Existence and nonexistence of least energy positive solutions to critical Schr\"{o}dinger systems with Hardy potential

TL;DR

This paper investigates the existence and nonexistence of least energy positive solutions for a critical Schrödinger system with Hardy potential, revealing dimension-dependent phenomena and introducing new analytical techniques.

Contribution

It establishes new results on solution existence for coupled Schrödinger systems with Hardy potential, highlighting differences across dimensions and cases of cooperation and competition.

Findings

Existence of ground states for N≥5 with nonnegative coupling.

Nonexistence of ground states in weakly cooperative cases when N=3,4.

Dimension-dependent complexity in solution behavior.

Abstract

We are concerned with the following coupled Schr\"{o}dinger system with Hardy potential in the critical case \begin{equation*} \begin{cases} -\Delta u_{i}-\frac{\lambda_{i}}{|x|^2}u_{i}=|u_i|^{2^*-2}u_i+\sum_{j\neq i}^{3}\beta_{ij}|u_{j}|^{\frac{2^*}{2}}|u_i|^{\frac{2^*}{2}-2}u_i, ~x\in \mathbb{R}^N, u_i\in D^{1,2}(\mathbb{R}^N),\,\, N\geq 3,\,\, i=1,2,3, \end{cases} \end{equation*} where $2^*=\frac{2N}{N-2}$, $\lambda_i\in (0,\Lambda_N), \Lambda_N:= \frac{(N-2)^2}{4}$, $\beta_{ij}=\beta_{ji}$ for $i \neq j$. By virtue of variational methods, we establish the existence and nonexistence of least energy solutions for the purely cooperative case ($\beta_{ij}> 0$ for any $i\neq j$) and the simultaneous cooperation and competition case ($\beta_{i_{1}j_{1}}>0$ and $\beta_{i_{2}j_{2}}<0$ for some $(i_{1}, j_{1})$ and $(i_{2}, j_{2})$). Moreover, it is shown that fully nontrivial ground state solutions exist when $\beta_{ij}\ge0$ and $N\ge5$, but {\bf NOT} in the weakly pure cooperative case ($\beta_{ij}>0$ and small, $i\neq j$) when $N=3,4$. We emphasize that this reveals that the existence of ground state solutions differs dramatically between $N=3, 4$ and higher dimensions $N\geq 5$. In particular, the cases of $N=3$ and $N\geq 5$ are more complicated than the case of $N=4$ and the proofs heavily depend on the dimension. Some novel tricks are introduced for $N=3$ and $N\ge5$.