First of all, for future reference, in Maple e is not 2.7128..., it's just another variable.  You probably want exp(-z[11]/t[1]) etc.

Second: equations involving different exponentials are generally hard to solve, but in this case the only dependence of the four equations on z[11], ..., z[22] is through exp(-z[11]/t[1]), exp(-z[21]/t[1]), exp(-z[22]/t[2]), exp(-z[12]/t[2]), so we might as well use those as variables instead of z[11] to z[22]: I'll call them x[11] to  x[22].  Now you tried to solve for z[11] to z[22], but u[11] to u[22] seem also to be variables.  Note that z[ij] = 0 corresponds to x[ij] = 1.  The constraints u[ij]*(x[ij]-1) = 0 can be included as equations too.  So:

> eqs:= { x[11]*(p[21]*x[21]+p[22]*x[22])=t[1]*(1-w-u[11])/w/p[11], 
          x[21]*(p[11]*x[11]+p[12]*x[12])=t[1]*(1-w-u[21])/w/p[21], 
          x[12]*(p[21]*x[21]+p[22]*x[22])=t[2]*(1-w-u[12])/w/p[12], 
          x[22]*(p[11]*x[11]+p[12]*x[12])=t[2]*(1-w-u[22])/w/p[22],
          u[11]*(x[11]-1)=0, u[12]*(x[12]-1)=0, u[21]*(x[21]-1)=0, u[22]*(x[22]-1)=0 };
> solve(eqs, {x[11],x[12],x[21],x[22],u[11],u[12],u[21],u[22]});

The result is a sequence of 12 solutions.  Whether any of them satisfies the nonnegativity constraints may depend on the constants w, p[11],p[12],p[21],p[22],t[1],t[2].  For example:

> eqs1:= eval(eqs, {w=1,p[11]=2,p[12]=3,p[21]=4,p[22]=5,t[1]=6,t[2]=7});
  S1:= {solve(eqs1, {x[11],x[12],x[21],x[22],u[11],u[12],u[21],u[22]})}; 

Remove the solutions where some u[ij] < 0 or some x[ij] <= 0 or some x[ij] > 1:
 

> remove(s -> ormap(is, subs(s, 
    {seq(u[ij]< 0, ij = [11,12,21,22]), seq(x[ij] <= 0, ij = [11,12,21,22]),
     seq(x[ij] > 1, ij = [11,12,21,22])})), S1);

{{u[11] = 0, u[12] = 0, u[21] = 0, u[22] = 0, x[11] = -3/2*x[12], x[12] = x[12], x[21] = -5/4*x[22], x[22] = x[22]}}

But that's not acceptable either, because if x[22] > 0 it would make x[11] and x[21]  < 0.