I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) Find a function $f:\mathbb R^3 \to \mathbb R$ parametrized in spherical coordinates s.t.
$$(f - 1) \Delta f + f^2 = 0$$
where I have suppressed the argument $(r, \theta, \phi)$ to $f$ for brevity.


If I try a spherical harmonic, I have that $\Delta f = 0$, so that $f^2 = 0$ and $f$ vanishes identically. Likewise, if $f$ solves the Helmholtz equation, we have $\Delta f = -k^2 f$ and $-k^2 f(f-1) + f^2 = 0$ which also fails to yield a useful solution.


Is there an ansatz that I may choose to reduce the problem into a linear equation. I remember from quantum physics that a certain nonlinear PDE arising from the time independent Schrodinger equation in a Harmonic potential could be made linear by multiplying an unknown function by the asymptotic behavior. In that case, $f(u) = g(u) \exp\left(-\frac{u^2}{2}\right)$ was the required solution. Here, however, I am having difficulty getting an idea of the asymptotic behavior. Could I assume invariance with respect to $\theta$ and $\phi$ and try to find asymptotic behavior for $r$? Can this equation be transformed into a well known nonlinear PDE? (I had actually noticed that in the 1D rectangular case, the equation bares some resemblance to a Sturm-Liouville problem.)


**An Instructive Mistake:** I have been trying to investigate if the original equation satisfies the Painleve criterion via Mathematica, but have been unsuccessful in getting a definitive result. I am willing to broaden my search for a solution $f:\mathbb R^3 \to \mathbb C$. I recall a [certain paper](https://hal.archives-ouvertes.fr/hal-00776157/document) that I read while studying quantum physics, which made a rather bold claim (given that no stipulations were put on $G$) about solving certain classes of PDEs.



> 
> Let $G : \mathbb R\_+ \to \mathbb R$ be any real valued function and consider the generalized Schrodinger equation
> $$i\hbar \frac{\partial \psi}{\partial t} + G(-\hbar^2\Delta)\psi = 0$$
> will have a solution
> $$\psi(x,t) = e^{-itG(-\hbar^2\Delta)} \psi\_0(x)$$
> because
> $$\mathcal F(G(-\hbar^2 \Delta)\psi)(p) = G(|p|^2) \hat \psi(p)$$
> acts as a Fourier multiplier.
> 
> 
> 


With this in mind, I could manipulate my initial equation as


$$
\begin{align}
(f-1)\Delta f + f^2 &= 0 \\
f\Delta f + f^2 &= \Delta f \\
\frac{\Delta f}{\Delta f + f} &= f \\
\underbrace{\left(\frac{\Delta}{\Delta + 1}\right)}\_{G(\Delta)} f &= f 
\end{align}
$$


and define $F(r,\theta,\phi,t) = \upsilon(t)f(r,\theta,\phi)$ after neglecting constants by


$$
\begin{align}
F(r,\theta,\phi,t) &= \exp\left(-it\left(\frac{\Delta}{\Delta+1}\right)\right)f\_0(r,\theta,\phi) \\
& = \exp\left(-itG(\Delta)\right)f\_0(r,\theta,\phi)
\end{align}
$$


Could I treat the above like a flow to prove that constant phase shifts of the equation remain valid if we allow it to take complex values? I did not think that this result would be trivial because the powers of $f$ are not the same throughout the equation.


**Correction:** As @Zachary mentioned in the comments, this method fails because $G$ is a nonlinear function.


**Update:** I numerically integrated this equation under a variety of initial conditions and observed horrible ill-conditioning and poor stability. I have decided to accept the answer that no desirable solution to the equation exists.



Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?
Some simple observations at least rule out certain types of "nice" solutions. First of all, if $f=1$ anywhere, then $\Delta f$ must be singular. Moreover, if either $f>1$ or $f<1$, then $\Delta f$ has a sign opposite to $f-1$. So, if $f>1$, there cannot be a minimum of $f$, and if $f<1$, there cannot be a maximum. What qualitative properties do you require or expect?


