When studying the blow-up for focusing non-linear Schrödinger equation (NLS) one often compares the initial-state to a stationary solution.

In the energy-critical case, this stationary solution is for example

$$ \Delta W+ \vert W\vert^{\frac{4}{N-2}}W=0 $$

see Kenig-Merle.

For the mass-critical case however this ground state satisfies

$$ \Delta W + \vert W\vert^{1+\frac{4}{N}}W =W.$$

What I would like to know. Why is there this additional term $W$ in the mass-critical case?


A useful perspective on this is given in

Weinstein, Michael I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87, 567-576 (1983). ZBL0527.35023.

As observed in that paper, the ground state for NLS with energy-critical or sub-critical exponent $2 \leq p \leq 1 + \frac{4}{N-2}$ can be viewed as the extremiser for the Gagliardo-Nirenberg inequality

$$ \frac{\int_{{\bf R}^N} |u|^{p+1}}{(\int_{{\bf R}^N} |u|^2)^{1-\frac{(N-2)(p-1)}{4}} (\int_{{\bf R}^N} |\nabla u|^2)^{N(p-1)/4}}.$$

Standard Euler-Lagrange calculations then shows in the energy-subcritical case $p < 1 + \frac{4}{N-2}$ that these extremisers (formally at least) obey the ground state equation $\Delta u + \alpha |u|^{p-1} u = \beta u$ with some Lagrange multipliers $\alpha,\beta$ that can be renormalised to $1$ by suitable rescaling. But in the energy critical case $p = 1 + \frac{4}{N-2}$, the mass term $\int_{{\bf R}^N} |u|^2$ in the above functional disappears, and so does the Lagrange multiplier $\beta$.

See also Appendix B of my textbook

Tao, Terence, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics 106. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4143-2/pbk). xv, 373 p. (2006). ZBL1106.35001.

for some further discussion of the energy-subcritical case.


In a physics context the nonlinear Schrödinger equation describes the ground state wave function $W(\mathbf{x})$ of a two-dimensional interacting atomic gas, with interaction potential $m(\mathbf{x})|W|^2$ in an external potential $V(\mathbf{x})$ at chemical potential $\mu$:

$$-\Delta W(\mathbf{x})+V(\mathbf{x})W(\mathbf{x})+\mu W(\mathbf{x})=m(\mathbf{x})|W(\mathbf{x})|^2 W(\mathbf{x}).$$

For the mass-critical case one has $m(\mathbf{x})=\pm 1$. The equation in the OP corresponds to the case $m=1$, $V=0$, $\mu=1$, so a mass-critical system at nonzero chemical potential. As explained in Threshold behavior and uniqueness of ground states for mass critical inhomogeneous Schrödinger equation, the solution for $\mu=1$ can be mapped onto the solution for other values of $\mu$.

  • $\begingroup$ so why do we now have the choices $\mu=1$ and $\mu=0$ in the two scenarios? $\endgroup$ – Sascha Aug 28 '19 at 10:47
  • $\begingroup$ $\mu\neq 0$ spoils the energy-criticality; for the mass-critical case you only want the mass term $m$ to be unaffected by the scaling, so you can allow for $\mu\neq 0$. $\endgroup$ – Carlo Beenakker Aug 28 '19 at 11:45

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