I'm interested to know if anyone can give me some information (name, references, etc.) about this PDE $$|\nabla \omega|^2= \alpha \omega^n+ \beta,$$ where $\omega$ is a scalar function of two or more variables, $\alpha$ and $\beta$ are arbitrary constants $\in$ $\mathbb{R}$.

  • $\begingroup$ I do not understand it. From the equation you get that $\omega=(-\beta/\alpha)^{1/n}$ is constant. There must be a mistake in your statement. $\endgroup$ – Piotr Hajlasz Nov 30 '18 at 13:31
  • $\begingroup$ @Piotr Hajlasz - Yes, there was a mistake, thank you! $\endgroup$ – Alexander Pigazzini Nov 30 '18 at 13:47
  • $\begingroup$ It looks like an eikonal equation with a nonlinear wave speed; that's what I would search for anyway as a start. $\endgroup$ – Willie Wong Nov 30 '18 at 14:33
  • $\begingroup$ @Willie Wong - Thank you very much! $\endgroup$ – Alexander Pigazzini Nov 30 '18 at 14:36
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    $\begingroup$ Actually, can you not transform this into a standard eikonal equation? If you let $F:\mathbb{R}\to\mathbb{R}$ be a primitive of $(\alpha z^n + \beta)^{-\frac12}$, then your equation is equivalent to $|\nabla F(\omega)| = 1$ which is the standard eikonal equation. $\endgroup$ – Willie Wong Nov 30 '18 at 14:42

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