在性：临界情形

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Hugo Tavares, Song You

备注：: 30 pages

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摘要：

本文研究了以下薛定谔方程组在临界情形下解的存在性：\begin{equation*} -Δu_{i}+λ_{i}u_{i}=u_{i}\sum_{j = 1}^{d}β_{ij}u_{j}^{2} \text{ in } Ω, \quad u_i=0 \text{ on } \partial Ω, \qquad i=1,...,d. \end{equation*}这里，$Ω\subset \mathbb{R}^{4}$是一个光滑有界区域，对于每个$i$，有$d\geq 2$，$-λ_{1}(Ω)<λ_{i}<0$和$β_{ii}>0$，对于$i\neq j$，有$β_{ij}=β_{ji}$，其中$λ_{1}(Ω)$是$-Δ$在Dirichlet边界条件下的第一特征值.在假设的组件被分为$m$组，和$β_{ij}\geq 0$（合作），只要组件$i$和$j$属于同一组，而$β_{ij}<0$或$β_{ij}$是积极的和小（竞争或弱合作）的组件$i$和$j$属于不同的群体，我们建立了非负解的存在性$m$非平凡组件，以及分类结果。此外，在$β_{ij}$的附加假设下，我们建立了混合合作和竞争的情况下最小能量正解的存在性。证明是通过归纳的数量组，并需要新的估计比较系统的能量水平与适当的子系统。在$Ω=\mathbb{R}^4$和$λ_1=\ldots=λ_m=0$的情况下，我们提出了新的不存在性结果。本文可以看作是[Soave-Tavares，J. Differential Equations 261（2016），505-537]在临界情形下的对应，同时推广和改进了[Chen-Zou，Arch. Ration.机械肛门205（2012），515--551]，[郭罗邹，非线性31（2018），314--339].

英文摘要：

In this paper we investigate the existence of solutions to the following Schrödinger system in the critical case \begin{equation*} -Δu_{i}+λ_{i}u_{i}=u_{i}\sum_{j = 1}^{d}β_{ij}u_{j}^{2} \text{ in } Ω, \quad u_i=0 \text{ on } \partial Ω, \qquad i=1,...,d. \end{equation*} Here, $Ω\subset \mathbb{R}^{4}$ is a smooth bounded domain, $d\geq 2$, $-λ_{1}(Ω)<λ_{i}<0$ and $β_{ii}>0$ for every $i$, $β_{ij}=β_{ji}$ for $i\neq j$, where $λ_{1}(Ω)$ is the first eigenvalue of $-Δ$ with Dirichlet boundary conditions. Under the assumption that the components are divided into $m$ groups, and that $β_{ij}\geq 0$ (cooperation) whenever components $i$ and $j$ belong to the same group, while $β_{ij}<0$ or $β_{ij}$ is positive and small (competition or weak cooperation) for components $i$ and $j$ belonging to different groups, we establish the existence of nonnegative solutions with $m$ nontrivial components, as well as classification results. Moreover, under additional assumptions on $β_{ij}$, we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case $Ω=\mathbb{R}^4$ and $λ_1=\ldots=λ_m=0$, we present new nonexistence results. This paper can be seen as the counterpart of [Soave-Tavares, J. Differential Equations 261 (2016), 505-537] in the critical case, while extending and improving some results from [Chen-Zou, Arch. Ration. Mech. Anal. 205 (2012), 515--551], [Guo-Luo-Zou, Nonlinearity 31 (2018), 314--339].

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