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- W4225559255 abstract "In the present paper, we consider the coupled Schr{o}dinger systems with critical exponent: begin{equation*} begin{cases} -Delta u_i+lambda_{i}u_i=sumlimits_{j=1}^{d} beta_{ij}|u_j|^{3}|u_i|u_i quad ~text{ in } Omega, u_i in H_0^1(Omega) ,quad i= 1,2,...,d. end{cases} end{equation*} Here, $Omegasubset mathbb{R}^{3}$ is a smooth bounded domain, $d geq 2$, $beta_{ii}>0$ for every $i$, and $beta_{ij}=beta_{ji}$ for $i neq j$. We study a Br'{e}zis-Nirenberg type problem: $-lambda_{1}(Omega)<lambda_{1},cdots,lambda_{d}<-lambda^*(Omega)$, where $lambda_{1}(Omega)$ is the first eigenvalue of $-Delta$ with Dirichlet boundary conditions and $lambda^*(Omega)in (0, lambda_1(Omega))$. We acquire the existence of least energy positive solutions to this system for weakly cooperative case ($beta_{ij}>0$ small) and for purely competitive case ($beta_{ij}leq 0$) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case $Ngeq 5$. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schr{o}dinger system in dimension three." @default.
- W4225559255 created "2022-05-05" @default.
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- W4225559255 date "2022-04-01" @default.
- W4225559255 modified "2023-10-12" @default.
- W4225559255 title "Least energy positive soultions for $d$-coupled Schr{o}dinger systems with critical exponent in dimension three" @default.
- W4225559255 doi "https://doi.org/10.48550/arxiv.2204.00748" @default.
- W4225559255 hasPublicationYear "2022" @default.
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