I have been trying to solve the following recursive equation for $0<x_c<1$ for few a days: $$ P(x) = 2\mathbf{1}_{0\leq x\leq x_c} + 2\int_x^1 dy P\left(\frac{x}{y}\right)y^{-1} \mathbf{1}_{x_c< y \leq 1 } $$
I know that the function has to be $0$ for $x>x_c$ and the only non-trivial part is $0\leq x \leq x_c$. I tried to translate the above recursion relation for $P(x)$ into an equation for the Mellin transform $M(s) = \int_0^{\infty} P(x) x^{s-1}$ which has the solution (?)
$$M(s) = \frac{x_c^s}{s-2+2x_c^s}$$
But I am not sure if I can trust this result and what the inverse Melin transform and therefore $P(x)$ is. Is there any other approach?
$P(x)$ is defined on the non-negative reals.