Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings

TL;DR

This paper studies positive solutions to coupled Sobolev critical Schrödinger equations with mass constraints, revealing existence of multiple solutions and their asymptotic behaviors for small coupling parameters.

Contribution

It establishes the existence of two positive solutions, including a local minimizer and a mountain pass solution, for the first time in the mass mixed case with Sobolev critical nonlinearity.

Findings

Existence of two positive solutions for small coupling parameter .

One solution is a local minimizer, the other a mountain pass solution.

As , solutions exhibit specific asymptotic behaviors.

Abstract

We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{2^*-2}u+\nu\alpha |u|^{\alpha-2}|v|^{\beta}u ~\hbox{in}~ \R^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{2^*-2}v+\nu\beta |u|^{\alpha}|v|^{\beta-2}v ~\hbox{in}~ \R^N \end{cases} $$ subject to the mass constraints $\int_{\mathbb{R}^N}|u|^2 \ud x=a^2$ and $\int_{\mathbb{R}^N}|v|^2 \ud x=b^2$, where, $a>0,\,b>0,\,N=3,4$ and $2^*:=\frac{2N}{N-2}$ is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i. e., $ \alpha>1,\beta>1,\alpha+\beta<2+\frac{4}{N}$. For some suitable small $\nu>0$, we show that the above system admits two positive solutions, one of which is a local minimizer, and another one is a mountain pass solution. Moreover, as $\nu\to0^+$, asymptotic behaviors of solutions are also considered. Our result gives an affirmative answer to a Soave's type open problem raised by Bartsch {\it et al.} (Calc. Var. Partial Differential Equations 62(1), Paper No. 9, 34, 2023).