Multiple positive solutions with prescribed masses for a coupled Schr\"odinger system: mass mixed and Sobolev critical coupled case

TL;DR

This paper proves the existence of multiple positive normalized solutions for a Sobolev critical coupled Schrödinger system, extending previous results and resolving an open problem in the field.

Contribution

It establishes the existence of two positive solutions in the mass mixed case for all dimensions N≥3, including local minimizer and mountain pass solutions, with new technical lemmas.

Findings

Existence of two positive solutions for small coupling parameter ν

Extension of results to all dimensions N≥3

Properties of the local minimizer including uniqueness and stability

Abstract

The aim of this paper is to establish multiple 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 involving Sobolev critical exponent: $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{p-2}u+\nu\alpha|u|^{\alpha-2}u|v|^\beta, x\in \mathbb{R}^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{q-2}v+\nu\beta|v|^{\beta-2}v|u|^\alpha, x\in \mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b, \end{cases} N\geq 3, $$ where $\mu_1,\mu_2, \nu, a, b>0$. We are particularly interested in the mass mixed case that $2<p, q<2+\frac{4}{N}, \alpha>1, \beta>1$, 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 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 $p,q\leq \alpha+\beta-\frac{2}{N}$ 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 limiting profiles for $\nu\rightarrow 0^+$.