My DG code run well on P<2,but it has spurious oscillations near discontinuities in high order, who knows how to build a high order limiter to treat with the discontinuity?

See Hesthaven's book. You'll find the answer, not only how to supress the wiggles but how to develop high order DG method p >> 1.

If memory serves, there are no higher order limiters. They all make the solution first order locally. I think there is a proof that shows that any scheme higher than order 1 is not monotonicity preserving. This causes the introduction of spurious waves. That being said, these limiters do not degrade the entire solution to a first order one. It only has a local effect. In addition, if you look at it carefully, I think it is easy to show that first order is the best you can hope for when trying to resolve a discontinuity.

Hope this helps.