Can you provide any additional constraint information?

In[1]:= s = 95;
u = 0.1;
x = 100;
k = 0.01;
Subscript[c, A] = (Subscript[p, A]*(s*(1 + u) - x)) + k;
Subscript[c, B] = (Subscript[p, B]*(s*(1 + u) - x)) - k;
Subscript[Z, A]=(1-(v-Subscript[c,A])/(Subscript[c,B]-Subscript[c,A]))^\[Beta] (v-Subscript[c,A]) ;
Subscript[Z, B]=(1-(v-Subscript[c,B])/(Subscript[c,B]-Subscript[c,A]))^\[Alpha] (Subscript[c,B]-v);
Maximize[{Norm[Subscript[Z, A]] + Norm[Subscript[Z, B]], Subscript[c, A] <= v <= Subscript[c, B]},
   {v, \[Alpha], \[Beta], Subscript[p, A], Subscript[p, B]}]

During evaluation of In[61]:=NMaximize::cvdiv:Failed to converge to a solution. The function may be unbounded.

Out[68]= {1.945590892669753*10^514, {v -> -48.2786, \[Alpha] -> 56.0364, \[Beta] -> -102.072,
   Subscript[p, A] -> -52.0972, Subscript[p, B] -> -10.7259}}

In[1]:= s = 95; u = 0.1; x = 100; k = 0.01;
Subscript[c, A] = (Subscript[p, A]*(s*(1 + u) - x)) + k;
Subscript[c, B] = (Subscript[p, B]*(s*(1 + u) - x)) - k;
Subscript[Z, A] = (1-(v-Subscript[c,A])/(Subscript[c,B]-Subscript[c,A]))^\[Beta](v-Subscript[c,A]);
Subscript[Z, B] = (1-(v-Subscript[c,B])/(Subscript[c,B]-Subscript[c,A]))^\[Alpha](Subscript[c,B]-v);
Maximize[{Norm[Subscript[Z,A]]+Norm[Subscript[Z,B]], Subscript[c,A]<=v<=Subscript[c,B] && 
   0<=Subscript[p,A]<=1 && 0<=Subscript[p,B]<= 1}, {v, \[Alpha], \[Beta], Subscript[p,A], Subscript[p,B]}]

During evaluation of In[1]:=NMaximize::cvdiv:Failed to converge to a solution. The function may be unbounded.

Out[9]= {3.97046*10^15, {v -> 0.101697, \[Alpha] -> 53.8285, \[Beta] -> -34.2923, Subscript[p, A] -> 0.0203771, 
  Subscript[p, B] -> 0.0799841}}

You may want to look at the values of each part of ZA and ZB to determine whether more constraints are needed or whether there is something about the way that I have done this which is incorrect.

From the initial post I was guessing that you needed to maximize both of the items.

In[1]:= s = 95; u = 0.1; x = 100; k = 0.01; Subscript[c, A] = (Subscript[p, A](s(1 + u) - x)) + k; Subscript[c, B] = (Subscript[p, B](s(1 + u) - x)) - k; Subscript[Z, A] = (1 - (v - Subscript[c, A])/(Subscript[c, B] - Subscript[c, A]))^[Beta] (v - Subscript[c, A]); Subscript[Z, B] = (1 - (v - Subscript[c, B])/(Subscript[c, B] - Subscript[c, A]))^[Alpha] (Subscript[c, B] - v); Maximize[{Subscript[Z, A], Subscript[c, A] <= v <= Subscript[c, B] && 0 <= Subscript[p, A] <= 1 && 0 <= Subscript[p, B] <= 1}, {v, [Alpha], [Beta], Subscript[p, A],Subscript[p, B]}]

During evaluation of In[1]:=NMaximize::nrnum:The function value -0.113773+0.104822 I is not a real number at {v,[Alpha],[Beta],Subscript[p, A],Subscript[p, B]} = {1.24375,0.162408,-0.0367399,0.0548333,0.081051}. >>

Out[6]= {0.863531, {v -> 0.754547, [Alpha] -> -0.523666, [Beta] -> -0.734744, Subscript[p, A] -> 0.0000800015, Subscript[p, B] -> 0.906934}}

In[7]:= Maximize[{Subscript[Z, B], Subscript[c, A] <= v <= Subscript[c, B] && 0 <= Subscript[p, A] <= 1 && 0 <= Subscript[p, B] <= 1}, {v, [Alpha], [Beta], Subscript[p, A],Subscript[p, B]}]

During evaluation of In[7]:=NMaximize::nrnum:The function value 1.613510^8-4.6536810^7 I is not a real number at {v,[Alpha],[Beta],Subscript[p, A],Subscript[p, B]} = {0.377249,18.8636,-19.7831,0.162586,0.150344}. >>

Out[7]= {2.99746*10^7, {v -> 0.454771, [Alpha] -> 16.1621, [Beta] -> -15.78, Subscript[p, A] -> 0.187606, Subscript[p, B] -> 0.2869}}