Existence and nonexistence of least energy positive solutions to critical Schrödinger systems with Hardy potential

Abstract

We are concerned with the following coupled Schrödinger system with Hardy potential in the critical case \begin{equation*} \begin{cases} -Δu_{i}-\frac{λ_{i}}{|x|^2}u_{i}=|u_i|^{2^*-2}u_i+\sum_{j\neq i}^{3}β_{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 $2^{\ast }=\frac{2N}{N-2}$, ${\lambda }_i\in (0,{\Lambda }_N),{\Lambda }_N:=\frac{(N-2{)}^2}{4}$, ${\beta }_{ij}={\beta }_{ji}$ for $i\ne j$. 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 $i\ne j$) 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 ${\beta }_{ij}\ge 0$ and $N\ge 5$, but {\bf NOT} in the weakly pure cooperative case (${\beta }_{ij}>0$ and small, $i\ne j$) 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 $N\ge 5$. In particular, the cases of $N=3$ and $N\ge 5$ 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 $N\ge 5$.

35pages