Look at the first equation.  LHS is never negative, so neither is RHS.  Similarly, Q, R, S are >=0 too.

Say you can order the unknowns P<Q<R<S.  Then substituting P for Q in first equation would make LHS<3P.  But the modified LHS=3Q by the second equation giving 3Q<3P, an impossibility.  By extension the same is true for the other unknowns.  Therefore they are all equal and the only solutions are the ones Jer noted:  P=0 and P=1/3.