Good morning. During my thesis, I have come to the following problem: suppose $(M, g)$ is a closed Riemannian manifold of dimensione greater than $2$. You have a function $\varphi \in C^0(M)$ s.t. $\varphi \ge 0$, $|| \varphi ||_\infty = 1$, $dV \{ \varphi = 0 \} = 0$ but $\{ \varphi = 0 \} \ne \varnothing$. I need to study the limit (up to subsequences!) $$ \frac{1 - \varphi^\varepsilon}{|| 1 - \varphi^\varepsilon||_p} $$ for $0<\varepsilon<1$, $p \in (1, +\infty)$, $\varepsilon \to 0$. One can easily see that if $\frac{\varepsilon}{|| 1 - \varphi^\varepsilon||_p} \to c \ne 0$, the weak limit must be different by $0$.
However, if $\frac{\varepsilon}{|| 1 - \varphi^\varepsilon||_p} \to 0$, you have that $\frac{1 - \varphi^\varepsilon}{|| 1 - \varphi^\varepsilon||_p}$ is $L^p$ limited (...) but tends to $0$ locally in $L^p(\{ \varphi > 0 \})$, so the weak limit must be $0$. My questions are:
1) Can I avoid this case? I think I can't, but I hope I am wrong.
2) Do in this case the measures $$ \frac{|1 - \varphi^\varepsilon|^p}{|| 1 - \varphi^\varepsilon||^p_p} \, dV $$ converge to a measure with support in $\{ \varphi = 0 \}$? How can I prove this? Thank you for the answers. Please feel free to use every technique you think can solve this problem.