The Newton's problem assuming non-constant density of the fluid

This paper investigates the Newton’s problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions $u(r)$ satisfying the initial conditions $u(0)=u^{\mathrm{\prime }}(0)=0$ using a fixed-point theorem. We show that the maximal domain of the solution is finite, $[0,r_M)$, terminating at a critical slope $u^{\mathrm{\prime }}(r_M)=\frac{1}{\sqrt{3}}$.