Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

TL;DR

This paper investigates the existence of normalized ground state solutions for a coupled semilinear elliptic system with critical and subcritical nonlinearities, analyzing the Pohozaev manifold and solving open problems in the field.

Contribution

It introduces new methods to establish the existence of positive normalized ground states for systems with Sobolev critical nonlinearities, addressing previously unresolved issues.

Findings

Existence of positive normalized ground states under certain conditions.

Analysis of the Pohozaev manifold's geometry.

Resolution of open problems in elliptic systems with critical nonlinearities.

Abstract

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -\Delta u+\lambda_1u=\mu_1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -\Delta v+\lambda_2v=\mu_2 |v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbb R^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $\beta$, we obtain the existence of the positive normalized ground state solution to the above system. We have solved some unsolved open problems in this area.