Multiple positive solutions with prescribed masses for a coupled Schrödinger system: mass mixed and Sobolev critical coupled case

Abstract: The aim of this paper is to establish multiple positive normalized solutions $(u,v,λ1,λ_2)\in H^1(\mathbb{R}^N,\mathbb{R}^2)\times \mathbb{R}^2$ to the following coupled Schrödinger system involving Sobolev critical exponent: $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{p-2}u+να|u|^{α-2}u|v|^β, x\in \mathbb{R}^N,\ -Δv+λ_2 v=μ_2|v|^{q-2}v+νβ|v|^{β-2}v|u|^α, x\in \mathbb{R}^N,\ \int{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b, \end{cases} N\geq 3, $$ where $μ_1,μ_2, ν, a, b>0$. We are particularly interested in the mass mixed case that $2<p, q\<2+\frac{4}{N}, α\>1, β>1$, and $α+β=2^*:=\frac{2N}{N-2}$. For sufficiently small $ν>0$, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to 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 extend the result of Gou and Jeanjean [{\it Nonlinearity}, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis ``either $p,q\leq α+β-\frac{2}{N}$ 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 $ν$, and the limiting profiles for $ν\rightarrow 0^+$.