Normalized solutions for a class of Sobolev critical Schrödinger systems

This paper focuses on the existence and multiplicity of normalized solutions for the following coupled Schrödinger system with Sobolev critical coupling term:$\{\begin{array}{cc}& -\Delta u+{\lambda }_{1}u={\omega }_1|u{|}^{p-2}u+\frac{\alpha \nu }{2^{⁎}}|u{|}^{\alpha -2}|v{|}^{\beta }u,\text{in }{R}^N,\\ & -\Delta v+{\lambda }_{2}v={\omega }_2|v{|}^{p-2}v+\frac{\beta \nu }{2^{⁎}}|u{|}^{\alpha }|v{|}^{\beta -2}v,\text{in }{R}^N,\\ & \underset{R^N}{\int }{u}^{2}dx=a^2,\underset{R^N}{\int }{v}^{2}dx=b^2,\end{array}$ where $N\ge 3$, $a,b>0$, ${\omega }_1,{\omega }_2,\nu \in R\setminus \left\{0\right\}$, and the exponents $p,\alpha ,\beta$ satisfy$\alpha >1,\beta >1,\alpha +\beta =2^{⁎},2<p\le 2^{⁎}=2N/(N-2).$ The parameters ${\lambda }_1,{\lambda }_2\in R$ will arise as Lagrange multipliers that are not prior given. This paper mainly presents several existence and multiplicity results under explicit conditions on $a,b$ for the focusing case ${\omega }_1,{\omega }_2>0$ and attractive case $\nu >0$:

(1) When $2<p<2+4/N$, we prove that there exist two solutions: one is a local minimizer, which serves as a normalized ground state, and the other is of mountain-pass type, which is a normalized excited state.

(2) When $2+4/N\le p<2^{⁎}$, we prove that there exists a mountain-pass type solution, which serves as a normalized ground state.

(3) When $p=2^{⁎}$, the existence and classification of normalized ground states are provided for and $N\ge 5$, alongside a non-existence result for $N=3,4$. These results reflect the properties of the Aubin-Talenti bubble, which attains the best Sobolev embedding constant.

Furthermore, we present a non-existence result for the defocusing case ${\omega }_1,{\omega }_2<0$. We believe our methods can also address the open problem of the multiplicity of normalized solutions for Schrödinger systems with Sobolev critical growth, with potential for future development and broader applicability.