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.](https://arxiv.org/pdf/math/0610266.pdf)


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?



Ground state for non-linear Schrödinger
A useful perspective on this is given in


*Weinstein, Michael I.*, [**Nonlinear Schrödinger equations and sharp interpolation estimates**](http://dx.doi.org/10.1007/BF01208265), Commun. Math. Phys. 87, 567-576 (1983). [ZBL0527.35023](https://zbmath.org/?q=an:0527.35023).


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](https://zbmath.org/?q=an:1106.35001).


for some further discussion of the energy-subcritical case.


