Concentration phenomena in the conformal Brezis-Nirenberg problem

Abstract

We study the blow up phenomena of least energy solutions to some semilinear elliptic boundary value problem $(P_{\varepsilon, a})$ below on domains of a manifold which has a metric pointwise conformal to the Euclidean metric. Typical examples of our problem are set on domains of spaces of constant positive or negative curvature. It is known that the least energy solutions concentrate at one point in the domain as a parameter involved tends to 0. We characterize the location of concentration point of the least energy solutions as the maximum point of some function, defined by the coefficient function, the conformal factor and the (Euclidean) Robin function on the domain.