ِDoes the equation $y\partial_x u(x,y)x\partial_y u(x,y)+cu=0$ have complexvalued compactsupported or vanishingatinfinity $C^1$ solution defined on the whole plane without any singularity? Here $c$ is a complex number. I have used the method of characteristics but the solution is $u(x,y)=g(\sqrt{x^2+y^2})exp(c arctg(y/x))$ where g is a given function. But this solution has singularity at x=0. More generally, a similar question for the equation $(A(x).\nabla)u+cu=0$ where $A(x)$ is a real vector field over $R^n$ and $c$ is a complex number. Is there a complexvalued compactsupported or vanishingatinfinity $C^1$ solution on whole $R^n$?
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$\begingroup$ Did you also want nontrivial? Since $u\equiv 0$is a solution. $\endgroup$– Willie WongJul 14, 2021 at 15:41

$\begingroup$ In any case, your equation can be written in polar coordinates $(r,\theta)$ as $$ \partial_\theta u + c u = 0 $$ and the existence of nonsingular solution boils down to asking whether the equation $g' = c g$ has a $2\pi$ periodic solution. $\endgroup$– Willie WongJul 14, 2021 at 15:49

$\begingroup$ Since you mentioned the similar equation and method of characteristics: it depends on the geometry of the integral curves of $A$. If $A$ has closed curves then there will be at most a discrete, countable set of $c$ that will work. OTOH, if all integral curves of $A$ are unbounded, then the only compactly supported solution will be the 0 solution. $\endgroup$– Willie WongJul 14, 2021 at 15:53

$\begingroup$ @ Willie Thanks for useful comments. $\endgroup$– E.AkramiJul 18, 2021 at 12:40
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