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We obtain the instability of Type (II) Lawson-Osserman cones in Euclideanspaces, and thus provide a family of (uncountably many) unstable solutions withsingularity to the Dirichlet problem for minimal graphs of high codimensionversus smooth unstable ones by Lawson-Osserman through a min-max technique. Toour knowledge, these are the first examples of non-smooth unstable minimalgraphs and unlikely detectible through the mean curvature flow or min-maxtheory.

On instability of Type (II) Lawson-Osserman Cones