Let $P,Q$ and $R$ are continuous functions of $x,y,u$ and consider the first order PDE: $$ Pu_x+Qu_y=R, $$ such that $u=g$ on a curve $\Gamma$ where $u=u(x,y)$ solves the above PDE.
Is there any condition when the above equation have no solution and have infinite no of solutions?
I know if $\Gamma$ satisfies the non-characteristic condition (or transversility condition), then it has unique solution.
But, I am unable to find any condition for which it has no or infinite no of solutions.
Any help would be appreciated.
Thank you.