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You are forgetting that, as part of the renormalization, the functions $\psi_s$ are uniformly bounded above by one. This together with the convergence in $C_{loc}^2(S^n \setminus \{ P \})$ implies that $\lVert \psi_s \rVert_p^p \to \lVert \psi \rVert_p^p$. Since $\lVert \varphi_s \rVert_s = 1$ by construction of $\varphi_s$, the conclusion follows.