Sharp Estimates for Bubbling Solutions of an Even-Order Mean Field Equation

In this article, we investigate a sequence of multi-bubble solutions [Formula: see text] to the following mean field equation: [Formula: see text] where [Formula: see text] is a bounded and smooth domain in [Formula: see text] with [Formula: see text] as a positive integer, [Formula: see text] is a [Formula: see text] positive function, [Formula: see text] are constants such that [Formula: see text], for some constant [Formula: see text] and [Formula: see text], [Formula: see text] stand for either Navier or Dirichlet boundary conditions. We show that (after passing to a subsequence if necessary) [Formula: see text] for some positive integer [Formula: see text]. Furthermore, we obtain the following sharp estimates of [Formula: see text]: [Formula: see text] where [Formula: see text] is a positive generic constant, [Formula: see text] is the Green function of [Formula: see text] with either Navier or Dirichlet boundary conditions, [Formula: see text] is the regular part of [Formula: see text], [Formula: see text] is the local maximum point of [Formula: see text] in a neighborhood of [Formula: see text] with [Formula: see text] as the blow-up point of [Formula: see text] for each [Formula: see text], and [Formula: see text]. Our approach extends the works of Chen-Lin [20] and Lin-Wei [37], which studied the second-order and fourth-order equation, to the general even-order equation. Moreover, this result also hold for other boundary condition, if boundary blow-up can be excluded.