Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

Abstract

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-δu+λ_1 u=|u|^{2^*-2}u+{να} |u|^{α-2}|v|^βu,\quad \text{in }\mathbb{R}^N, &-δv+λ_2 v=|v|^{2^*-2}v+{νβ} |u|^α|v|^{β-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $\alpha ,\beta >1$, $2<\alpha +\beta <2^{\ast }=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $\nu <0$. When $\nu >0$ and $\alpha +\beta \le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<\nu <{\nu }_0$, the constant ${\nu }_0$ will be explicitly given. In the case $\alpha +\beta >2+\frac{4}{N}$ we prove the existence of a threshold ${\nu }_1\ge 0$ such that a normalized ground state solution exists for $\nu >{\nu }_1$, and does not exist for $\nu <{\nu }_1$. We also give conditions for ${\nu }_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $\nu \to 0^{+}$ or $\nu \to +\infty$.

References in corpus (3)

• Bose-Einstein condensation: Twenty years after
• Superfluid Fermi-Fermi mixture: phase diagram, stability, and soliton formation
• Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

Cited by in corpus (1)

• Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities