Does this what you want?
{maxA, pmax} = {#1, p /. #2} & @@ FindMaximum[Subscript[A, max], p];
Plot[{Subscript[A, max], Subscript[A, n0], Subscript[A, n2], Subscript[A, n3]}, {p, 0, 200},
PlotStyle -> {{Thick}, {Dashed}, {Dashed}, {Dashed}},
AxesLabel -> {p, A}, PlotRange -> {0, 0.007},
PlotLegends -> LineLegend[{n = 15, n = 0, n = 30, n = 60}],
GridLines -> {{pmax}, {maxA}}]