ClearAll["Global`*"] // Simplify
lhs = D[\[Rho][s]^\[Gamma], s];
sol = \[Rho][s] /.
   DSolve[{lhs == -c1 \[Rho][s] s, \[Rho][0] == \[Rho]0}, \[Rho][s],
     s][[1]];
assume = s > 0 && \[Rho]0 > 0 && c1 > 0 && \[Gamma] > 1;
smax = Solve[sol == 0, s] // Simplify[#, Assumptions -> assume] &
If[Refine[(smax /. #) > 0, assume], #, ## &[]] & /@ smax