as $\varepsilon \to 0$, where we used the Definition 3.2 of Weyl$_g \otimes B$. Using (4.35) and plugging (5.11), (5.12), (5.14), (5.17), (5.18) in (5.10) proves (1.5) and concludes the proof of Theorem 1.1.

5.2. **The case** $n = 4$. It follows from (2.4), (4.4), (5.2) and (5.5) that

$$ (5.19) \quad \lim_{R \to +\infty} \limsup_{\varepsilon \to 0} \sup_{x \in B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} \left| \frac{|x|^2 v_\varepsilon(x)}{\mu_\varepsilon} - \lambda(V) \right| \quad \text{and} \\ \lim_{R \to +\infty} \limsup_{\varepsilon \to 0} \sup_{x \in B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} \left| \frac{|x|^3 \nabla v_\varepsilon(x)}{\mu_\varepsilon} + 2\lambda(V) \frac{x}{|x|} \right| = 0. $$

For any fixed $R > 0$ we have, by (5.5) and (5.19),

$$ \int_{B(0, \delta)} \hat{h}_{\varepsilon} v_{\varepsilon}^2 dx = O(R^4 \mu_{\varepsilon}^2) + (\hat{h}_{\varepsilon}(0) + \varepsilon_R) \mu_{\varepsilon}^2 \int_{B(0, R^{-1}) \setminus B(0, R\mu_{\varepsilon})} \frac{\lambda(V)^2}{|x|^4} dx $$

where $\varepsilon_R \to 0$ as $R \to +\infty$. As a consequence,

$$ \int_{B(0, \delta)} \hat{h}_{\varepsilon} v_{\varepsilon}^2 dx = (\lambda(V)^2 \omega_3 \hat{h}_{\varepsilon}(0) + o(1)) \mu_{\varepsilon}^2 \ln \frac{1}{\mu_{\varepsilon}} $$

as $\varepsilon \to 0$. By (5.8) and going back to (5.14) we therefore have, when $n = 4$,

$$ (5.20) \quad \int_{B(0, \delta)} (x^k \partial_k v_\varepsilon + v_\varepsilon) \hat{h}_\varepsilon v_\varepsilon dx \\ = -\lambda(V)^2 \omega_3 \left(h(x_0) - \frac{1}{6} S_g(x_0)\right) \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon} + o\left(\mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon}\right) $$

as $\varepsilon \to 0$, where as before we have let $x_0 = \lim_{\varepsilon \to 0} x_\varepsilon$. We are left with the term (5.16). Integrating by parts and using (5.5) shows that

$$ (5.21) \quad \int_{B(0, \delta)} x^p x^q \partial_{ij}^2 v_\varepsilon \left( x^k \partial_k v_\varepsilon + \frac{n-2}{2} v_\varepsilon \right) dx \\ = \int_{B(0, \delta)} x^p x^q \partial_i v_\varepsilon \partial_j v_\varepsilon dx + O(\mu_\varepsilon^2). $$

Arguing as above we get by (5.5) and (5.19) that

$$ \begin{align*} & \text{Weyl}_g(x_0)_{ipqj} \int_{B(0, \delta)} x^p x^q \partial_i v_\varepsilon \partial_j v_\varepsilon \, dx \\ &= O(R^4 \mu_\varepsilon^2) + \text{Weyl}_g(x_0)_{ipqj} \int_{B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} x^p x^q \partial_i v_\varepsilon \partial_j v_\varepsilon \, dx \\ &= 4\lambda(V)^2 \mu_\varepsilon^2 \text{Weyl}_g(x_0)_{ipqj} \int_{B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} x^p x^q \frac{x_i}{|x|^4} \frac{x_j}{|x|^4} \, dx \\ &\quad + O(R^4 \mu_\varepsilon^2) + O\left(\varepsilon_R \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon}\right) \\ &= O(R^4 \mu_\varepsilon^2) + O\left(\varepsilon_R \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon}\right) \end{align*} $$

where as before $\lim_{R \to +\infty} \varepsilon_R = 0$ and where the last term involving $\text{Weyl}_g(x_0)$ vanishes by antisymmetry of $\text{Weyl}_g(x_0)$. Going back to (5.16) with (5.21) we have
as $\varepsilon \to 0$, where we used the Definition 3.2 of Weyl$_g \otimes B$. Using (4.35) and plugging (5.11), (5.12), (5.14), (5.17), (5.18) in (5.10) proves (1.5) and concludes the proof of Theorem 1.1.

5.2. **The case** $n = 4$. It follows from (2.4), (4.4), (5.2) and (5.5) that

$$ (5.19) \quad \lim_{R \to +\infty} \limsup_{\varepsilon \to 0} \sup_{x \in B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} \left| \frac{|x|^2 v_\varepsilon(x)}{\mu_\varepsilon} - \lambda(V) \right| \quad \text{and} \\ \quad \lim_{R \to +\infty} \limsup_{\varepsilon \to 0} \sup_{x \in B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} \left| \frac{|x|^3 \nabla v_\varepsilon(x)}{\mu_\varepsilon} + 2\lambda(V) \frac{x}{|x|} \right| = 0. $$

For any fixed $R > 0$ we have, by (5.5) and (5.19),

$$ \int_{B(0,\delta)} \hat{h}_{\varepsilon} v_{\varepsilon}^2 dx = O(R^4 \mu_{\varepsilon}^2) + (\hat{h}_{\varepsilon}(0) + \varepsilon_R) \mu_{\varepsilon}^2 \int_{B(0, R^{-1}) \setminus B(0, R\mu_{\varepsilon})} \frac{\lambda(V)^2}{|x|^4} dx $$

where $\varepsilon_R \to 0$ as $R \to +\infty$. As a consequence,

$$ \int_{B(0,\delta)} \hat{h}_{\varepsilon} v_{\varepsilon}^2 dx = (\lambda(V)^2 \omega_3 \hat{h}_{\varepsilon}(0) + o(1)) \mu_{\varepsilon}^2 \ln \frac{1}{\mu_{\varepsilon}} $$

as $\varepsilon \to 0$. By (5.8) and going back to (5.14) we therefore have, when $n = 4$,

$$ (5.20) \quad \int_{B(0,\delta)} (x^k \partial_k v_\varepsilon + v_\varepsilon) \hat{h}_\varepsilon v_\varepsilon dx \\ = -\lambda(V)^2 \omega_3 \left(h(x_0) - \frac{1}{6} S_g(x_0)\right) \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon} + o\left(\mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon}\right) $$

as $\varepsilon \to 0$, where as before we have let $x_0 = \lim_{\varepsilon \to 0} x_\varepsilon$. We are left with the term (5.16). Integrating by parts and using (5.5) shows that

$$ (5.21) \quad \int_{B(0,\delta)} x^p x^q \partial_{ij}^2 v_\varepsilon \left( x^k \partial_k v_\varepsilon + \frac{n-2}{2} v_\varepsilon \right) dx \\ = \int_{B(0,\delta)} x^p x^q \partial_i v_\varepsilon \partial_j v_\varepsilon dx + O(\mu_\varepsilon^2). $$

Arguing as above we get by (5.5) and (5.19) that

$$ \begin{align*} & \text{Weyl}_g(x_0)_{ipqj} \int_{B(0,\delta)} x^p x^q \partial_i v_\varepsilon \partial_j v_\varepsilon dx \\ &= O(R^4 \mu_\varepsilon^2) + \text{Weyl}_g(x_0)_{ipqj} \int_{B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} x^p x^q \partial_i v_\varepsilon \partial_j v_\varepsilon dx \\ &= 4\lambda(V)^2 \mu_\varepsilon^2 \text{Weyl}_g(x_0)_{ipqj} \int_{B(0, R^{-1}) \setminus B(0, R\mu_\varepsilon)} x^p x^q \frac{x_i}{|x|^4} \frac{x_j}{|x|^4} dx \\ &\quad + O(R^4 \mu_\varepsilon^2) + O\left(\varepsilon_R \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon}\right) \\ &= O(R^4 \mu_\varepsilon^2) + O\left(\varepsilon_R \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon}\right) \end{align*} $$

where as before $\lim_{R \to +\infty} \varepsilon_R = 0$ and where the last term involving $\text{Weyl}_g(x_0)$ vanishes by antisymmetry of $\text{Weyl}_g(x_0)$. Going back to (5.16) with (5.21) we have
proven that

$$ (5.22) \qquad \int_{B(0,\delta)} \left( x^k \partial_k v_\varepsilon + \frac{n-2}{2} v_\varepsilon \right) A_\varepsilon(v_\varepsilon) dx = o\left( \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon} \right) $$

as $\varepsilon \to 0$. Remark that by (2.9) we have $2\omega_3\lambda(V) = \int_{\mathbb{R}^4} |V|^2 V dx$. With this observation, using (4.35) and plugging (5.11), (5.12), (5.14), (5.20) and (5.22) in (5.10), we get (1.6), which proves Theorem 1.2 for $n = 4$.

5.3. The case $n = 3$. We are left with the case $n = 3$. Using (5.11) and (5.13) the Pohozaev identity (5.10) writes as

$$ (5.23) \qquad \begin{aligned} & \int_{\partial B(0,\delta)} \left( \frac{1}{2} v_\varepsilon \partial_\nu v_\varepsilon - \frac{\delta}{2} |\nabla v_\varepsilon|^2 + \delta(\partial_\nu v_\varepsilon)^2 + \frac{1}{p_\varepsilon} \Lambda_{x_\varepsilon}^{p_\varepsilon-6} \delta |v_\varepsilon|^{p_\varepsilon} \right) d\sigma \\ & = \frac{1}{12} (6-p_\varepsilon) \int_{\mathbb{R}^n} |V|^6 dx + o(|p_\varepsilon - 6|) \\ & + \int_{B(0,\delta)} \left( x^k \partial_k v_\varepsilon + \frac{1}{2} v_\varepsilon \right) \tilde{A}_\varepsilon(x) dx \end{aligned} $$

with

$$ \tilde{A}_\varepsilon(x) := (\Delta_{g_\varepsilon} - \Delta_\xi) v_\varepsilon + \left( \frac{1}{8} S_{g_\varepsilon} + \hat{h}_\varepsilon \right) v_\varepsilon, $$

where $\hat{h}_\varepsilon$ is given by (5.8) and where $v_\varepsilon$ is defined in (5.3). First, by using (5.5) together with (5.15), we see that

$$ |\tilde{A}_\varepsilon(x)| \lesssim \frac{\mu_\varepsilon^{\frac{1}{2}}}{\mu_\varepsilon + |x|} \quad \text{for } x \in B(0, \delta). $$

As a consequence, straightforward computations show that

$$ (5.24) \qquad \left| \int_{B(0,\delta)} \left( x^k \partial_k v_\varepsilon + \frac{1}{2} v_\varepsilon \right) \tilde{A}_\varepsilon(x) dx \right| = O(\delta \mu_\varepsilon) $$

as $\varepsilon \to 0$. Independently, (5.5) shows that

$$ (5.25) \qquad \left| \int_{\partial B(0,\delta)} \frac{1}{p_\varepsilon} \Lambda_{x_\varepsilon}^{p_\varepsilon-6} \delta |v_\varepsilon|^{p_\varepsilon} d\sigma \right| = o(\mu_\varepsilon) $$

as $\varepsilon \to 0$. We now compute the remaining boundary term. It follows from (4.4), (5.2), (5.5), (5.7) and standard elliptic theory that

$$ (5.26) \qquad \lim_{\varepsilon \to 0} \frac{v_\varepsilon(x)}{\mu_\varepsilon^{\frac{1}{2}}} = \lambda \hat{G}(x) \quad \text{in } C_{\text{loc}}^2(B_{2\delta}(0)\setminus\{0\}), $$

where we have let

$$ (5.27) \qquad \lambda := 4\pi\lambda(V) $$

and

$$ \hat{G}(x) := \frac{G_h(x_0, \exp_{x_0}^{g_{x_0}}(x))}{\Lambda_{x_0}(\exp_{x_0}^{g_{x_0}}(x))} \quad \text{for all } x \in B_{2\delta}(0)\setminus\{0\}. $$

Here we recall that $\lambda(V)$ is the first term in the expansion at infinity of $V$ (see (2.4)), $G_h$ is the Green's function for the operator $\Delta_g + h$ in $M$, $\Lambda_{x_0}$ is as in (5.2)
proven that

$$ (5.22) \qquad \int_{B(0,\delta)} \left( x^k \partial_k v_\varepsilon + \frac{n-2}{2} v_\varepsilon \right) A_\varepsilon(v_\varepsilon) dx = o\left( \mu_\varepsilon^2 \ln \frac{1}{\mu_\varepsilon} \right) $$

as $\varepsilon \to 0$. Remark that by (2.9) we have $2\omega_3\lambda(V) = \int_{\mathbb{R}^4} |V|^2 V dx$. With this observation, using (4.35) and plugging (5.11), (5.12), (5.14), (5.20) and (5.22) in (5.10), we get (1.6), which proves Theorem 1.2 for $n = 4$.

5.3. The case $n = 3$. We are left with the case $n = 3$. Using (5.11) and (5.13) the Pohozaev identity (5.10) writes as

$$ (5.23) \qquad \begin{aligned} & \int_{\partial B(0, \delta)} \left( \frac{1}{2} v_{\varepsilon} \partial_{\nu} v_{\varepsilon} - \frac{\delta}{2} |\nabla v_{\varepsilon}|^{2}_{\xi} + \delta(\partial_{\nu} v_{\varepsilon})^{2} + \frac{1}{p_{\varepsilon}} \Lambda_{x_{\varepsilon}}^{p_{\varepsilon}-6} \delta |v_{\varepsilon}|^{p_{\varepsilon}} \right) d\sigma \\ & = \frac{1}{12} (6 - p_{\varepsilon}) \int_{\mathbb{R}^{n}} |V|^{6} dx + o(|p_{\varepsilon} - 6|) \\ & + \int_{B(0, \delta)} (x^{k} \partial_{k} v_{\varepsilon} + \frac{1}{2} v_{\varepsilon}) \tilde{A}_{\varepsilon}(x) dx \end{aligned} $$

with

$$ \tilde{A}_{\varepsilon}(x) := (\Delta_{g_{\varepsilon}} - \Delta_{\xi}) v_{\varepsilon} + \left( \frac{1}{8} S_{g_{\varepsilon}} + \hat{h}_{\varepsilon} \right) v_{\varepsilon}, $$

where $\hat{h}_{\varepsilon}$ is given by (5.8) and where $v_{\varepsilon}$ is defined in (5.3). First, by using (5.5) together with (5.15), we see that

$$ |\tilde{A}_{\varepsilon}(x)| \lesssim \frac{\mu_{\varepsilon}^{\frac{1}{2}}}{\mu_{\varepsilon} + |x|} \quad \text{for } x \in B(0, \delta). $$

As a consequence, straightforward computations show that

$$ (5.24) \qquad \left| \int_{B(0, \delta)} \left( x^k \partial_k v_\varepsilon + \frac{1}{2} v_\varepsilon \right) \tilde{A}_\varepsilon(x) dx \right| = O(\delta \mu_\varepsilon) $$

as $\varepsilon \to 0$. Independently, (5.5) shows that

$$ (5.25) \qquad \left| \int_{\partial B(0, \delta)} \frac{1}{p_{\varepsilon}} \Lambda_{x_{\varepsilon}}^{p_{\varepsilon}-6} \delta |v_{\varepsilon}|^{p_{\varepsilon}} d\sigma \right| = o(\mu_{\varepsilon}) $$

as $\varepsilon \to 0$. We now compute the remaining boundary term. It follows from (4.4), (5.2), (5.5), (5.7) and standard elliptic theory that

$$ (5.26) \qquad \lim_{\varepsilon \to 0} \frac{v_{\varepsilon}(x)}{\mu_{\varepsilon}^{\frac{1}{2}}} = \lambda \hat{G}(x) \quad \text{in } C_{\text{loc}}^{2}(B_{2\delta}(0)\setminus\{0\}), $$

where we have let

$$ (5.27) \qquad \lambda := 4\pi\lambda(V) $$

and

$$ \hat{G}(x) := \frac{G_h(x_0, \exp_{x_0}^{g_{x_0}}(x))}{\Lambda_{x_0}(\exp_{x_0}^{g_{x_0}}(x))} \quad \text{for all } x \in B_{2\delta}(0)\setminus\{0\}. $$

Here we recall that $\lambda(V)$ is the first term in the expansion at infinity of $V$ (see (2.4)), $G_h$ is the Green's function for the operator $\Delta_g + h$ in $M$, $\Lambda_{x_0}$ is as in (5.2)
and $\exp_{x_0}^{g_{x_0}}$ is the exponential map at $x_0$ with respect to $g_{x_0}$. It first follows from (5.26) that

$$ (5.28) \qquad \begin{aligned} & \lim_{\epsilon \to 0} \mu_{\epsilon}^{-1} \int_{\partial B(0, \delta)} \left( \frac{1}{2} v_{\epsilon} \partial_{\nu} v_{\epsilon} - \frac{\delta}{2} |\nabla v_{\epsilon}|_{\xi}^2 + \delta(\partial_{\nu} v_{\epsilon})^2 \right) d\sigma \\ & = \lambda^2 \int_{\partial B(0, \delta)} \left( \frac{1}{2} \hat{G} \partial_{\nu} \hat{G} - \frac{\delta}{2} |\nabla \hat{G}|_{\xi}^2 + \delta(\partial_{\nu} \hat{G})^2 \right) d\sigma. \end{aligned} $$

By (5.2) and (5.4), and since $G_h(x_0, \cdot)$ satisfies the expansion (3.8), there exists a continuous function $\hat{\beta} : B(0, 2\delta) \to \mathbb{R}$ such that

$$ \hat{G}(x) = \frac{1}{4\pi|x|} + \hat{\beta}(x) \quad \text{for all } x \in B_{2\delta}(0) \setminus \{0\}. $$

It follows from Definition 3.3 that $\hat{\beta}(0) = m_h(x_0)$ and (3.9) and (5.4) show that we have

$$ |\hat{\beta}(x)| \le C; \quad |\nabla \hat{\beta}(x)| \le C(1 + |\ln |x||) \quad \text{for all } x \in B_{2\delta}(0) \setminus \{0\}. $$

Straightforward computations then show that

$$ (5.29) \qquad \int_{\partial B(0, \delta)} \left( \frac{1}{2} \hat{G} \partial_{\nu} \hat{G} - \frac{\delta}{2} |\nabla \hat{G}|_{\xi}^2 + \delta(\partial_{\nu} \hat{G})^2 \right) d\sigma = -\frac{1}{2} m_h(x_0) + \varepsilon_{\delta}, $$

where $\lim_{\delta \to 0} \varepsilon_{\delta} = 0$. Using (4.35), plugging (5.24), (5.25), (5.28) and (5.29) in (5.23) and using (5.27) finally shows that

$$ (5.30) \qquad \lim_{\epsilon \to 0} \frac{6 - p_{\epsilon}}{\mu_{\epsilon}} = \frac{-96\pi^2 \lambda(V)^2 m_h(x_0)}{\int_{\mathbb{R}^3} |V|^6 dx}. $$

This proves (1.7) and concludes the proof of Theorem 1.2. $\square$

## 6. A NON-BLOWUP SITUATION

In this last Section we show how Theorem 1.1 can be used to rule out the existence of one-bubble blowing-up solutions of equations like:

$$ (6.1) \qquad \Delta_g u_{\epsilon} + h u_{\epsilon} = |u_{\epsilon}|^{2^* - 2 - \epsilon} u_{\epsilon} $$

at a given point, in some cases. In [8], Deng-Musso-Wei have proposed a creative approach to construct families of sign-changing solutions $(u_{\epsilon})_{\epsilon>0}$ to the equation (6.1) that blow-up like a single bubble as in (1.4). The bubble that the authors choose in [8] is one of the symmetric bubbles constructed in [7]. In their arguments, Deng-Musso-Wei have introduced the function

$$ \varphi_{\ell} := h - c_n \left(1 + \frac{n-4}{3n}\ell\right) S_g, $$

where $\ell$ is a parameter associated to the familie of bubbles in [7]. The function $\varphi_{\ell}$ plays a crucial role in the results of [8]. We show below how, under some conditions, it is possible to use $\varphi_{\ell}$ to rule out one-bubble blow-up.

We let $(M, g)$ be a smooth, closed, connected Riemannian manifold of dimension $n \ge 5$ that is locally conformally flat and has constant scalar curvature equal to $-1$. This is for instance satisfied if $(M, g)$ is a closed hyperbolic manifold. We define

$$ c_n = \frac{n-2}{4(n-1)} \quad \text{and we let } 1 < t < 1 + \frac{n-4}{3n}. $$

Let $\xi_0 \in M$ be any point.
and $\exp_{x_0}^{g_{x_0}}$ is the exponential map at $x_0$ with respect to $g_{x_0}$. It first follows from (5.26) that

$$ (5.28) \qquad \begin{aligned} & \lim_{\epsilon \to 0} \mu_{\epsilon}^{-1} \int_{\partial B(0, \delta)} \left( \frac{1}{2} v_{\epsilon} \partial_{\nu} v_{\epsilon} - \frac{\delta}{2} |\nabla v_{\epsilon}|_{\xi}^2 + \delta(\partial_{\nu} v_{\epsilon})^2 \right) d\sigma \\ & = \lambda^2 \int_{\partial B(0, \delta)} \left( \frac{1}{2} \hat{G} \partial_{\nu} \hat{G} - \frac{\delta}{2} |\nabla \hat{G}|_{\xi}^2 + \delta(\partial_{\nu} \hat{G})^2 \right) d\sigma. \end{aligned} $$

By (5.2) and (5.4), and since $G_h(x_0, \cdot)$ satisfies the expansion (3.8), there exists a continuous function $\hat{\beta}: B(0, 2\delta) \to \mathbb{R}$ such that

$$ \hat{G}(x) = \frac{1}{4\pi|x|} + \hat{\beta}(x) \quad \text{for all } x \in B_{2\delta}(0)\setminus\{0\}. $$

It follows from Definition 3.3 that $\hat{\beta}(0) = m_h(x_0)$ and (3.9) and (5.4) show that we have

$$ |\hat{\beta}(x)| \le C; \quad |\nabla \hat{\beta}(x)| \le C(1 + |\ln |x||) \quad \text{for all } x \in B_{2\delta}(0)\setminus\{0\}. $$

Straightforward computations then show that

$$ (5.29) \qquad \int_{\partial B(0, \delta)} \left( \frac{1}{2} \hat{G} \partial_{\nu} \hat{G} - \frac{\delta}{2} |\nabla \hat{G}|_{\xi}^2 + \delta(\partial_{\nu} \hat{G})^2 \right) d\sigma = -\frac{1}{2} m_h(x_0) + \varepsilon_{\delta}, $$

where $\lim_{\delta \to 0} \varepsilon_{\delta} = 0$. Using (4.35), plugging (5.24), (5.25), (5.28) and (5.29) in (5.23) and using (5.27) finally shows that

$$ (5.30) \qquad \lim_{\epsilon \to 0} \frac{6 - p_{\epsilon}}{\mu_{\epsilon}} = \frac{-96\pi^2 \lambda(V)^2 m_h(x_0)}{\int_{\mathbb{R}^3} |V|^6 dx}. $$

This proves (1.7) and concludes the proof of Theorem 1.2. $\square$

## 6. A NON-BLOWUP SITUATION

In this last Section we show how Theorem 1.1 can be used to rule out the existence of one-bubble blowing-up solutions of equations like:

$$ (6.1) \qquad \Delta_g u_{\epsilon} + h u_{\epsilon} = |u_{\epsilon}|^{2^* - 2 - \epsilon} u_{\epsilon} $$

at a given point, in some cases. In [8], Deng-Musso-Wei have proposed a creative approach to construct families of sign-changing solutions $(u_{\epsilon})_{\epsilon>0}$ to the equation (6.1) that blow-up like a single bubble as in (1.4). The bubble that the authors choose in [8] is one of the symmetric bubbles constructed in [7]. In their arguments, Deng-Musso-Wei have introduced the function

$$ \varphi_{\ell} := h - c_n \left(1 + \frac{n-4}{3n}\ell\right) S_g, $$

where $\ell$ is a parameter associated to the familie of bubbles in [7]. The function $\varphi_{\ell}$ plays a crucial role in the results of [8]. We show below how, under some conditions, it is possible to use $\varphi_{\ell}$ to rule out one-bubble blow-up.

We let $(M, g)$ be a smooth, closed, connected Riemannian manifold of dimension $n \ge 5$ that is locally conformally flat and has constant scalar curvature equal to $-1$. This is for instance satisfied if $(M, g)$ is a closed hyperbolic manifold. We define

$$ c_n = \frac{n-2}{4(n-1)} \quad \text{and we let } 1 < t < 1 + \frac{n-4}{3n}. $$

Let $\xi_0 \in M$ be any point.
**Step 6.1:** We claim that there exists $h \in C^\infty(M)$ such that

$$ (6.2) \qquad \left\{ \begin{array}{l} h(\xi_0) = -tc_n \\ \nabla h(\xi_0) = 0 \\ \nabla^2 h(x_0) \text{ is positive-definite} \\ \Delta_g + h \text{ is coercive in } H^1(M). \end{array} \right. $$

Under (6.2), $\xi_0$ is in particular a local strict minimum point of $h$.

We prove the Claim. Let for instance $\chi \in C_c^\infty(\mathbb{R}^n)$ be a cutoff function with $0 \le \chi \le 1$, $\chi(x) = 1$ in $B(0, 1)$ and $\chi(x) = 0$ in $\mathbb{R}^n \setminus B(0, 2)$. Let, for $0 < \delta < \frac{i_g(M)}{2}$ and $x \in M$,

$$ h_{\delta}(x) = 1 - \chi\left(\frac{d_g(\xi_0, x)}{\delta}\right) + \chi\left(\frac{d_g(\xi_0, x)}{\delta}\right)\left(-tc_n + d_g(\xi_0, x)^2\right). $$

We claim that $h_\delta$ satisfies (6.2) for $\delta$ small enough (but fixed). Indeed, in $B_g(\xi_0, \delta)$, we have $h_\delta(x) = -tc_n + d_g(\xi_0, x)^2$, and straightforward computations show that

$$ \|h_{\delta} - 1\|_{L^{\frac{n}{2}}(M)} \le C\delta^2, $$

so that $\Delta_g + h_\delta$ is coercive for $\delta$ small enough. This proves the claim.

**Step 6.2:** We now let $h \in C^\infty(M)$ satisfy (6.2). For $\ell \ge 1$ and $\xi \in M$ we define

$$ \varphi_{\ell}(\xi) = h(\xi) - c_n \left(1 + \frac{n-4}{3n}\ell\right) S_g(\xi) = h(\xi) + c_n \left(1 + \frac{n-4}{3n}\ell\right), $$

We claim that

$$ (6.3) \qquad \left\{ \begin{array}{l} \Delta_g + h \text{ is coercive;} \\ \varphi_{\ell}(\xi_0) > 0 \text{ for all } l \ge 1; \\ \xi_0 \text{ is a } C^1\text{-stable critical point of } \varphi_{\ell} \text{ uniformly in } \ell \ge 1. \end{array} \right. $$

We prove the claim. We have that

$$ \varphi_{\ell}(\xi_0) = c_n \left(1 + \frac{n-4}{3n}\ell - t\right) \ge c_n \left(1 + \frac{n-4}{3n} - t\right) > 0 $$

for any $\ell \ge 1$. It is easily seen with (6.2) that, for any $\ell \ge 1$, $\xi_0$ is a critical point of $\varphi_\ell$ satisfying $\varphi_\ell(\xi_0) > 0$ and $\varphi_\ell(\xi_0) \to +\infty$ as $\ell \to +\infty$. Since $\xi_0$ has non-zero degree, it is a $C^1$-stable critical point of $h$ (following the terminology of [19]). That is, there is $r_0 > 0$ that satisfies the following: for any $0 < r \le r_0$ there exists $\alpha > 0$ such that any $\tilde{h} \in C^1(\overline{B_g(\xi_0, r)})$ satisfying $\|\tilde{h} - h\|_{C^1(\overline{B_g(\xi_0, r)})} \le \alpha$ has at least one critical point in $B_g(\xi_0, r)$. By definition of $\varphi_\ell$ we observe that $\xi_0$ then remains a $C^1$-stable critical point of $\varphi_\ell$ uniformly in $\ell \ge 1$: that is, for any $0 < r \le r_0$ and $\alpha$ as above, and for any $\ell \ge 1$, any function $\tilde{\varphi} \in C^1(\overline{B_g(\xi_0, r)})$ satisfying $\|\tilde{\varphi} - \varphi_\ell\|_{C^1(\overline{B_g(\xi_0, r)})} \le \delta$ has at least one critical point in $B_g(\xi_0, r)$. This proves (6.3).

**Step 6.3:** We claim that there is no family $(u_\epsilon)_{0<\epsilon\le 1} \in C^2(M)$ of solutions of (6.1) such that

$$ (6.4) \qquad u_{\epsilon} = B_{\epsilon} + o(1) \text{ in } H^{1}(M), $$

where $B = (B_\epsilon)_{\epsilon>0}$ is a bubble centered at $(\xi_\epsilon)_\epsilon \in M$, $\lim_{\epsilon\to 0} \xi_\epsilon = \xi_0$ as in Definition 3.1.
**Step 6.1:** We claim that there exists $h \in C^\infty(M)$ such that

$$ (6.2) \qquad \left\{ \begin{array}{l} h(\xi_0) = -tc_n \\ \nabla h(\xi_0) = 0 \\ \nabla^2 h(x_0) \text{ is positive-definite} \\ \Delta_g + h \text{ is coercive in } H^1(M). \end{array} \right. $$

Under (6.2), $\xi_0$ is in particular a local strict minimum point of $h$.

We prove the Claim. Let for instance $\chi \in C_c^\infty(\mathbb{R}^n)$ be a cutoff function with $0 \le \chi \le 1$, $\chi(x) = 1$ in $B(0, 1)$ and $\chi(x) = 0$ in $\mathbb{R}^n \setminus B(0, 2)$. Let, for $0 < \delta < \frac{i_g(M)}{2}$ and $x \in M$,

$$ h_{\delta}(x) = 1 - \chi\left(\frac{d_g(\xi_0, x)}{\delta}\right) + \chi\left(\frac{d_g(\xi_0, x)}{\delta}\right)\left(-tc_n + d_g(\xi_0, x)^2\right). $$

We claim that $h_\delta$ satisfies (6.2) for $\delta$ small enough (but fixed). Indeed, in $B_g(\xi_0, \delta)$, we have $h_\delta(x) = -tc_n + d_g(\xi_0, x)^2$, and straightforward computations show that

$$ \|h_{\delta} - 1\|_{L^{\frac{n}{2}}(M)} \le C\delta^2, $$

so that $\Delta_g + h_\delta$ is coercive for $\delta$ small enough. This proves the claim.

**Step 6.2:** We now let $h \in C^\infty(M)$ satisfy (6.2). For $\ell \ge 1$ and $\xi \in M$ we define

$$ \varphi_{\ell}(\xi) = h(\xi) - c_n \left(1 + \frac{n-4}{3n}\ell\right) S_g(\xi) = h(\xi) + c_n \left(1 + \frac{n-4}{3n}\ell\right), $$

We claim that

$$ (6.3) \qquad \left\{ \begin{array}{l} \Delta_g + h \text{ is coercive;} \\ \varphi_{\ell}(\xi_0) > 0 \text{ for all } l \ge 1; \\ \xi_0 \text{ is a } C^1\text{-stable critical point of } \varphi_{\ell} \text{ uniformly in } \ell \ge 1. \end{array} \right. $$

We prove the claim. We have that

$$ \varphi_{\ell}(\xi_0) = c_n \left(1 + \frac{n-4}{3n}\ell - t\right) \ge c_n \left(1 + \frac{n-4}{3n} - t\right) > 0 $$

for any $\ell \ge 1$. It is easily seen with (6.2) that, for any $\ell \ge 1$, $\xi_0$ is a critical point of $\varphi_\ell$ satisfying $\varphi_\ell(\xi_0) > 0$ and $\varphi_\ell(\xi_0) \to +\infty$ as $\ell \to +\infty$. Since $\xi_0$ has non-zero degree, it is a $C^1$-stable critical point of $h$ (following the terminology of [19]). That is, there is $r_0 > 0$ that satisfies the following: for any $0 < r \le r_0$ there exists $\alpha > 0$ such that any $\tilde{h} \in C^1(\overline{B_g(\xi_0, r)})$ satisfying $\|\tilde{h} - h\|_{C^1(\overline{B_g(\xi_0, r)})} \le \alpha$ has at least one critical point in $B_g(\xi_0, r)$. By definition of $\varphi_\ell$ we observe that $\xi_0$ then remains a $C^1$-stable critical point of $\varphi_\ell$ uniformly in $\ell \ge 1$: that is, for any $0 < r \le r_0$ and $\alpha$ as above, and for any $\ell \ge 1$, any function $\tilde{\varphi} \in C^1(\overline{B_g(\xi_0, r)})$ satisfying $\|\tilde{\varphi} - \varphi_\ell\|_{C^1(\overline{B_g(\xi_0, r)})} \le \delta$ has at least one critical point in $B_g(\xi_0, r)$. This proves (6.3).

**Step 6.3:** We claim that there is no family $(u_\varepsilon)_{0<\varepsilon\le 1} \in C^2(M)$ of solutions of (6.1) such that

$$ (6.4) \qquad u_{\varepsilon} = B_{\varepsilon} + o(1) \text{ in } H^1(M), $$

where $B = (B_\varepsilon)_{\varepsilon>0}$ is a bubble centered at $(\xi_\varepsilon)_{\varepsilon \in M}$, $\lim_{\varepsilon \to 0} \xi_\varepsilon = \xi_0$ as in Definition 3.1.
We prove the claim by contradiction. Since $(M, g)$ is locally conformally flat, then the Weyl tensor vanishes. It then follows from Theorem 1.1 of the present paper that if there is blow-up like in (6.4), then equation (1.5) in particular tells us that

$$ (6.5) \qquad h(\xi_0) + c_n = h(\xi_0) - c_n S_g(\xi_0) \geq 0. $$

But this is an obvious contradiction with (6.2) since $h(\xi_0) = -tc_n$ and $t > 1$. This proves the claim.

## REFERENCES

[1] B. Ammann and E. Humbert, *The second Yamabe invariant*, J. Funct. Anal. **235** (2006), no. 2, 377-412. MR2225458

[2] N. Aronszajn, *A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order*, J. Math. Pures Appl. (9) **36** (1957), 235-249. MR92067

[3] T. Aubin, *Équations du type Monge-Ampère sur les variétés kählériennes compactes*, Bull. Sci. Math. (2) **102** (1978), no. 1, 63-95.

[4] Arthur L. Besse, *Einstein manifolds*, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR2371700

[5] Denis Bonheure, Jean-Baptiste Castéras, and Bruno Premoselli, *Classification of radial blow-up at the first critical exponent for the lin-ni-takagi problem in the ball* (2022). Preprint.

[6] Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, *Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth*, Comm. Pure Appl. Math. **42** (1989), no. 3, 271-297. MR982351 (90c:35075)

[7] Manuel del Pino, Monica Musso, Frank Pacard, and Angela Pistoia, *Large energy entire solutions for the Yamabe equation*, J. Differential Equations **251** (2011), no. 9, 2568-2597. MR2825341 (2012k:35176)

[8] Shengbing Deng, Monica Musso, and Juncheng Wei, *New type of sign-changing blow-up solutions for scalar curvature type equations*, Int. Math. Res. Not. IMRN **13** (2019), 4159-4197. MR3978436

[9] Wei Yue Ding, *On a conformally invariant elliptic equation on $\mathbb{R}^n$*, Comm. Math. Phys. **107** (1986), no. 2, 331-335. MR863646

[10] Olivier Druet, *From one bubble to several bubbles: the low-dimensional case*, J. Differential Geom. **63** (2003), no. 3, 399-473. MR2015469

[11] ——, *Compactness for Yamabe metrics in low dimensions*, Int. Math. Res. Not. **23** (2004), 1143-1191. MR2041549 (2005b:53056)

[12] Olivier Druet, Emmanuel Hebey, and Frédéric Robert, *Blow-up theory for elliptic PDEs in Riemannian geometry*, Mathematical Notes, vol. 45, Princeton University Press, Princeton, NJ, 2004. MR2063399 (2005g:53058)

[13] Thomas Duyckaerts, Carlos Kenig, and Frank Merle, *Solutions of the focusing nonradial critical wave equation with the compactness property*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) **15** (2016), 731-808. MR3495446

[14] Nassif Ghoussoub, Saikat Mazumdar, and Frédéric Robert, *Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity*, Mem. Amer. Math. Soc. **285** (2023), no. 1415, v+126. MR4599209

[15] Emmanuel Hebey, *Introduction à l'analyse non linéaire sur les variétés*, Diderot Editeur Arts Sciences, 1997.

[16] ——, *Compactness and stability for nonlinear elliptic equations*, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. MR3235821

[17] Emmanuel Hebey and Michel Vaugon, *From best constants to critical functions*, Math. Z. **237** (2001), no. 4, 737-767.

[18] John M. Lee and Thomas H. Parker, *The Yamabe problem*, Bull. Amer. Math. Soc. (N.S.) **17** (1987), no. 1, 37-91. MR888880 (88f:53001)

[19] YanYan Li, *On a singularly perturbed elliptic equation*, Adv. Differential Equations **2** (1997), no. 6, 955-980. MR1606351

[20] Yanyan Li and Meijun Zhu, *Yamabe type equations on three-dimensional Riemannian manifolds*, Commun. Contemp. Math. **1** (1999), no. 1, 1-50. MR1681811 (2000m:53051)
We prove the claim by contradiction. Since $(M, g)$ is locally conformally flat, then the Weyl tensor vanishes. It then follows from Theorem 1.1 of the present paper that if there is blow-up like in (6.4), then equation (1.5) in particular tells us that

$$ (6.5) \qquad h(\xi_0) + c_n = h(\xi_0) - c_n S_g(\xi_0) \geq 0. $$

But this is an obvious contradiction with (6.2) since $h(\xi_0) = -tc_n$ and $t > 1$. This proves the claim.

## REFERENCES

[1] B. Ammann and E. Humbert, *The second Yamabe invariant*, J. Funct. Anal. **235** (2006), no. 2, 377-412. MR2225458

[2] N. Aronszajn, *A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order*, J. Math. Pures Appl. (9) **36** (1957), 235-249. MR92067

[3] T. Aubin, *Équations du type Monge-Ampère sur les variétés kählériennes compactes*, Bull. Sci. Math. (2) **102** (1978), no. 1, 63-95.

[4] Arthur L. Besse, *Einstein manifolds*, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR2371700

[5] Denis Bonheure, Jean-Baptiste Castéras, and Bruno Premoselli, *Classification of radial blow-up at the first critical exponent for the lin-ni-takagi problem in the ball* (2022). Preprint.

[6] Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, *Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth*, Comm. Pure Appl. Math. **42** (1989), no. 3, 271-297. MR982351 (90c:35075)

[7] Manuel del Pino, Monica Musso, Frank Pacard, and Angela Pistoia, *Large energy entire solutions for the Yamabe equation*, J. Differential Equations **251** (2011), no. 9, 2568-2597. MR2825341 (2012k:35176)

[8] Shengbing Deng, Monica Musso, and Juncheng Wei, *New type of sign-changing blow-up solutions for scalar curvature type equations*, Int. Math. Res. Not. IMRN **13** (2019), 4159-4197. MR3978436

[9] Wei Yue Ding, *On a conformally invariant elliptic equation on $\mathbb{R}^n$*, Comm. Math. Phys. **107** (1986), no. 2, 331-335. MR863646

[10] Olivier Druet, *From one bubble to several bubbles: the low-dimensional case*, J. Differential Geom. **63** (2003), no. 3, 399-473. MR2015469

[11] ——, *Compactness for Yamabe metrics in low dimensions*, Int. Math. Res. Not. **23** (2004), 1143-1191. MR2041549 (2005b:53056)

[12] Olivier Druet, Emmanuel Hebey, and Frédéric Robert, *Blow-up theory for elliptic PDEs in Riemannian geometry*, Mathematical Notes, vol. 45, Princeton University Press, Princeton, NJ, 2004. MR2063399 (2005g:53058)

[13] Thomas Duyckaerts, Carlos Kenig, and Frank Merle, *Solutions of the focusing nonradial critical wave equation with the compactness property*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) **15** (2016), 731-808. MR3495446

[14] Nassif Ghoussoub, Saikat Mazumdar, and Frédéric Robert, *Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity*, Mem. Amer. Math. Soc. **285** (2023), no. 1415, v+126. MR4599209

[15] Emmanuel Hebey, *Introduction à l'analyse non linéaire sur les variétés*, Diderot Editeur Arts Sciences, 1997.

[16] ——, *Compactness and stability for nonlinear elliptic equations*, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. MR3235821

[17] Emmanuel Hebey and Michel Vaugon, *From best constants to critical functions*, Math. Z. **237** (2001), no. 4, 737-767.

[18] John M. Lee and Thomas H. Parker, *The Yamabe problem*, Bull. Amer. Math. Soc. (N.S.) **17** (1987), no. 1, 37-91. MR888880 (88f:53001)

[19] YanYan Li, *On a singularly perturbed elliptic equation*, Adv. Differential Equations **2** (1997), no. 6, 955-980. MR1606351

[20] Yanyan Li and Meijun Zhu, *Yamabe type equations on three-dimensional Riemannian manifolds*, Commun. Contemp. Math. **1** (1999), no. 1, 1-50. MR1681811 (2000m:53051)
[21] Fernando Coda Marques, *A priori estimates for the Yamabe problem in the non-locally conformally flat case*, J. Differential Geom. **71** (2005), no. 2, 315-346. MR2197144 (2006i:53046)

[22] Maria Medina and Monica Musso, *Doubling nodal solutions to the Yamabe equation in $R^n$ with maximal rank*, J. Math. Pures Appl. (9) **152** (2021), 145-188. MR4280834

[23] Maria Medina, Monica Musso, and Juncheng Wei, *Desingularization of Clifford torus and nonradial solutions to the Yamabe problem with maximal rank*, J. Funct. Anal. **276** (2019), no. 8, 2470-2523. MR3926123

[24] Hussein Mesmar and Frédéric Robert, *Concentration analysis for elliptic critical equations with no boundary control: Ground-state blow-up*, Discrete and Continuous Dynamical Systems - S **17** (2024), no. 4, 1599-1620.

[25] Anna Maria Micheletti, Angela Pistoia, and Jérôme Vétois, *Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds*, Indiana Univ. Math. J. **58** (2009), no. 4, 1719-1746. MR2542977

[26] Morio Obata, *The conjectures on conformal transformations of Riemannian manifolds*, J. Differential Geometry **6** (1971/72), 247-258. MR303464

[27] Angela Pistoia and Jérôme Vétois, *Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds*, J. Differential Equations **254** (2013), no. 11, 4245-4278. MR3035432

[28] Bruno Premoselli, *A priori estimates for finite-energy sign-changing solutions blowing-up solutions of critical elliptic equations*, Int. Math. Res. Not. IMRN (2021). To appear.

[29] Bruno Premoselli and Jérôme Vétois, *Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part*, J. Differential Equations **266** (2019), no. 11, 7416-7458. MR3926106

[30] ————, *Sign-changing blow-up for the Yamabe equation at the lowest energy level*, Adv. Math. **410** (2022), Paper No. 108769, 50. MR4509411

[31] Bruno Premoselli and Jérôme Vétois, *Stability and instability results for sign-changing solutions to second-order critical elliptic equations*, Journal de Mathématiques Pures et Appliquées **167** (2022), 257-293.

[32] Frédéric Robert, *Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d'ordre deux*, 2010. https://iecl.univ-lorraine.fr/files/2021/04/ConstrucGreen.pdf.

[33] Frédéric Robert and Jérôme Vétois, *Sign-changing blow-up for scalar curvature type equations*, Comm. Partial Differential Equations **38** (2013), no. 8, 1437-1465. MR3169751

[34] ————, *Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold*, Calc. Var. Partial Differential Equations **54** (2015), no. 1, 693-716. MR3385177

[35] Richard M. Schoen, *Lecture notes from courses at stanford* (1988). Written by D. Pollack, preprint.

[36] Michael Struwe, *A global compactness result for elliptic boundary value problems involving limiting nonlinearities*, Math. Z. **187** (1984), no. 4, 511-517. MR760051

[37] Neil S. Trudinger, *Remarks concerning the conformal deformation of Riemannian structures on compact manifolds*, Ann. Scuola Norm. Sup. Pisa (3) **22** (1968), 265-274. MR0240748

BRUNO PREMOSELLI, UNIVERSITÉ LIBRE DE BRUXELLES, SERVICE D'ANALYSE, CP 218, BOULEVARD DU TRIOMPHE, B-1050 BRUXELLES, BELGIQUE.

Email address: bruno.premoselli@ulb.be

FRÉDÉRIC ROBERT, UNIVERSITÉ DE LORRAINE, CNRS, IECL, F-54000 NANCY, FRANCE

Email address: frederic.robert@univ-lorraine.fr
[21] Fernando Coda Marques, *A priori estimates for the Yamabe problem in the non-locally conformally flat case*, J. Differential Geom. **71** (2005), no. 2, 315-346. MR2197144 (2006i:53046)

[22] Maria Medina and Monica Musso, *Doubling nodal solutions to the Yamabe equation in $\mathbb{R}^n$ with maximal rank*, J. Math. Pures Appl. (9) **152** (2021), 145-188. MR4280834

[23] Maria Medina, Monica Musso, and Juncheng Wei, *Desingularization of Clifford torus and nonradial solutions to the Yamabe problem with maximal rank*, J. Funct. Anal. **276** (2019), no. 8, 2470-2523. MR3926123

[24] Hussein Mesmar and Frédéric Robert, *Concentration analysis for elliptic critical equations with no boundary control: Ground-state blow-up*, Discrete and Continuous Dynamical Systems - S **17** (2024), no. 4, 1599-1620.

[25] Anna Maria Micheletti, Angela Pistoia, and Jérôme Vétois, *Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds*, Indiana Univ. Math. J. **58** (2009), no. 4, 1719-1746. MR2542977

[26] Morio Obata, *The conjectures on conformal transformations of Riemannian manifolds*, J. Differential Geometry **6** (1971/72), 247-258. MR303464

[27] Angela Pistoia and Jérôme Vétois, *Sign-changing bubble towers for asymptotically critical elliptic equations on Riemannian manifolds*, J. Differential Equations **254** (2013), no. 11, 4245-4278. MR3035432

[28] Bruno Premoselli, *A priori estimates for finite-energy sign-changing solutions blowing-up solutions of critical elliptic equations*, Int. Math. Res. Not. IMRN (2021). To appear.

[29] Bruno Premoselli and Jérôme Vétois, *Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part*, J. Differential Equations **266** (2019), no. 11, 7416-7458. MR3926106

[30] ————, *Sign-changing blow-up for the Yamabe equation at the lowest energy level*, Adv. Math. **410** (2022), Paper No. 108769, 50. MR4509411

[31] Bruno Premoselli and Jérôme Vétois, *Stability and instability results for sign-changing solutions to second-order critical elliptic equations*, Journal de Mathématiques Pures et Appliquées **167** (2022), 257-293.

[32] Frédéric Robert, *Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d'ordre deux*, 2010. https://iecl.univ-lorraine.fr/files/2021/04/ConstrucGreen.pdf.

[33] Frédéric Robert and Jérôme Vétois, *Sign-changing blow-up for scalar curvature type equations*, Comm. Partial Differential Equations **38** (2013), no. 8, 1437-1465. MR3169751

[34] ————, *Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold*, Calc. Var. Partial Differential Equations **54** (2015), no. 1, 693-716. MR3385177

[35] Richard M. Schoen, *Lecture notes from courses at stanford* (1988). Written by D. Pollack, preprint.

[36] Michael Struwe, *A global compactness result for elliptic boundary value problems involving limiting nonlinearities*, Math. Z. **187** (1984), no. 4, 511-517. MR760051

[37] Neil S. Trudinger, *Remarks concerning the conformal deformation of Riemannian structures on compact manifolds*, Ann. Scuola Norm. Sup. Pisa (3) **22** (1968), 265-274. MR0240748

BRUNO PREMOSELLI, UNIVERSITÉ LIBRE DE BRUXELLES, SERVICE D'ANALYSE, CP 218, BOULEVARD DU TRIOMPHE, B-1050 BRUXELLES, BELGIQUE.

Email address: bruno.premoselli@ulb.be

FRÉDÉRIC ROBERT, UNIVERSITÉ DE LORRAINE, CNRS, IECL, F-54000 NANCY, FRANCE

Email address: frederic.robert@univ-lorraine.fr