Lazer–McKenna conjecture: The critical case

We consider an elliptic problem of Ambrosetti–Prodi type involving critical Sobolev exponent on a bounded smooth domain of dimension six or higher. By constructing solutions with many sharp peaks near the boundary of the domain, but not on the boundary, we prove that the number of solutions for this problem is unbounded as the parameter tends to infinity, thereby proving the Lazer–McKenna conjecture in the critical case.