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.