Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\{\varphi_k\}_k \in C^\infty(M)$ such that $\{\varphi_k\}_k$ satisfy the basic concentration compactness assumption:
$$\forall k: \hspace{0.5cm} \Vert \varphi_k\Vert_{L^p} = \text{constant.} \text{ }$$
And if that helps, assume that $p$ is the critical Sobolev constant, i.e. $p = 2^* = \frac{2n}{n - 2}.$
After passing to a subsequence, which we still call $\{\varphi_k\}_k$, we have weak$^*$ convergence: $$\varphi_k^p dM \to d\mu,\: k \rightarrow \infty,$$
where $d\mu$ is a Radon measure.
My question is: Are there any known additional sufficient conditions on $\{\varphi_k\}_k$ or on the manifold $M$ that rule out concentration and dichotomy? This is mainly a reference request.
I am particularly interested in the sufficient conditions that rule out concentration. In other words, I want to avoid the following scenario: $$d\mu = c\delta_x,$$ where $x \in M$ and $c$ is a constant.
When I say dichotomy, I mean: there exists $x \in M$ and an open set $U \subset M$, $x \notin U$ such that: $$d\mu(\{ x\}) > 0, \hspace{1cm} d\mu (U) > 0.$$
Such problems come up in variational characterizations to nonlinear elliptic PDEs on manifolds with power-type nonlinearity.
