题目：On positive normalized solutions to a mass mixed coupled Schrödinger system with Sobolev critical exponent
摘要：I will report a recent work (Joint with Qing Guo, Qihan He and Wei Shuai) concerning positive normalized solutions $(u,v,\lambda_1,\lambda_2)\in H^1(\mathbb{R}^N,\mathbb{R}^2)\times \mathbb{R}^2$ to the following coupled Schr\"odinger system
$$
\begin{cases}
-\Delta u+\lambda_1 u=\mu_1|u|^{p-2}u+\nu\alpha|u|^{\alpha-2}u|v|^\beta, x\in \mathbb{R}^N,\\
-\Delta v+\lambda_2 v=\mu_2|v|^{q-2}v+\nu\beta|v|^{\beta-2}v|u|^\alpha, x\in \mathbb{R}^N,\\
\end{cases} N\geq 3,
$$
subject to the normalization constraint
$$\int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b.$$
Here, $\mu_1,\mu_2, \nu>0$ are given parameters, and $a,b>0$ denote the masses. We are particularly interested in the mass mixed with a Sobolev critical coupled case where $2<p, q<2+\frac{4}{N}, \alpha>1, \beta>1$, and $\alpha+\beta=2^*:=\frac{2N}{N-2}$. For sufficiently small $\nu>0$, we demonstrate that the above system admits two positive solutions, one of which acts as a local minimizer, and the other as a mountain pass solution. This result resolves Soave's open problem [{\it J. Funct. Anal.}, 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions $N\geq 3$. Our results also significantly extending the result of Gou and Jeanjean[{\it Nonlinearity}, 2018, Theorem 1.1] to the Sobolev critical coupled case and by removing the constraint ``either $p,q\leq \alpha+\beta-\frac{2}{N}$ or $|p-q|\leq \frac{2}{N}$" for $N\geq 5$. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter $\nu$, and the asymptotic behavior as $\nu\rightarrow 0^+$.
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