Help with Plot commands?

Questions
1

I need to identify the intersection point of the blue graph where A is equal to the maximum value. And display the intersection with two lines. Is it possible?

Open in Wolfram Cloud ↗
2

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}}]