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ContourPlot[-8 a^2 x + 9 x^2 - 12 x^3 + 4 x^4 == 
  0, {a, -.6, .6}, {x, -1, 3}, FrameLabel -> Automatic, 
 MaxRecursion -> 4, PlotPoints -> 100, 
 RegionFunction -> Function[{a, x}, x > 1*^-16]]

I am not getting how to set the desire range in this plot. The required range is shown in this attached figure.

Required range

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Your code works fine for me (v11.0):

enter image description here

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