# $|\nabla \omega|^2= \alpha \omega^n+ \beta$ - References

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 $$\in$$ $$\mathbb{R}$$.

• 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. – Piotr Hajlasz Nov 30 '18 at 13:31
• @Piotr Hajlasz - Yes, there was a mistake, thank you! – Alexander Pigazzini Nov 30 '18 at 13:47
• It looks like an eikonal equation with a nonlinear wave speed; that's what I would search for anyway as a start. – Willie Wong Nov 30 '18 at 14:33
• @Willie Wong - Thank you very much! – Alexander Pigazzini Nov 30 '18 at 14:36
• 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. – Willie Wong Nov 30 '18 at 14:42