Sharp Brezis--Merle Inequalities for Polyharmonic Dirichlet Problems: Extremal Measures and Critical Asymptotics
We establish the Brezis--Merle inequality for very weak solutions of polyharmonic Dirichlet problems with integrable data on smooth bounded domains in conformal dimensions. The inequality is sharp both with respect to the admissible range of the exponent and to the quantity appearing on the right-hand side, which is characterized as the maximum of an associated one-pole Green functional. The supremum defining this quantity is not attained by any integrable datum, but remains unchanged when the admissible class is extended to Radon probability measures; in this enlarged class, the extremals are Dirac masses, and their concentration points are characterized in terms of the associated Green function. On the unit ball, we prove uniqueness of the extremal measure and obtain an explicit optimal bound extending the sharp result for the Laplacian. Finally, we determine the critical asymptotics of the sharp constant on the right-hand side and show that the possible limiting concentration points are selected by the maximum points of the polyharmonic Robin function.