# Estimation of Appearance and Occupancy Information in Bird’s Eye View from
Surround Monocular Images

Sarthak Sharma1, Unnikrishnan R. Nair1, Udit Singh Parihar1, Midhun Menon S1
and Srikanth Vidapanakal1 1Authors are with the Ola Electric AI,
Bengaluru,India. {sarthak.sharma1, unnikrishnan.r, udit.parihar, midhun.s,
<EMAIL_ADDRESS>
###### Abstract

Autonomous driving requires efficient reasoning about the location and
appearance of the different agents in the scene, which aids in downstream
tasks such as object detection, object tracking, and path planning. The past
few years have witnessed a surge in approaches that combine the different
task-based modules of the classic self-driving stack into an End-to-End(E2E)
trainable learning system. These approaches replace perception, prediction,
and sensor fusion modules with a single contiguous module with shared latent
space embedding, from which one extracts a human-interpretable representation
of the scene. One of the most popular representations is the Bird’s-eye View
(BEV), which expresses the location of different traffic participants in the
ego vehicle frame from a top-down view. However, a BEV does not capture the
chromatic appearance information of the participants.

To overcome this limitation, we propose a novel representation that captures
various traffic participants’ appearance and occupancy information from an
array of monocular cameras covering $360^{\circ}$ field of view (FOV). We use
a learned image embedding of all camera images to generate a BEV of the scene
at any instant that captures both appearance and occupancy of the scene, which
can aid in downstream tasks such as object tracking and executing language-
based commands.

We test the efficacy of our approach on synthetic dataset generated from
CARLA. The code, data set, and results can be found at
https://rebrand.ly/APP_OCC-results.

##  I Introduction

Autonomous driving is one of the most active areas of research. The software
stack for autonomous driving has evolved a lot since the first DARPA Grand
Challenge 2004. From a modular architecture comprising a cascade of task-
specific blocks (sensor fusion, perception, planning, and control), it has
evolved to an E2E system that learns to generate driving behaviors from
perceived sensor inputs. E2E systems can produce diverse and complex driving
behaviors because it learns by mimicking expert behaviors. However, driving is
a highly evolved and context-sensitive task, and hence, a direct regression
over behaviors does not converge quickly or easily. Researchers have
successfully used shared embeddings with trained auxiliary interpretable
hierarchical tasks to overcome this problem. One of such promising
interpretable intermediate representations is BEV[2, 3, 4, 5, 6, 7, 8].

Figure 2:  Architecture: Proposed architecture for estimating color and
occupancy from monocular camera inputs.

  * •

We pass each camera image $I_{t}^{i}\in\mathbb{R}^{H\times W\times 3}$ through
an Efficient-net-B0[9] backbone to obtain feature embedding
$\varepsilon_{t}^{i}$.

  * •

We then split these camera features into a context vector
${}^{\text{context}}\varepsilon_{t}^{i}$ and a depth distribution
${}^{\text{depth}}\varepsilon_{t}^{i}$.

  * •

We derive a self-attended discrete depth weighed features on the $1:16$ image
using the outer product
${}^{\text{context}}\varepsilon_{t}^{i}\otimes^{\text{depth}}\varepsilon_{t}^{i}$.

  * •

Once all the N cameras have been processed, we perform voxel pooling at pre-
defined grid resolution $\text{grid}_{X}\times\text{grid}_{Y}$ to obtain a
tensor of dimension $C\times\text{grid}_{X}\times\text{grid}_{Y}$.

  * •

We finally output Occupancy $\mathcal{O}$ and appearance $\mathcal{A}$ using a
Res-Net[10] backbone, followed by up-sampling layers. Cross-Entropy loss
supervises the occupancy prediction and $L_{1}$ loss supervises the appearance
prediction.

BEV is a top-down orthographic view of the space around the Self-Driving
Vehicle (SDV) in an egocentric frame of reference. It is averse to occlusion
effects (a significant drawback of perspective projection), and being a
representation in metric space, it is amenable to planning. The existing
approaches in BEV [5, 3, 4, 6] aim to capture the occupancy information of
different traffic participants in the scene, such as vehicles, roads, and lane
markings. While this information provides semantic occupancy and shape
understanding (in the form of object dimensions, orientation), it lacks
appearance information.

Appearance and location are vital cues that humans infer from the environment.
We use them to reason about the environment and its different actors heavily.
We use them to label/tag the actors and track their temporal behavior. With
steeringless and pedalless cars around the corner, [11, 12, 13], it is only
natural that Multi-Object Tracking (MOT)[14, 15, 16] and language-based
navigation [17, 18, 19, 20, 21] will be inevitable features for any future
SDV. Complementary appearance cues present in the BEV occupancy space and the
RGB image space provide strong priors for the above-mentioned tasks.

In MOT, existing work focuses mainly on costs derived from occupancy
information such as position, shape, and orientation [14, 15, 16]. In [16], to
exploit the appearance cues, the authors back-project different object
instances in BEV into image space. This can lead to significant overlap of
regions and weak correlation during occluded object scenarios.

For tasks such as language-based navigation [17, 18, 19, 20, 21] (e.g. You can
park up ahead behind the silver car, next to the lamp post with the orange
sign on it), appearance and semantic knowledge are critical to create
associations. The image space captures appearance information, but it lacks
metric understanding of the scene due to perspective distortion. BEV captures
the semantic information in metric space in the form of an occupancy grid but
lacks appearance priors.

In this work, we present, to the best of our knowledge, a novel BEV
representation that captures the appearance and occupancy information of the
traffic participants. Focusing on the relevant classes of traffic participants
$S\in$ {road, vehicles and lane markings}, we present a representation that
can densely reason about the appearance and color of these participants from
multiple monocular cameras that cover a $360^{\circ}$ FOV. The occupancy
information determines the probability of belonging to the classes in $S$, for
each cell of the grid. We learn to capture the information about the chromatic
appearance as RGB triplets for the classes in $S$.

Our core contributions are as follows.

  * •

We propose, to the best of our knowledge, the first such unified dense BEV
representation that captures the appearance and occupancy of different traffic
participants in the scene. The proposed architecture injests images from
monocular cameras with overlapping FOV and the respective camera parameters
(extrinsic and intrinsic).

  * •

We demonstrate the positive coupling between the occupancy and appearance
training tasks. Using ablation studies, we quantitatively show how the capture
of the image information of the scene improves the estimate of the occupancy
information.

  * •

We test our method on a synthetic data set generated from Carla [1] and report
qualitative and quantitative results for occupancy estimation. We compare our
method with two SOTA approaches [5, 3] and report a significant performance
improvement.

  * •

We propose a pipeline for generating training data for the proposed
architecture from the NuScenes dataset. [22] and show qualitative results for
the same.

##  II Related Work

###  II-A Occupancy information in BEV space

There has been a recent surge in approaches that reason about the occupancy in
a BEV space using images from a monocular camera array with combined surround
FOV. In [5], the architecture learns an implicit representation of the scene
semantics in BEV. The network trains on the data captured from an array of
monocular cameras facing the front of the SDV with a combined 180∘ FOV. They
assist learning by using an auxiliary task of waypoint prediction. In
approaches such as [3, 4], the authors do dense semantic reasoning in the BEV
grid space using images from similar monocular camera rigs. The networks in
[8, 7] learn to predict the spatiotemporal dynamics of the scene on a BEV grid
using LiDAR point clouds as input. However, these approaches do not capture
the color information of the different traffic participants.

###  II-B Appearance information in BEV space

In [23], the authors propose a method to obtain the BEV of a scene from a
single perspective image. A Convolutional Neural Network (CNN) estimates the
image’s vertical vanishing point and the ground plane vanishing line
(horizon). The vanishing point and the vanishing line of the ground plane
determine the homography matrix $H$ that maps the image to the overhead view
after removing the perspective distortion. However, the approach depends on
the FOV of the camera. Furthermore, it does not reason about the semantic
classes of different agents in the top-down view.

Classical methods like Inverse Perspective Mapping (IPM) [24] reason about the
BEV of the scene by estimating the homography matrix based on the road
surface. While this method leads to a plausible BEV of the planar road
surface(Fig 3, Top), it heavily distorts the other traffic participants, as
shown in Fig 3. The semantic information of the scene is also not available.

Figure 3: Estimating BEV using IPM: We show results of obtaining BEV using IPM
on NuScenes [22], where we calculate the homography matrix using the planar
road points. Top When the world is mainly planar around the ego-vehicle, the
BEV estimate of the scene captures the information well. Bottom Once the
planarity assumption of the road gets violated, the BEV estimate accuracy
drops drastically.

##  III Approach

###  III-A Synthetic data generation

We,

  * •

Collect monocular RGB camera images using an expert driving agent in CARLA
[1]. We collect driving data on routes from the eight publicly available
virtual towns for training and test data generation, randomly spawning
scenarios at several locations along each route.

  * •

Replicate the NuScenes [22] data set for the intrinsic and extrinsic
parameters of the camera. We place six cameras - five of them {front-left,
front, front-right, rear-left and rear-right} with a FOV of 70∘ and one (rear)
with an FOV of 110∘. The camera orientations are
$55^{\circ},0^{\circ},-55^{\circ},110^{\circ},-110^{\circ},180^{\circ}$
respectively. We show the sensor setup in Fig. 4. Images have a resolution of
$1920\times 1080$.

  * •

To capture the color BEV of the scene, we place a camera at the height of
$100$ meters above the car, pitching down at $90^{\circ}$. This BEV camera has
a FOV of $90$ ∘ and captures top-down RGB and semantic views. We capture these
images at a resolution of $1000\times 1000$ pixels, focusing on an area of
$200m\times 200m$ around the vehicle. We use a $400\times 400$ center-crop
resized to a $200\times 200$ resolution image, finally capturing an area
$80m\times 80m$ around the vehicle.

###  III-B Model Architecture

Motivated by [3], our architecture infers the occupancy and appearance
information from multiple monocular images $180^{\circ}$ FOV. Inputs to our
model are the camera images and camera parameters (extrinsic and intrinsic).
The model then deduces the appearance and occupancy of the scene in BEV. We
show the model architecture in Figure 2.

Encoding image features: At the current time $t$, we have RGB images
$I=\left\\{I_{t}^{i}|i=\\{1...N\\}\right\\}$, where N=number of cameras, and
$I_{t}^{i}\in\mathbb{R}^{H\times W\times 3}$ ($W$ and $H$ are the width and
height of each image). To extract features, we pass each image $I_{t}^{i}$
through a standard convolution encoder $E$ (using Efficient-net-B0 [9]). The
encoder then down-samples the input by a factor of $16$ and generates a
feature embedding ${}^{\text{context}}\varepsilon_{t}^{i}$ =
$E(I_{t}^{i})\in\mathbb{R}^{C\times H^{\prime}\times W^{\prime}}$ where
$\displaystyle{H^{\prime}=\frac{H}{16}}$ and
$\displaystyle{W^{\prime}=\frac{W}{16}}$ and $C$ = number of image features.
We also reason about a discrete probability distribution of metric depth
$d\in[D_{\text{min}},D_{\text{max}}]$ of each projected pixel (to a resolution
of $\Delta d$) on the down sampled image. We denote the depth embedding as
${}^{\text{depth}}\varepsilon_{t}^{i}\in\mathbb{R}^{D\times H^{\prime}\times
W^{\prime}}$ where
$\displaystyle{D=\frac{D_{\text{max}}-D_{\text{min}}}{\Delta d}}$. At each
pixel index $(i,j)$ in the $H^{\prime}\times W^{\prime}$ downsampled image,
there exists a context vector $c_{ij}\in\mathbb{R}^{C}$ and corresponding
discrete depth distribution $d_{ij}\in\mathbb{R}^{D}$. The network first
learns the stacked tensor embedding
$\varepsilon_{t}^{i}=\left[{}^{\text{context}}\varepsilon_{t}^{i},~{}^{\text{depth}}\varepsilon_{t}^{i}\right]$.
Next, using the outer product
$\gamma_{t}^{i}=^{\text{context}}\varepsilon_{t}^{i}\otimes^{\text{depth}}\varepsilon_{t}^{j}\in\mathbb{R}^{C\times
D\times H^{\prime}\times W^{\prime}}$, we capture the camera features
modulated by the discrete depth probabilities to form an approximation for
self-attention for context features at each down-sampled image pixel. For each
camera, we lift the set of outer product vectors
$\left\\{\gamma_{1},\gamma_{2},\dots,\gamma_{N}\right\\}$ of the weighed
features for all image pixels using the known intrinsic and extrinsic
parameters of the camera to bring them to the ego-frame.

Figure 4: Sensor setup for data collection for appearance and occupancy
reasoning: Left We place $6$ RGB cameras on the ego vehicle, covering full
$360^{\circ}$ surroundings. Out of the $6$ cameras, $5$ (front-left, front,
front-right, rear-left and rear-right) are of $70^{\circ}$ FOV, and $1$ (rear)
is of $110^{\circ}$ FOV. We show the FOV in red and the orientations of the
camera in black. Right We show the placement of our BEV camera, at the height
of $100m$ above the ground, with a FOV of $90^{\circ}$. The BEV camera
provides us with information regarding the appearance and occupancy of the
scene. Figure 5: Qualitative results of our method, trained on the Carla [1]
dataset: We could capture the BEV image with occupancy and complete appearance
information for different turning and intersection scenarios. We display the
longitudinal distance of the vehicle centroid in the occupancy grid from the
center of the ego vehicle. Best viewed digitally.

BEV projection: We transform the lifted set of features
$\left\\{\gamma_{1},\gamma_{2},\dots,\gamma_{N}\right\\}$ into the BEV. We
define the extent of the BEV grid as $\epsilon_{X}\times\epsilon_{Y}$,
centered on the ego vehicle. We discretize this area into a unit grid of
resolution $\nabla_{X}\times\nabla_{Y}$, resulting in a grid of dimension
$\text{grid}_{X}\times\text{grid}_{Y}$. The above point cloud obtained from
the camera encoder is then Voxel pooled, as described in [25] to obtain a
tensor of dimension $C\times\text{grid}_{X}\times\text{grid}_{Y}$.

Appearance and occupancy information: From the above-pooled tensor, we extract
the appearance and occupancy information. To obtain the occupancy tensor
$\mathcal{O}$ with dimension
$|S|\times\text{grid}_{X}\times\text{grid}_{Y}$(where $|.|$ indicates
cardinality), we pass the above tensor of dimensions
$C\times\text{grid}_{X}\times\text{grid}_{Y}$ through the first three layers
of the ResNet-18 [10] backbone, followed by two up-sampling layers. We follow
the same paradigm for the appearance reasoning tensor $\mathcal{A}$ \- passing
the projected BEV tensor through the first three layers of ResNet-18 [14],
followed by two up-sampling layers to get
$n_{c}\times\text{grid}_{X}\times\text{grid}_{Y}$ tensor (where $n_{c}$ =
#color channels).

While the appearance tensor $\mathcal{A}$ is penalized for $L_{1}$ loss, the
occupancy tensor $\mathcal{O}$ is penalized for cross-entropy (CE) loss.

##  IV Experiments and Results

We carried out extensive experiments of our approach on data collected from
Carla [1]. We record $30$ sequences comprising over $12K$ scene instances. A
scene instance comprises a set of N RGB images at any instant with
$360^{\circ}$ FOV, along with their corresponding occupancy and appearance
information from the BEV camera image. We make a $80-20$ split for training
and test data. For our experiments, parameters we chose are
$N=6,H=128,W=352,C=64,\epsilon_{X}=80m,\epsilon_{Y}=80m,\nabla_{X}=0.4,\nabla_{Y}=0.4,grid_{X}=200,grid_{Y}=200$.
We sample depth values $d$ in the range $D_{min}=4$ and $D_{max}=45$, spaced
$1.0m$ apart. We train our method, as described in Section III, using PyTorch
[26] framework. We use Adam optimizer [27] with a learning rate of $1e-3$ and
a batch size of $20$.

###  IV-A Appearance and Occupancy information

We show quantitative and qualitative results of our approach in the following
section.

Intersection-over-Union (IoU) Road Vehicle Lane Lift-Splat-Shoot [3] $83.4$
$37.67$ $11.29$ Ours $\mathbf{86.77}$ $\mathbf{38.99}$ $\mathbf{18.05}$

TABLE I: BEV occupancy prediction on Carla [1] in full surround $360\degree$
($6$ cameras setting).

In Table I and Table II, we show the quantitative numbers for the Intersection
over Union (IoU) metric for the prediction of occupancy on our test split. We
compare our prediction of BEV occupancy with [3, 5] using Intersection-over-
Union (IoU) as the metric. Lift-Splat-Shoot[3] reports the metrics on NuScenes
[22], hence we re-train the method of [3] for occupancy prediction for classes
$S\in$ {road, vehicles and lane markings} on our data collected from Carla
[1]. For NEAT[5], the authors provide a pre-trained model trained on Carla [1]
based dataset.

Intersection-over-Union (IoU) Road Vehicle Lane NEAT [5] $58.1$
$\mathbf{76.3}$ $--$ Lift-Splat-Shoot [3] $63.53$ $28.57$ $12.27$ Ours
$\mathbf{68.48}$ $32.49$ $\mathbf{17.23}$

TABLE II: BEV occupancy prediction on Carla [1] for $180\degree$ towards the
front of the ego-vehicle ($3$ cameras setting).

In Table I, we compare our method with [3]. The approaches listed in Table I
take as input $6$ monocular camera images (along with their extrinsic and
intrinsic parameters), covering full surround $360\degree$ of the ego vehicle
and reason about the information in an $80m\times 80m$ grid around the ego
vehicle. We train our method for both appearance and occupancy prediction,
whereas we train [3] for occupancy prediction. We report better performance
for all the classes $S\in$ {road, vehicles and lane markings}, improving by
$3.37,1.32$ and $6.76$ respectively.

In Table II, we compare our method with [5] and [3]. For [5], the authors
provide a pre-trained model trained on Carla [1], which simultaneously reasons
about occupancy and waypoint prediction for roads and vehicles. As the FOV for
observation is $180^{\circ}$ front-facing, we re-train the method of [3] to
take as input $3$ monocular camera images (along with their extrinsic and
intrinsic parameters), covering front surround $180\degree$ of the ego vehicle
and reason about the information in a $40m\times 80m$ ego centric grid. We
train our method in a similar setting, reasoning about appearance and
occupancy information. Compared with [3], our model exhibits a better
performance for all the classes $S\in$ {road, vehicles and lane markings}, by
$4.95$, $3.92$ and $4.96$ respectively. Compared with [5], our performance
improves by $10.38$ on the road, but our performance degrades by $43.81$ for
the vehicle class. NEAT [5] does not report results on the class of lanes. We
get an appearance loss of $3.046$ for the $360\degree$ surround setting and
$2.89$ for the front $180\degree$ setting. We show the loss plot in Fig 6.

In Fig. LABEL:fig:teaser and Fig. 5, we show the qualitative results of our
method in different traffic and intersection scenarios. We could capture
appearance and occupancy information on straight roads, intersections, and
turn scenarios. For each vehicle in the occupancy grid, we also display the
estimated longitudinal distance of its centroid, computed from the center of
the ego vehicle (shown in green).

Figure 6: Quantitative Result for Appearance and Occupancy Loss: Here, we show
the loss values for appearance (color) and occupancy (IoU) for our val split
for surround $360\degree$ setting. The occupancy loss decreases from an
initial value of $1.2$ to $0.05$ after 7K iterations. The appearance loss
decreases from an initial value of $20.12$ to $3.046$ after the same number of
iterations.

Intersection-over-Union (IoU) Road Vehicle Lane occupancy ($180\degree$)
$63.53$ $28.57$ $12.27$ occupancy + appearance ($180\degree$) $\mathbf{68.48}$
$\mathbf{32.49}$ $\mathbf{17.23}$ occupancy ($360\degree$) $83.4$ $37.67$
$11.29$ occupancy + appearance ($360\degree$) $\mathbf{86.77}$
$\mathbf{38.99}$ $\mathbf{18.05}$

TABLE III: Ablation study. Comparing reasoning about occupancy and appearance
information in (a). $180\degree$ front-facing ($3$ cameras setting) (b).
$360\degree$ surround ($6$ cameras setting).

###  IV-B Ablation Studies

We perform an ablation study of our network trained on two camera
configurations to validate how reasoning about appearance aids in reasoning
about occupancy. In configuration 1, we reason in front of the ego vehicle
(covering $180\degree$ FOV). In configuration 2, we reason in a surround
setting around the ego-vehicle (covering $360\degree$). We summarize the
results in Table III. For both the settings, we train two models-one that
argues only about the occupancy and the other that argues about both occupancy
and appearance. We consistently observe that incorporating appearance improves
the reasoning about the occupancy of all classes $S\in$ {road, vehicles, and
lane markings} in both the settings.

Figure 7: Pipeline for generating data from a real-world dataset, for, e.g.,
NuScenes [22] to capture appearance and occupancy information in BEV: Top We
colorize LiDAR point clouds by projecting them into the camera images
assigning each point the nearest pixel color. Bottom We project the available
annotated vehicle cuboids in the camera images, and for each of the $(R,G,B)$
channels, assign the post binning mode value. We could closely capture the
colors of different vehicles and the detailed appearance of roads and lanes.

###  IV-C Real-world data generation

For real-world datasets like [22], we split the generation of appearance
information for BEV into two stages. For static classes of interest (like
road, lanes), we colorize the LiDAR point clouds by projecting them into the
time-synced $6$ camera images and assigning every $(x,y,z)$ point in the point
cloud the nearest $(R,G,B)$ image value. This step captures the static classes
with reasonable fidelity. However, it cannot capture dynamic classes of
interest (like vehicles) because of the inherent nature of the operation of
temporal aggregation of point clouds. For dynamic classes of interest (like
vehicles), we paint the vehicle polygons. To get the color for each vehicle
polygon, we project the available annotated vehicle cuboids in the camera
images. Assign the mode value from a coarse color intensity histogram for each
color channel. We bin $8$-Bit colors into $25$ bins of length $10$ each. We
show the pipeline in Fig. 7 For occupancy information in BEV, we follow the
approach of [3, 4] and use the available map information, cuboid annotations
for vehicles, sensor and ego-vehicle poses to generate the occupancy map for
the scene.

##  V Conclusion

This work proposes a method to estimate appearance and occupancy information
in a Bird’s-eye View (BEV) of the scene, centered on the ego vehicle.

We propose an architecture that can reason about the appearance and occupancy
information in BEV from a set of $N$ monocular images with a $360^{\circ}$
total FOV and known intrinsic and extrinsic parameters. We train our method on
data generated from Carla [1]. We carry out extensive qualitative and
quantitative experiments to capture the efficacy of our system in estimating
appearance and occupancy information in different traffic scenarios. We
compare our method with SOTA approaches like [3, 5] and report significant
performance improvement. We also carry out ablation studies in different
camera configurations(front $180\degree$ and surround $360\degree$). Ablation
studies show how appearance can improve occupancy estimation of the scene. We
also present a data generation pipeline for real-world datasets like
NuScenes[22] and show representative generated data using the same.

##  VI Future work

Foremost, we plan to transfer the network to real-world datasets like NuScenes
[22], KITTI [28], and Lyft [29], and conduct detailed experiments and ablation
studies. We also plan to explore the application of color BEV for Multi-Object
Tracking (MOT), where we reason about appearance and occupancy in the same
output space to associate objects. We also have plans to prospect the
potential for using color BEV to develop a language-based navigation system.

## References

  * [1] A. Dosovitskiy, G. Ros, F. Codevilla, A. Lopez, and V. Koltun, “Carla: An open urban driving simulator,” in _Conference on robot learning_. PMLR, 2017, pp. 1–16.
  * [2] W. Luo, B. Yang, and R. Urtasun, “Fast and furious: Real time end-to-end 3d detection, tracking and motion forecasting with a single convolutional net,” in _Proceedings of the IEEE conference on Computer Vision and Pattern Recognition_ , 2018, pp. 3569–3577.
  * [3] J. Philion and S. Fidler, “Lift, splat, shoot: Encoding images from arbitrary camera rigs by implicitly unprojecting to 3d,” in _European Conference on Computer Vision_. Springer, 2020, pp. 194–210.
  * [4] A. Hu, Z. Murez, N. Mohan, S. Dudas, J. Hawke, V. Badrinarayanan, R. Cipolla, and A. Kendall, “Fiery: Future instance prediction in bird’s-eye view from surround monocular cameras,” _arXiv preprint arXiv:2104.10490_ , 2021.
  * [5] K. Chitta, A. Prakash, and A. Geiger, “Neat: Neural attention fields for end-to-end autonomous driving,” in _Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)_ , October 2021, pp. 15 793–15 803.
  * [6] U. R. Nair, S. Sharma, M. S. Menon, and S. Vidapanakal, “Nmr: Neural manifold representation for autonomous driving,” 2022. [Online]. Available: https://arxiv.org/abs/2205.05551
  * [7] S. Casas, A. Sadat, and R. Urtasun, “Mp3: A unified model to map, perceive, predict and plan,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 14 403–14 412.
  * [8] A. Sadat, S. Casas, M. Ren, X. Wu, P. Dhawan, and R. Urtasun, “Perceive, predict, and plan: Safe motion planning through interpretable semantic representations,” in _European Conference on Computer Vision_. Springer, 2020, pp. 414–430.
  * [9] M. Tan and Q. Le, “Efficientnet: Rethinking model scaling for convolutional neural networks,” in _International conference on machine learning_. PMLR, 2019, pp. 6105–6114.
  * [10] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
  * [11] M. Wayland, “U.s. clears way for truly driverless vehicles without steering wheels,” _CNBC_. [Online]. Available: https://www.cnbc.com/2022/03/11/us-clears-way-for-driverless-vehicles-without-steering-wheels.html
  * [12] D. Rufiange, “Vehicles with steering wheel optional, as early as 2025?” _Auto123.com_. [Online]. Available: https://www.auto123.com/en/news/vehicles-without-steering-2025-autonomous-driving-level-3/68472/
  * [13] M. Wayland, “Steeringless and pedalless vehicles become legal in the us,” _newstextarea.com_. [Online]. Available: https://newstextarea.com/steeringless-and-pedalless-vehicles-become-legal-in-the-us/
  * [14] Ö. Erkent, D. S. Gonzalez, A. Paigwar, and C. Laugier, “Gridtrack: Detection and tracking of multiple objects in dynamic occupancy grids,” in _International Conference on Computer Vision Systems_. Springer, 2021, pp. 180–194.
  * [15] C. Gómez-Huélamo, J. Del Egido, L. M. Bergasa, R. Barea, M. Ocana, F. Arango, and R. Gutiérrez-Moreno, “Real-time bird’s eye view multi-object tracking system based on fast encoders for object detection,” in _2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC)_. IEEE, 2020, pp. 1–6.
  * [16] S. Sharma, J. A. Ansari, J. K. Murthy, and K. M. Krishna, “Beyond pixels: Leveraging geometry and shape cues for online multi-object tracking,” in _2018 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2018, pp. 3508–3515.
  * [17] T. Deruyttere, S. Vandenhende, D. Grujicic, Y. Liu, L. V. Gool, M. Blaschko, T. Tuytelaars, and M.-F. Moens, “Commands 4 autonomous vehicles (c4av) workshop summary,” in _European Conference on Computer Vision_. Springer, 2020, pp. 3–26.
  * [18] T. Deruyttere, G. Collell, and M.-F. Moens, “Giving commands to a self-driving car: A multimodal reasoner for visual grounding,” _arXiv preprint arXiv:2003.08717_ , 2020.
  * [19] T. Deruyttere, S. Vandenhende, D. Grujicic, L. Van Gool, and M.-F. Moens, “Talk2car: Taking control of your self-driving car,” _arXiv preprint arXiv:1909.10838_ , 2019.
  * [20] N. Rufus, K. Jain, U. K. R. Nair, V. Gandhi, and K. M. Krishna, “Grounding linguistic commands to navigable regions,” in _2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2021, pp. 8593–8600.
  * [21] N. Rufus, U. K. R. Nair, K. M. Krishna, and V. Gandhi, “Cosine meets softmax: A tough-to-beat baseline for visual grounding,” in _European Conference on Computer Vision_. Springer, 2020, pp. 39–50.
  * [22] H. Caesar, V. Bankiti, A. H. Lang, S. Vora, V. E. Liong, Q. Xu, A. Krishnan, Y. Pan, G. Baldan, and O. Beijbom, “nuscenes: A multimodal dataset for autonomous driving,” in _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2020, pp. 11 621–11 631.
  * [23] S. Ammar Abbas and A. Zisserman, “A geometric approach to obtain a bird’s eye view from an image,” in _Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops_ , 2019, pp. 0–0.
  * [24] R. Hartley and A. Zisserman, _Multiple view geometry in computer vision_. Cambridge university press, 2003\.
  * [25] A. H. Lang, S. Vora, H. Caesar, L. Zhou, J. Yang, and O. Beijbom, “Pointpillars: Fast encoders for object detection from point clouds,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 12 697–12 705.
  * [26] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, “Pytorch: An imperative style, high-performance deep learning library,” in _Advances in Neural Information Processing Systems 32_ , H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, Eds. Curran Associates, Inc., 2019, pp. 8024–8035. [Online]. Available: http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf
  * [27] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _arXiv preprint arXiv:1412.6980_ , 2014.
  * [28] A. Geiger, P. Lenz, C. Stiller, and R. Urtasun, “Vision meets robotics: The kitti dataset,” _The International Journal of Robotics Research_ , vol. 32, no. 11, pp. 1231–1237, 2013.
  * [29] J. Houston, G. Zuidhof, L. Bergamini, Y. Ye, L. Chen, A. Jain, S. Omari, V. Iglovikov, and P. Ondruska, “One thousand and one hours: Self-driving motion prediction dataset,” _arXiv preprint arXiv:2006.14480_ , 2020.


# Exposure effects are not automatically useful for policymaking

Eric Auerbach111Department of Economics, Northwestern University, 2211 Campus
Drive, Evanston IL, 60208<EMAIL_ADDRESS>Jonathan
Auerbach222Department of Statistics, George Mason University, 4400 University
Drive, MS 4A7, Fairfax, VA 22030<EMAIL_ADDRESS>Max Tabord-
Meehan333Department of Economics, University of Chicago 1126 East 59th Street,
Chicago, IL, 60637<EMAIL_ADDRESS>
##  1 Introduction

We thank Sävje (2023) for a thought-provoking article and appreciate the
opportunity to share our perspective as social scientists. In his article,
Sävje recommends misspecified exposure effects as a way to avoid strong
assumptions about interference when analyzing the results of an experiment. In
this invited discussion, we highlight a limitation of Savje’s recommendation:
exposure effects are not generally useful for evaluating social policies
without the strong assumptions that Savje seeks to avoid.

Our discussion is organized as follows. Section 2 summarizes our position,
Section 3 provides a concrete example, and Section 4 concludes. Proof of
claims are in an appendix.

##  2 Exposure effects are not automatically useful

Exposure effects are commonly used as an organizational tool. The idea is to
decompose a complicated interference pattern into individual units that are
easier to manage separately. For instance, consider an experimenter who wants
to learn whether assigning a treatment to every unit is expected to increase
outcomes on average. The effect of this policy may be difficult to estimate
directly when there is interference. However, it can often be written as a
combination of exposure effects, each of which may be straightforward to
estimate using standard techniques. See for instance Manski (2013) and Aronow
and Samii (2017).

But an individual exposure effect is not automatically relevant or
interpretable in isolation. For example, finding that one exposure effect is
large and positive does not mean that the treatment is expected to increase
outcomes for any collection of units. There may be other negative exposure
effects that together effectively cancel out the positive one. Without
accounting for all of the interdependencies between units, an experimenter can
never be sure that they have considered every interaction that is relevant for
the policy question at hand. It is this accounting exercise that requires
strong assumptions about the interference structure.

This is why we disagree with Sävje’s recommendation. We do not dispute his
technical contribution, which is to show that it is possible to consistently
estimate a large class of exposure effects under relatively weak assumptions.
However, Sävje goes on to argue that, as a consequence, experimenters can
safely misspecify the interference structure when analyzing the results of an
experiment. Our disagreement is that we do not see how the results of an
experiment can generally be useful for evaluating social policies outside the
context of a correctly specified interference structure.

##  3 A concrete example: the effect of additional policing on crime

To make our position concrete, we consider a hypothetical experiment in which
a police department randomly assigns additional officers (the treatment) to
city blocks (the units). The goal of the department is to determine the
expected number of crimes prevented by the marginal officer. Interference
occurs because although the officer may only interact with the people that
live and work on that block, those people will travel to other blocks where
they may commit or fall victim to a crime. It is important for the department
to account for this behavior because it may be, for instance, that the
additional officer does not actually prevent any crimes. Rather criminals
simply move their operations from the treated blocks to other less-policed
areas. If these interactions are ignored, the department may incorrectly
conclude that the misplaced crimes were prevented and overstate the expected
impact of additional policing.

In our example, we show that the experimenter can learn about the expected
amount of crime prevented by additional policing using a specific combination
of exposure effects determined by the interference structure. But without any
assumptions, an arbitrary exposure effect can take essentially any sign or
magnitude. As a result, it can not possibly be useful for evaluating the
impact of additional policing on crime.

###  3.1 Model and policy question

The amount of crime reported in block $i=1,\ldots,n$ is given by the model

$\displaystyle
y_{i}(\boldsymbol{z})=\alpha_{i}+\beta_{i}z_{i}+\sum_{r=1}^{R}\gamma_{ir}\sum_{j\in
N_{i}^{r}}z_{j}$ (1)

where $z_{i}\in\\{0,1\\}$ indicates whether an additional officer (treatment)
is assigned to block $i$,
$\boldsymbol{z}=(z_{1},\ldots,z_{n})\in\mathcal{Z}:=\\{0,1\\}^{n}$,
$\alpha_{i}$ is the amount of crime reported in block $i$ when no blocks are
treated, $\beta_{i}$ is the change in crime caused by assigning treatment to
block $i$, $N_{i}^{r}\subseteq\\{1,\ldots,n\\}$ is the collection of blocks at
distance $r$ from block $i$, $\sum_{j\in N_{i}^{r}}z_{j}$ is the number of
treated blocks at distance $r$ from block $i$, and $\gamma_{ir}$ is the change
in crime caused by assigning treatment to an additional block of distance $r$
from $i$. In words, $\beta_{i}$ is a direct effect and $\gamma_{ir}$ is a
spillover effect of treatment. The interference structure is determined by the
sets $N_{i}^{r}$. We assume $i\not\in N_{i}^{r}$ and $N_{i}^{r}\cap
N_{i}^{r^{\prime}}=\varnothing$ for any $r,r^{\prime}=1,...,R$ with $r\neq
r^{\prime}$. Following Sävje, $y_{i}(\boldsymbol{z})$ is nonstochastic.

The police department wants to know whether an additional officer is expected
to reduce crime. Specifically, they are interested in a treatment policy that
assigns the treatment to one block drawn uniformly at random from the city.
Under (1), the expected amount of crime without the policy is
$\sum_{i}\alpha_{i}$. The expected amount of crime with the policy is
$\sum_{i}\alpha_{i}+\bar{\beta}+\bar{\gamma}$ where
$\bar{\beta}:=\frac{1}{n}\sum_{i}\beta_{i}$ and
$\bar{\gamma}:=\frac{1}{n}\sum_{i=1}^{n}\sum_{r=1}^{R}\gamma_{ir}|N_{i}^{r}|$.
The policy is expected to reduce crime if $\bar{\beta}+\bar{\gamma}<0$. In
words, $\bar{\beta}+\bar{\gamma}$ is the average policy effect, $\bar{\beta}$
is the average direct effect, and $\bar{\gamma}$ is the average spillover
effect.

###  3.2 The policy question can be addressed using correctly specified
exposure effects

To estimate the average policy effect we express $\bar{\beta}$ and
$\bar{\gamma}$ as sums of exposure effects. We show in the supplement that
$d_{i}(\boldsymbol{z}):\mathcal{Z}\rightarrow\Delta$ is a correctly specified
exposure mapping where $d_{i}(\boldsymbol{z})=(z_{i},\sum_{j\in
N_{i}^{1}}z_{j},...,\sum_{j\in N_{i}^{R}}z_{j})$ and
$\Delta:=\mathbb{R}^{R+1}$. We also show that

$\displaystyle\bar{\beta}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{y}_{i}(e_{1})-\tilde{y}_{i}(e_{0})\right)\text{
and }$ (2) $\displaystyle\bar{\gamma}$
$\displaystyle=\sum_{r=1}^{R}\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{y}_{i}(e_{r+1})-\tilde{y}_{i}(e_{0})\right)|N_{i}^{r}|$
(3)

where $\tilde{y}_{i}:\Delta\rightarrow\mathbb{R}$ is such that
$\tilde{y}_{i}(d_{i}(\boldsymbol{z}))=y_{i}(\boldsymbol{z})$,
$e_{r}\in\mathbb{R}^{R+1}$ has a $1$ in the $r$th entry and a $0$ in every
other entry, and $e_{0}\in\mathbb{R}^{R+1}$ is a vector of all $0$s. The
exposure effects on the right-hand side of (2) and (3) can be directly
estimated using data from a randomized experiment. For example, suppose
treatment is assigned to block $i$ with probability $p\in(0,1)$ independently
across blocks. The random variable $Z_{i}\in\\{0,1\\}$ indicates whether the
treatment is assigned to block $i$ in the experiment and
$\boldsymbol{Z}:=(Z_{1},...,Z_{n})$. The Horvitz-Thompson estimators of
$\bar{\beta}$ and $\bar{\gamma}$ are

$\displaystyle\hat{\beta}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}\mathbbm{1}\\{d_{i}(\boldsymbol{Z})=e_{1}\\}}{\pi_{i}(e_{1})}-\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}\mathbbm{1}\\{d_{i}(\boldsymbol{Z})=e_{0}\\}}{\pi_{i}(e_{0})}$
(4) $\displaystyle\hat{\gamma}$
$\displaystyle=\sum_{r=1}^{R}\left(\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}\mathbbm{1}\\{d_{i}(\boldsymbol{Z})=e_{r+1}\\}}{\pi_{i}(e_{r+1})}-\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}\mathbbm{1}\\{d_{i}(\boldsymbol{Z})=e_{0}\\}}{\pi_{i}(e_{0})}\right)|N_{i}^{r}|$
(5)

where $Y_{i}=y_{i}(\boldsymbol{Z})$ and
$\pi_{i}(d)=\text{pr}(d_{i}(\boldsymbol{Z})=d)$. See Aronow and Samii (2017).
This gives an estimator for the average policy effect
$\hat{\beta}+\hat{\gamma}$ which can be used to make inferences about the sign
of $\bar{\beta}+\bar{\gamma}$ and evaluate whether an additional officer is
expected to reduce crime.

###  3.3 Sävje recommends misspecified exposure effects

The policy evaluation strategy outlined above requires knowledge of the sets
$N_{i}^{r}$. In practice, however, the experimenter may not know this
interference structure. Sävje argues that experimenters should in such cases
shift their focus to exposure effects that they deem relevant and
interpretable, even if they are induced by misspecified exposure mappings.

To understand Sävje’s recommendation in the context of our example, we suppose
the experimenter misspecifies $N_{i}^{r}$. That is, they assume the
interference is given by $\breve{N}_{i}^{r}\subseteq\\{1,2,\ldots,n\\}$ that
may be different from $N_{i}^{r}$, but still satisfy
$i\not\in\breve{N}_{i}^{r}$ and
$\breve{N}_{i}^{r}\cap\breve{N}_{i}^{r^{\prime}}=\varnothing$ for any
$r,r^{\prime}=1,...,R$ with $r\neq r^{\prime}$. Following Sävje, this
misspecification induces the misspecified exposure mappings
$\breve{d}_{i}(z)=(z_{i},\sum_{j\in\breve{N}_{i}^{1}}z_{j},...,\sum_{j\in\breve{N}_{i}^{R}}z_{j})\in\Delta$
and expected potential outcome functions
$\bar{y}_{i}(d):\Delta\rightarrow\mathbb{R}$ where
$\bar{y}_{i}(d)=E[y_{i}(\boldsymbol{Z})|\breve{d}_{i}(\boldsymbol{Z})=d]$. A
misspecified average spillover effect is

$\displaystyle\breve{\gamma}$
$\displaystyle:=\sum_{r=1}^{R}\frac{1}{n}\sum_{i=1}^{n}\left(\bar{y}_{i}(e_{r+1})-\bar{y}_{i}(e_{0})\right)|\breve{N}_{i}^{r}|.$
(6)

###  3.4 The policy question can not be addressed using misspecified exposure
effects

Our issue with Sävje’s recommendation is that we do not see how one can
meaningfully interpret an exposure effect outside the context of a correctly
specified interference structure. In our example, without assumptions about
how $\breve{N}_{i}^{r}$ is related to $N_{i}^{r}$, the misspecified average
spillover effect $\breve{\gamma}$ can essentially take any sign or magnitude.
Specifically, we show in the appendix that $\breve{\gamma}$ is a weighted
combination of spillover effects

$\displaystyle\breve{\gamma}=\frac{1}{n}\sum_{i=1}^{n}\sum_{r=1}^{R}\gamma_{ir}|N_{i}^{r}|\breve{w}_{ir}\text{
where
}\breve{w}_{ir}=\sum_{s=1}^{R}\frac{|N_{i}^{r}\cap\breve{N}_{i}^{s}||\breve{N}_{i}^{r}|}{|\breve{N}_{i}^{s}||N_{i}^{r}|}.$
(7)

The weights are nonnegative but do not necessarily sum to one. Their magnitude
can be arbitrarily large or small, depending on the discrepancy between
$N_{i}^{r}$ and $\breve{N}_{i}^{r}$. They can also be arbitrarily related to
the sign of $\gamma_{ir}$, so that the sign and magnitude of $\breve{\gamma}$
may be completely unrelated to the sign or magnitude of the average policy,
direct, or spillover effect.

We can think of two specific settings where $\breve{\gamma}$, or any other
misspecified exposure effect, would be useful. The first setting is when the
researcher is narrowly interested in testing the hypothesis that all of the
spillover effects $\gamma_{ir}$ are zero. Estimating a nonzero
$\breve{\gamma}$ provides evidence against this hypothesis. The second setting
is the case where the spillover effects all have the same sign. Under this
strong assumption, the sign of $\breve{\gamma}$ reveals the sign of the
spillovers.

However, we do not see how misspecified exposure effects can generally be
useful for policy making. They can be small in magnitude when the expected
impact of the policy is large. They can also be large when the impact of the
policy on every collection of units is small. Only with knowledge of the
interference structure can the experimenter take exposure effects and, using
formulas akin to (2) and (3), say something concrete about the policy of
interest.

##  4 Conclusion

Our main purpose in writing this discussion is to point out that exposure
effects are not automatically useful without assumptions on the interference
structure. However, misspecified exposure effects may still serve as useful
approximations for some policy effects of interest. For instance, in our own
work, Auerbach and Tabord-Meehan (2021) develop an exposure mapping “sieve”
based on rooted networks. We show that a large class of network interference
patterns can be well approximated by a specific sequence of exposure mappings.
These exposure mappings are misspecified in finite samples, but because they
are correctly specified asymptotically, we are able to consistently estimate
various policy effects. However, the quality of our approximation depends
crucially on how closely our exposure map sieve approximates the true
interference structure and this plays a role in determining the statistical
properties of our proposed estimators.

Ultimately, there is no free lunch. Causal inference with interference is a
complex problem. Experimenters must take a stance on the structure of the
interference in their experiments if they want to characterize the impact of a
treatment in a way that is useful for policy making.

## Acknowledgement

Research is supported by NSF grants SES-2149408 and SES-2149422.

## References

  * Aronow and Samii (2017) Aronow, P.M. & Samii, C. (2017).  Estimating average causal effects under general interference.  Annals of Applied Statistics 11, 1912-1947.
  * Auerbach and Tabord-Meehan (2021) Auerbach, E., & Tabord-Meehan, M.(2021).  The local approach to causal inference under network interference.  arXiv preprint arXiv:2105.03810.
  * Manski (2013) Manski, C. F. (2013).  Identification of treatment response with social interactions.  The Econometrics Journal 16(1), S1-S23.
  * Sävje (2023) Sävje, F. (2023).  Causal inference with misspecified exposure mappings: separating definitions and assumptions.  Biometrika forthcoming

## Proof of claims

Recall that

$\displaystyle y_{i}(\boldsymbol{z})$
$\displaystyle=\alpha_{i}+\beta_{i}z_{i}+\sum_{r=1}^{R}\gamma_{ir}\sum_{j\in
N_{i}^{r}}z_{j},$ $\displaystyle\bar{\beta}$
$\displaystyle:=\frac{1}{n}\sum_{i}\beta_{i},\text{ and }$
$\displaystyle\bar{\gamma}$
$\displaystyle:=\frac{1}{n}\sum_{i=1}^{n}\sum_{r=1}^{R}\gamma_{ir}|N_{i}^{r}|.$

Our first claim is

Claim 1: $d_{i}(\boldsymbol{z})=(z_{i},\sum_{j\in
N_{i}^{1}}z_{j},...,\sum_{j\in N_{i}^{R}}z_{j})\in\mathbb{R}^{R+1}$ is an
exposure map.

Proof of claim 1: $y_{i}(\boldsymbol{z})$ is a deterministic function of
$d_{i}(\boldsymbol{z})$ so that for any
$\boldsymbol{z},\boldsymbol{z}^{\prime}\in\\{0,1\\}^{n}$,
$d_{i}(\boldsymbol{z})=d_{i}(\boldsymbol{z}^{\prime})$ implies that
$y_{i}(\boldsymbol{z})=y_{i}(\boldsymbol{z}^{\prime})$. $\square$

Our second claim is

Claim 2:
$\bar{\beta}=\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{y}_{i}(e_{1})-\tilde{y}_{i}(e_{0})\right)$
and
$\bar{\gamma}=\sum_{r=1}^{R}\frac{1}{n}\sum_{i=1}^{n}\left(\tilde{y}_{i}(e_{r+1})-\tilde{y}_{i}(e_{0})\right)|N_{i}^{r}|.$

Proof of claim 2: $d_{i}(\boldsymbol{z})=e_{0}$ implies $z_{i}=0$ and
$\sum_{j\in N_{i}^{s}}z_{j}=0$ for $s=1,...,R$, so that
$\tilde{y}_{i}(e_{0})=\alpha_{i}$. $d_{i}(\boldsymbol{z})=e_{1}$ implies
$z_{i}=1$ and $\sum_{j\in N_{i}^{s}}z_{j}=0$ for $s=1,...,R$ so that
$\tilde{y}_{i}(e_{1})=\alpha_{i}+\beta_{i}$. $d_{i}(\boldsymbol{z})=e_{r+1}$
for $r=1,...,R$ implies $z_{i}=0$, $\sum_{j\in N_{i}^{s}}z_{j}=1$ for $s=r$,
and $\sum_{j\in N_{i}^{s}}z_{j}=0$ for $s\not=r$ so that
$\tilde{y}_{i}(e_{r+1})=\alpha_{i}+\gamma_{ir}$. $\square$

Our third claim is

Claim 3:
$\breve{\gamma}=\frac{1}{n}\sum_{i=1}^{n}\sum_{r=1}^{R}\gamma_{ir}|N_{i}^{r}|\breve{w}_{ir}$
where
$\breve{w}_{ir}=\sum_{s=1}^{R}\frac{|N_{i}^{r}\cap\breve{N}_{i}^{s}||\breve{N}_{i}^{r}|}{|\breve{N}_{i}^{s}||N_{i}^{r}|}$.

Proof of claim 3: Define $\breve{N}_{i}:=\cup_{r=1}^{R}\breve{N}_{i}^{r}$ and
write $y_{i}(\boldsymbol{Z})$ as

$\displaystyle
y_{i}(\boldsymbol{Z})=\alpha_{i}+\beta_{i}Z_{i}+\sum_{r=1}^{R}\gamma_{ir}\sum_{j\in
N_{i}^{r}}Z_{j}\mathbbm{1}\\{j\not\in\breve{N}_{i}\\}+\sum_{r=1}^{R}\gamma_{ir}\sum_{j\in
N_{i}^{r}}Z_{j}\mathbbm{1}\\{j\in\breve{N}_{i}\\}.$

The event $\breve{d}_{i}(\boldsymbol{Z})=e_{0}$ implies that $Z_{j}=0$ for
every $j\in\breve{N}_{i}\cup\\{i\\}$ and so
$\sum_{r=1}^{R}\gamma_{ir}\sum_{j\in
N_{i}^{r}}Z_{j}\mathbbm{1}\\{j\in\breve{N}_{i}\\}=0$. It follows that

$\displaystyle
E[y_{i}(\boldsymbol{Z})|\breve{d}_{i}(\boldsymbol{Z})=e_{0}]=\alpha_{i}+\sum_{r=1}^{R}\gamma_{ir}\sum_{j\in
N_{i}^{r}}\mathbbm{1}\\{j\not\in\breve{N}_{i}\\}p$

because the entries of $\boldsymbol{Z}$ are iid Bernoulli($p$). It follows
from the same logic that

$\displaystyle
E[y_{i}(\boldsymbol{Z})|\breve{d}_{i}(\boldsymbol{Z})=e_{s}]=\alpha_{i}+\sum_{r=1}^{R}\gamma_{ir}\sum_{j\in
N_{i}^{r}}\mathbbm{1}\\{j\not\in\breve{N}_{i}\\}p+\sum_{r=1}^{R}\gamma_{ir}\breve{w}_{irs}$

for $s=2,...,R+1$ where $\breve{w}_{irs}:=E\left[\sum_{j\in
N_{i}^{r}}Z_{j}\mathbbm{1}\\{j\in\breve{N}_{i}\\}|\breve{d}_{i}(\boldsymbol{Z})=e_{s}\right]$.
As a result,
$\breve{\gamma}=\frac{1}{n}\sum_{i=1}^{n}\sum_{r=1}^{R}\gamma_{ir}|N_{i}^{r}|\breve{\omega}_{ir}$
where
$\breve{\omega}_{ir}:=\sum_{s=1}^{R}\breve{w}_{irs}|\breve{N}_{i}^{r}|/|N_{i}^{r}|$.
What remains to be shown is that
$\breve{w}_{irs}=|N_{i}^{r}\cap\breve{N}_{i}^{s}|/|\breve{N}_{i}^{s}|$. To
demonstrate this, we note that that under the event
$\breve{d}_{i}(\boldsymbol{Z})=e_{s}$ for $s=2,...,R+1$, exactly one unit in
$\breve{N}_{i}^{s}$ is treated and no units in $\breve{N}_{i}^{s^{\prime}}$
are treated for $s^{\prime}\neq s$. $\breve{w}_{irs}=E\left[\sum_{j\in
N_{i}^{r}}Z_{j}\mathbbm{1}\\{j\in\breve{N}_{i}\\}|\breve{d}_{i}(\boldsymbol{Z})=e_{s}\right]$
is then just the probability that the one treated unit in $\breve{N}_{i}^{s}$
is also an element of $N_{i}^{r}$. Since the entries of $\boldsymbol{Z}$ are
iid, each element in $\breve{N}_{i}^{s}$ is equally likely to be the treated
one, and so the probability that the treated unit is in $N_{i}^{r}$ is
proportional to $|N_{i}^{r}\cap\breve{N}_{i}^{s}|$. It follows that
$\breve{w}_{irs}=|N_{i}^{r}\cap\breve{N}_{i}^{s}|/|\breve{N}_{i}^{s}|$.
$\square$


$\big{(}\triangle_{g}+h_{\alpha}-f^{\prime}(\mathcal{W}_{\alpha})\big{)}\varphi_{\alpha}=\tilde{R}_{\alpha}+N_{\alpha}(\varphi_{\alpha})+\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha},$

where $\tilde{R}_{\alpha}$ is as in (4.7). Since
$\varphi_{\alpha}\in\mathcal{S}_{\alpha}$ we have, by (4.5),
$\|\varphi_{\alpha}\|_{L^{2^{*}}(M)}=o(1)$ and

$\|N_{\alpha}(\varphi_{\alpha})\|_{L^{\frac{2n}{n+2}}(M)}\lesssim\|\varphi_{\alpha}\|_{L^{2^{*}}(M)}^{1+\tau}.$

Independently, we also have with (4.1) and (4.7) that

$\|\tilde{R}_{\alpha}\|_{L^{\frac{2n}{n+2}}(M)}\lesssim\varepsilon_{\alpha}+\sigma_{\alpha}.$

Sobolev’s inequality and Proposition 3.1 then show that

$\displaystyle\|\varphi_{\alpha}\|_{L^{2^{*}}(M)}\lesssim\|\varphi_{\alpha}\|_{H^{1}(M)}$
$\displaystyle\lesssim\|(\triangle_{g}+1)^{-1}(\tilde{R}_{\alpha}+N_{\alpha}(\varphi_{\alpha}))\|_{H^{1}(M)}$
$\displaystyle\lesssim\varepsilon_{\alpha}+\sigma_{\alpha}+\|\varphi_{\alpha}\|_{L^{2^{*}}(M)}^{1+\tau},$

which proves (4.3). ∎

###  4.2. A priori estimates for solutions of (4.2)

We now prove a slightly more general result than Theorem 1.1, that states that
solutions of (4.2) inherit pointwise estimates:

######  Theorem 4.3.

Let $(h_{\alpha})_{\alpha}$ be a sequence of functions converging towards $h$
in $L^{\infty}(M)$, $k\geq 1$ be an integer, $u_{0}\in C^{1,\beta}(M)$ for
$0<\beta<1$ be a solution of (2.15), $V_{1},\dots,V_{k}$ in
$D^{1,2}(\mathbb{R}^{n})$ be $k$ solutions of (2.1) and let
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ be
$k$ sequences satisfying (1.4). Let $(\mathcal{W}_{\alpha})_{\alpha}$ be a
sequence of continuous functions in $M$ satisfying (3.5) and (4.1) for some
$(\varepsilon_{\alpha})_{\alpha}\in\mathcal{E}$ and let $\varphi_{\alpha}\in
K_{\alpha}^{\perp}$ satisfy

$\Pi_{K_{\alpha}^{\perp}}\Bigg{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}-(\triangle_{g}+1)^{-1}\Big{[}(1-h_{\alpha})(\mathcal{W}_{\alpha}+\varphi_{\alpha})+f\big{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}\big{)}\Big{]}\Bigg{)}=0$

and

$\|\varphi_{\alpha}\|_{H^{1}(M)}=o(1)$

as $\alpha\to+\infty$. Let $u_{\alpha}=\mathcal{W}_{\alpha}+\varphi_{\alpha}$.
Up to passing to a subsequence for $(u_{\alpha})_{\alpha}$ we have, for any
sequence $(x_{\alpha})_{\alpha}$ of points of $M$,

$u_{\alpha}(x_{\alpha})=u_{0}(x_{\alpha})+\sum_{i=1}^{k}V_{i,\alpha}(x_{\alpha})+o\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha})\Big{)},$
(4.10)

where $V_{i,\alpha}$ is as in (2.14) and $B_{i,\alpha}$ as in (3.4).

Passing to a subsequence for $(u_{\alpha})_{\alpha}$ amounts to passing to a
subsequence for $(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq
i\leq k$, $(h_{\alpha})_{\alpha}$, $(\varepsilon_{\alpha})_{\alpha}$ and
$(\varphi_{\alpha})_{\alpha}$. As will be momentarily proven, Theorem 1.1
follows from Theorem 4.3. Theorem 4.3 shows in particular that for approximate
solutions $\mathcal{W}_{\alpha}+\varphi_{\alpha}$ to (1.1), ie satisfying only
(4.2) and not necessarily (1.1), the remainder $\varphi_{\alpha}$ is globally
pointwise small with respect to $u_{0}+\sum_{i=1}^{k}V_{i,\alpha}$. In
particular, approximate solution obtained by a Lyapunov-schmidt method satisfy
(4.10). This applies to the examples of blowing-up solutions to second-order
critical elliptic equations obtained in Ambrosetti-Malchiodi [2], Brendle [6],
Brendle-Marques [7], Esposito-Pistoia-Vétois [23], Morabito-Pistoia-Vaira
[34], Pistoia-Vaira [38], Robert-Vétois [47, 48, 49], Thizy [51] and Thizy-
Vétois [52]. For strongly coupled systems or in nonsymmetric situations,
pointwise estimates for approximate solutions constructed by a Lyapunov-
schmidt method have proven more effective than $H^{1}(M)$ estimates to obtain
the existence of blowing-up solutions, see e.g. Premoselli [39, 40, 41] or
Premoselli-Thizy [42].

The approach that we follow here to prove Theorem 4.3 (via the linear result
of Theorem 3.2 and the nonlinear perturbation argument of Proposition 4.2) is
particularly robust and bypasses many difficulties that are usually
encountered when trying to obtain pointwise bounds for solutions of critical
nonlinear equations.

###### Proof.

We first claim that exists $C>0$ and a sequence $(\sigma_{\alpha})_{\alpha}$
satisfying (3.7) such that, up to passing to a subsequence,

$|\varphi_{\alpha}(x)|\leq
C(\varepsilon_{\alpha}+\sigma_{\alpha})^{\frac{1}{2}}\sum_{i=0}^{k}B_{i,\alpha}(x)+C\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)$
(4.11)

holds for all $x\in M$. Indeed, since $\mathcal{W}_{\alpha}$ satisfies (3.5)
and (4.1), Propositions 4.1 and 4.2 both apply. On the one side, Proposition
4.2 states, up to passing to a subsequence, the existence of $C>0$, of a
sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7) and of a unique

$\displaystyle\check{\varphi}_{\alpha}\in
K_{\alpha}^{\perp}\cap\Big{\\{}\varphi\in C^{0}(M),$
$\displaystyle|\varphi(x)|\leq
C(\varepsilon_{\alpha}+\sigma_{\alpha})^{\frac{1}{2}}\sum_{i=0}^{k}B_{i,\alpha}(x)$
$\displaystyle+C\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\quad\textrm{
for all }x\in M\Big{\\}}$

that solves the equation

$\Pi_{K_{\alpha}^{\perp}}\Bigg{(}\mathcal{W}_{\alpha}+\check{\varphi}_{\alpha}-(\triangle_{g}+1)^{-1}\Big{[}(1-h_{\alpha})(\mathcal{W}_{\alpha}+\check{\varphi}_{\alpha})+f\big{(}\mathcal{W}_{\alpha}+\check{\varphi}_{\alpha}\big{)}\Big{]}\Bigg{)}=0$
(4.12)

and satisfies in addition $\|\check{\varphi}_{\alpha}\|_{H^{1}(M)}\to 0$. On
the other side, for some fixed $\delta>0$ small enough, Proposition 4.1 states
the existence of a unique $\hat{\varphi}_{\alpha}\in
K_{\alpha}^{\perp}\cap\\{\varphi\in
H^{1}(M),\|\varphi\|_{H^{1}(M)}\leq\delta\\}$ satisfying (4.12) up to a
subsequence. Since $\|\varphi_{\alpha}\|_{H^{1}(M)}\to 0$ as
$\alpha\to+\infty$ by assumption we therefore have
$\varphi_{\alpha}=\hat{\varphi}_{\alpha}$ and
$\check{\varphi}_{\alpha}=\hat{\varphi}_{\alpha}$. Hence
$\varphi_{\alpha}=\check{\varphi}_{\alpha}$ and $\varphi_{\alpha}$ satisfies
(4.11).

We now prove (4.10). We let $u_{\alpha}=\mathcal{W}_{\alpha}+\varphi_{\alpha}$
and we will prove that, up to passing to a subsequence for
$(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$,
$(h_{\alpha})_{\alpha}$, $(\varepsilon_{\alpha})_{\alpha}$ and
$(\varphi_{\alpha})_{\alpha}$, we have

$\Bigg{|}\Bigg{|}\frac{u_{\alpha}-u_{0}-\sum_{i=1}^{k}V_{i,\alpha}}{\sum_{i=0}^{k}B_{i,\alpha}}\Bigg{|}\Bigg{|}_{L^{\infty}(M)}\to
0$

as $\alpha\to+\infty$. We proceed by contradiction and assume, up to passing
to a subsequence, that there exists a sequence $(x_{\alpha})_{\alpha}$ of
points of $M$ and $\eta_{0}>0$ such that

$\Big{|}u_{\alpha}(x_{\alpha})-u_{0}(x_{\alpha})-\sum_{i=1}^{k}V_{i,\alpha}(x_{\alpha})\Big{|}\geq\eta_{0}\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha}).$
(4.13)

We still assume, up to renumbering the bubbles, that
$\mu_{1,\alpha}\geq\mu_{2,\alpha}\geq\dots\geq\mu_{k,\alpha}$. By (3.5) and
(4.11) such a sequence $(x_{\alpha})_{\alpha}$ then satisfies

$\frac{1}{\mu_{1,\alpha}}\min_{i=1\dots
k}d_{g}(x_{i,\alpha},x_{\alpha})\to+\infty$ (4.14)

as $\alpha\to+\infty$. By definition of $\varphi_{\alpha}$ there exist real
numbers $\lambda_{i,j}^{\alpha}$ such that $\varphi_{\alpha}$ satisfies

$\big{(}\triangle_{g}+h_{\alpha}-f^{\prime}(\mathcal{W}_{\alpha})\big{)}\varphi_{\alpha}=\tilde{R}_{\alpha}+N_{\alpha}(\varphi_{\alpha})+\sum\limits_{\underset{j=1..n_{i}}{i=1..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}$
(4.15)

where $\tilde{R}_{\alpha}$ and $N_{\alpha}(\varphi_{\alpha})$ are as in (4.7).
By (4.1), $\|\tilde{R}_{\alpha}\|_{H^{1}(M)}\leq
C(\varepsilon_{\alpha}+\sigma_{\alpha})$ for some $C>0$. Since
$\varphi_{\alpha}=\check{\varphi}_{\alpha}$ we have
$\|\varphi_{\alpha}\|_{H^{1}(M)}\leq C(\varepsilon_{\alpha}+\sigma_{\alpha})$
by (4.3). Integrating (4.15) against each $Z_{i,j,\alpha}$ and using (3.8)
yields:

$|\lambda_{i,j}^{\alpha}|\leq C(\varepsilon_{\alpha}+\sigma_{\alpha})$ (4.16)

for $0\leq i\leq k$ and $1\leq j\leq n_{j}$. Now, $u_{\alpha}$ satisfies

$\triangle_{g}u_{\alpha}+h_{\alpha}u_{\alpha}=|u_{\alpha}|^{2^{*}-2}u_{\alpha}+\sum\limits_{\underset{j=1..n_{i}}{i=1..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}.$

Let $G_{h}$ be the Green’s function of $\triangle_{g}+h$. Since
$\|h_{\alpha}-h\|_{L^{\infty}(M)}\to 0$ as $\alpha\to+\infty$ a representation
formula for $\triangle_{g}+h$ together with (4.16) yields, with (A.1):

$\displaystyle
u_{\alpha}(x_{\alpha})-\Pi_{K_{h}}(u_{\alpha})(x_{\alpha})=\int_{M}G_{h}(x_{\alpha},\cdot)|u_{\alpha}|^{2^{*}-2}u_{\alpha}dv_{g}$
(4.17)
$\displaystyle+O\Big{(}(\varepsilon_{\alpha}+\sigma_{\alpha})\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}$

where we have let $K_{h}=\ker(\triangle_{g}+h)$. Assume first that
$B_{0,\alpha}\equiv 0$, that is $u_{0}\equiv 0$ and
$\ker(\triangle_{g}+h)=\\{0\\}$. Then $\Pi_{K_{h}}(u_{\alpha})\equiv 0$ and,
as a consequence of (4.11) and using (A.7), we have

$\int_{M\backslash\bigcup_{i=1}^{k}B_{g}(x_{j,\alpha},\frac{r_{i,\alpha}}{R_{0}})}G_{h}(x_{\alpha},\cdot)^{2-n}|u_{\alpha}|^{2^{*}-2}u_{\alpha}dv_{g}=O\Big{(}\sigma_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}.$

Assume now that $B_{0,\alpha}\equiv 1$. Then, with (4.11) we have that

$\Pi_{K_{h}}(u_{\alpha})(x_{\alpha})=\Pi_{K_{h}}(u_{0})(x_{\alpha})+O\Big{(}\sigma_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}.$

On
$M\backslash\bigcup_{i=1}^{k}B_{g}(x_{j,\alpha},\frac{r_{i,\alpha}}{R_{0}})$
we have by (3.5), (3.72) and (4.11) that

$\Big{|}|u_{\alpha}|^{2^{*}-2}u_{\alpha}-|u_{0}|^{2^{*}-2}u_{0}\Big{|}\lesssim\sum_{i=1}^{k}B_{i,\alpha}+\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}+\sigma_{\alpha}B_{0,\alpha}$

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7). Since
$B_{0,\alpha}\equiv 1$ we have $r_{i,\alpha}\to 0$ for any
$i\in\\{1,\dots,k\\}$ by definition, so using a representation formula for
(2.15), (3.72) and (A.1) below gives

$\displaystyle\int_{M\backslash\bigcup_{i=1}^{k}B_{g}(x_{j,\alpha},\frac{r_{i,\alpha}}{R_{0}})}G_{h}(x_{\alpha},\cdot)^{2-n}|u_{\alpha}|^{2^{*}-2}u_{\alpha}dv_{g}$
$\displaystyle=u_{0}(x_{\alpha})-\Pi_{K_{h}}(u_{0})(x_{\alpha})$
$\displaystyle+O\Big{(}\sigma_{\alpha}\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}.$

Independently, if $i\in\\{1,\dots,k\\}$ we have, mimicking the proof of
(3.36):

$\displaystyle\int_{\bigcup_{j\in\mathcal{B}_{i}}\Big{(}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})\Big{)}}$
$\displaystyle
G_{h}(x_{\alpha},\cdot)^{2-n}|u_{\alpha}|^{2^{*}-2}u_{\alpha}dv_{g}$
$\displaystyle=O\Big{(}\sigma_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}.$

By (3.17), $B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$ and
$B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})$ are disjoint if
$\mu_{i,\alpha}\asymp\mu_{j,\alpha}$. If $\mu_{j,\alpha}=o(\mu_{i,\alpha})$
and $j\in\mathcal{B}_{i}$ then, since $\rho_{j,i,\alpha}\geq s_{j,i,\alpha}$
(see (3.22)) and $R_{0}>2$, $\Omega_{i,\alpha}$ and
$B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})$ have empty intersection.
Finally, if $\mu_{j,\alpha}=o(\mu_{i,\alpha})$ and
$d_{g}(x_{j,\alpha},x_{i,\alpha})>\frac{2r_{i,\alpha}}{R_{0}}$ we have
$r_{j,\alpha}=o(d_{g}(x_{i,\alpha},x_{j,\alpha}))$ by (3.16). Hence
$B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$ and
$B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})$ are disjoint. This shows that
the sets $\Omega_{i,\alpha}$ defined in (3.37) are pairwise disjoint. Coming
back to (4.17) we have thus shown that

$\displaystyle
u_{\alpha}(x_{\alpha})=u_{0}(x_{\alpha})+\sum_{i=1}^{k}\int_{\Omega_{i,\alpha}}G_{h}(x_{\alpha},\cdot)|u_{\alpha}|^{2^{*}-2}u_{\alpha}dv_{g}$
(4.18)
$\displaystyle+O\Big{(}(\varepsilon_{\alpha}+\sigma_{\alpha})\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}.$

Let $R>0$ and let $i\in\\{1,\dots,k\\}$. Let $y\in
B(0,R)\cap\hat{\Omega}_{i,\alpha}$, where $\hat{\Omega}_{i,\alpha}$ is as in
(3.50). By (4.14) we have

$(n-2)\omega_{n-1}d_{g}(x_{i,\alpha},x_{\alpha})^{n-2}G_{h}\big{(}x_{\alpha},\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\to
1\quad\textrm{ if }d_{g}(x_{i,\alpha},x_{\alpha})\to 0$

and

$G_{h}\big{(}x_{\alpha},\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}=G_{h}(x_{\alpha},x_{i,\alpha})+o(1)\quad\textrm{
if }d_{g}(x_{i,\alpha},x_{\alpha})\not\to 0$

as $\alpha\to+\infty$. Since

$(n-2)\omega_{n-1}d_{g}(x_{i,\alpha},x_{\alpha})^{n-2}G_{h}(x_{i,\alpha},x_{\alpha})\to
1$

as $\alpha\to+\infty$ if $d_{g}(x_{i,\alpha},x_{\alpha})\to 0$ the dominated
convergence theorem gives

$\displaystyle\int_{\Omega_{i,\alpha}\cap
B_{g}(x_{i,\alpha},R\mu_{i,\alpha})}G_{h}(x_{\alpha},\cdot)|u_{\alpha}|^{2^{*}-2}u_{\alpha}dv_{g}$
(4.19)
$\displaystyle=(1+o(1))G_{h}(x_{\alpha},x_{i,\alpha})\mu_{i,\alpha}^{\frac{n-2}{2}}\int_{B(0,R)}|V_{i}|^{2^{*}-2}V_{i}dx$
$\displaystyle=(1+o(1))G_{h}(x_{\alpha},x_{i,\alpha})(n-2)\omega_{n-1}\mu_{i,\alpha}^{\frac{n-2}{2}}\big{(}\lambda_{i}+O(R^{-2})\big{)}$
$\displaystyle=V_{i,\alpha}(x_{\alpha})+o(B_{i,\alpha}(x_{\alpha}))+O(R^{-2}B_{i,\alpha}(x_{\alpha}))$

where the third line follows from (2.5) and the last one by (2.14).
Independently, we have with (3.24), (4.11) and (A.7) that

$\Big{|}\int_{\Omega_{i,\alpha}\backslash
B_{g}(x_{i,\alpha},R\mu_{i,\alpha})}G_{h}(x_{\alpha},\cdot)|u_{\alpha}|^{2^{*}-2}u_{\alpha}dv_{g}\Big{|}\lesssim
R^{-2}B_{i,\alpha}(x_{\alpha}).$

The last two lines together with (4.18) show that, for any fixed $R>0$ and for
any $\alpha$ large enough we have

$u_{\alpha}(x_{\alpha})=u_{0}(x_{\alpha})+\sum_{i=1}^{k}V_{i,\alpha}(x_{\alpha})+o\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}+O\Big{(}R^{-2}\sum_{i=1}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}.$

Up to picking $R$ large enough and passing again to a subsequence this
contradicts (4.13) and concludes the proof of Theorem 4.3. ∎

###  4.3. Proof of Theorem 1.1

In this last subsection we prove Theorem 1.1. We start with a simple
consideration. Let $(M,g)$ be a closed manifold of dimension $n\geq 3$, let
$(h_{\alpha})_{\alpha}$ be a sequence of functions that converges in
$L^{\infty}(M)$ towards $h$ as $\alpha\to+\infty$, $k\geq 1$ be an integer,
$u_{0}\in C^{1,\beta}(M)$ for $0<\beta<1$ be a solution of (2.15),
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ be
$k$ sequences satisfying (1.4) and let $V_{1},\dots,V_{k}$ in
$D^{1,2}(\mathbb{R}^{n})$ be $k$ solutions of (2.1).

Let $(\varepsilon_{\alpha})_{\alpha}\in\mathcal{E}$ and let $c_{i,j,\alpha}$,
$0\leq i\leq k$, $1\leq i\leq n_{i}$, be real numbers satisfying

$\sum_{i=0}^{k}\sum_{j=1}^{n_{i}}|c_{i,j,\alpha}|\leq\varepsilon_{\alpha}.$

We define, for any $\alpha\geq 1$,

$\displaystyle u_{0,\alpha}$
$\displaystyle=u_{0}+\sum_{j=1}^{n_{0}}c_{0,j,\alpha}Z_{0,j}$ (4.20)
$\displaystyle\ W_{i,\alpha}$
$\displaystyle=V_{i,\alpha}+\sum_{j=1}^{n_{i}}c_{i,j,\alpha}Z_{i,j,\alpha}.$

Here $V_{i,\alpha}$ is given by (2.14), $(Z_{i,j,\alpha})_{1\leq j\leq n_{i}}$
for $1\leq i\leq k$ are given by (2.16) and span $K_{i,\alpha}$ and
$(Z_{0,j})_{1\leq j\leq n_{0}}$ span $K_{0}$ (see (2.17) and (2.18)). If
$u_{0}\equiv 0$ and $\ker(\triangle_{g}+h)=\\{0\\}$ then $u_{0,\alpha}\equiv
0$. We let

$\mathcal{W}_{\alpha}=u_{0,\alpha}+\sum_{i=1}^{k}W_{i,\alpha}.$

By (2.3), (4.20) and since $\varepsilon_{\alpha}\to 0$, this
$\mathcal{W}_{\alpha}$ satisfies (3.5). As the following lemma shows it also
satisfies (4.1):

######  Lemma 4.4.

There exists a constant $C$ only depending on $(M,g),n,V_{1},\dots,V_{k}$ and
$u_{0}$ such that

$\displaystyle\Big{|}\triangle_{g}\mathcal{W}_{\alpha}+h_{\alpha}\mathcal{W}_{\alpha}$
$\displaystyle-|\mathcal{W}_{\alpha}|^{2^{*}-2}\mathcal{W}_{\alpha}\Big{|}\leq
C(\varepsilon_{\alpha}+\|h_{\alpha}-h\|_{L^{\infty}(M)})B_{0,\alpha}$
$\displaystyle+C\sum_{i=1}^{k}B_{i,\alpha}+C\sum\limits_{\underset{i\neq
j}{i,j=0..k}}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}+C\varepsilon_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}$

in $M$.

###### Proof.

Let $\tilde{\mathcal{W}_{\alpha}}=u_{0}+\sum_{i=1}^{k}V_{i,\alpha}$. Using
(2.16) we have

$|\triangle_{g}Z_{j,m,\alpha}|\lesssim B_{j,\alpha}^{2^{*}-1}+B_{j,\alpha}$

for $1\leq j\leq k$ and $1\leq m\leq n_{j}$, while if $j=0$ we have, for all
$1\leq m\leq n_{0}$, $|(\triangle_{g}+1)Z_{0,m,\alpha}|\lesssim 1$. With (3.5)
and (4.20) we then have, since $\varepsilon_{\alpha}\leq 1$

$\big{|}(\triangle_{g}+h_{\alpha})\mathcal{W}_{\alpha}-(\triangle_{g}+h_{\alpha})\tilde{\mathcal{W}_{\alpha}}\big{|}\lesssim\varepsilon_{\alpha}B_{0,\alpha}+\sum_{j=1}^{k}B_{j,\alpha}+\varepsilon_{\alpha}\sum_{j=1}^{k}B_{j,\alpha}^{2^{*}-1}.$

Independently, and since $\varepsilon_{\alpha}\leq 1$,

$\big{|}|\mathcal{W}_{\alpha}|^{2^{*}-2}\mathcal{W}_{\alpha}-|\tilde{\mathcal{W}_{\alpha}}|^{2^{*}-2}\tilde{\mathcal{W}_{\alpha}}\big{|}\lesssim\varepsilon_{\alpha}B_{0,\alpha}+\varepsilon_{\alpha}\sum_{j=1}^{k}B_{j,\alpha}^{2^{*}-1}.$

Finally, (2.12) and (2.13) show that

$\displaystyle\Big{|}\triangle_{g}\tilde{\mathcal{W}_{\alpha}}+h_{\alpha}\tilde{\mathcal{W}_{\alpha}}-|\tilde{\mathcal{W}_{\alpha}}|^{2^{*}-2}\mathcal{W}_{\alpha}\Big{|}$
$\displaystyle\lesssim\|h_{\alpha}-h\|_{L^{\infty}(M)}B_{0,\alpha}$
$\displaystyle+\sum\limits_{\underset{i\neq
j}{i,j=0..k}}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}+\sum_{j=1}^{k}B_{j,\alpha}.$

Gathering these estimates proves the lemma. ∎

We can now conclude the proof of Theorem 1.1.

###### Proof of Theorem 1.1..

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$ and
$(h_{\alpha})_{\alpha}$ be a sequence of functions that converges in
$L^{\infty}(M)$ towards $h$ as $\alpha\to+\infty$. Let $(u_{\alpha})_{\alpha}$
be a sequence of (possibly sign-changing) solutions of

$\triangle_{g}u_{\alpha}+h_{\alpha}u_{\alpha}=|u_{\alpha}|^{2^{*}-2}u_{\alpha}$

in $M$, satisfying (1.2). Struwe’s decomposition (1.3) shows that there exist
$u_{0}\in C^{1,\beta}(M)$ for some $0<\beta<1$ solving

$\triangle_{g}u_{0}+hu_{0}=|u_{0}|^{2^{*}-2}u_{0},$

and that there exist $k\geq 1$, $k$ sequences $(x_{i,\alpha})_{\alpha}$,
$1\leq i\leq k$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ satisfying
(1.4) and $k$ solutions $V_{1},\dots,V_{k}$ in $D^{1,2}(\mathbb{R}^{n})$ of
(2.1) such that

$u_{\alpha}=u_{0}+\sum_{i=1}^{k}V_{i,\alpha}+\Psi_{\alpha},$

where $\|\Psi_{\alpha}\|_{H^{1}(M)}\to 0$ as $\alpha\to+\infty$ and where
$V_{i,\alpha}=V_{i}^{\mu_{i,\alpha},x_{i,\alpha}}$ is given by (2.10). Let

$\psi_{\alpha}^{\perp}=\Pi_{K_{\alpha}^{\perp}}(\Psi_{\alpha})\quad\textrm{
and }\quad\ \psi_{\alpha}^{||}=\Pi_{K_{\alpha}}(\Psi_{\alpha}),$

where $K_{\alpha}$ is given by (2.18). By (3.8) the functions
$Z_{i,j,\alpha}$, $0\leq i\leq k$, $1\leq j\leq n_{i}$ form an almost-
orthonormal family of $K_{\alpha}$ for the $H^{1}(M)$ scalar product, and we
therefore have

$\psi_{\alpha}^{||}=\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}c_{i,j,\alpha}Z_{i,j,\alpha}$
(4.21)

for some real numbers $c_{i,j,\alpha}$, $0\leq i\leq k$, $1\leq i\leq n_{i}$,
satisfying

$\sum_{i=0}^{k}\sum_{j=1}^{n_{i}}|c_{i,j,\alpha}|\leq
C\|\Psi_{\alpha}\|_{H^{1}(M)}$

for some $C>0$ independent of $\alpha$. Let

$\varepsilon_{\alpha}=C\|\Psi_{\alpha}\|_{H^{1}(M)},$ (4.22)

so that $\varepsilon_{\alpha}\to 0$ as $\alpha\to+\infty$. We can then rewrite
Struwe’s decomposition as

$u_{\alpha}=\mathcal{W}_{\alpha}+\psi_{\alpha}^{\perp},$

where $\psi_{\alpha}^{\perp}\in K_{\alpha}^{\perp}$,
$\mathcal{W}_{\alpha}=u_{0,\alpha}+\sum_{i=1}^{k}W_{i,\alpha}$ and
$u_{0,\alpha}$ and $W_{i,\alpha}$ are given by (4.20) with the value of
$c_{i,j,\alpha}$ given by (4.21). In particular, by Lemma 4.4,
$\mathcal{W}_{\alpha}$ satisfies (3.5) and (4.1). Since
$\|\psi_{\alpha}^{\perp}\|_{H^{1}(M)}\to 0$, Theorem 4.3 applies and concludes
the proof. ∎

######  Remark 4.5.

The proof of Theorem 4.3 requires to prove the intermediate estimate (4.11),
which is not as precise as (4.10) at finite distances from the points
$(x_{i,\alpha})_{1\leq i\leq k}$. The additional terms
$\Theta_{i,\alpha}B_{i,\alpha}$ that have to be taken into account in
Proposition 4.2 come from the error $O(B_{\mu_{\alpha},x_{\alpha}})$ in the
right-hand side of (2.13). If $V$ is a sign-changing solution of (2.1),
indeed, one only has in general

$|(\triangle_{g}+h)V^{\mu_{\alpha},x_{\alpha}}-|V^{\mu_{\alpha},x_{\alpha}}|^{2^{*}-2}V^{\mu_{\alpha},x_{\alpha}}|\lesssim
B_{\mu_{\alpha},x_{\alpha}}\quad\textrm{ in }M$

where $V^{\mu_{\alpha},x_{\alpha}}$ is given by (2.10) and this error estimate
does not seem to improve. This contrasts with what happens for positive
solutions, where radiality ensures that

$\triangle_{g}B_{\mu_{\alpha},x_{\alpha}}=B_{\mu_{\alpha},x_{\alpha}}^{2^{*}-1}$

at any order in conformal normal coordinates.

######  Remark 4.6.

Theorem 1.1 requires no assumption on $h$. If
$\ker(\triangle_{g}+h)\neq\\{0\\}$ and $u_{0}\equiv 0$ one expects
$u_{\alpha}$ to possess a non-zero projection on $\ker(\triangle_{g}+h)$.
Under these assumptions we have $B_{0,\alpha}\equiv 1$ and this is the reason
why $u_{\alpha}-\sum_{i=1}^{k}V_{i,\alpha}$ is estimated by
$o\big{(}\sum_{i=0}^{k}B_{i,\alpha}\big{)}$ in this case.

######  Remark 4.7.

When all the bubbles $V_{1},\dots,V_{k}$ appearing in the Struwe decomposition
of $u_{\alpha}$ are _non-degenerate_ solutions of (2.1) it is likely that the
estimates of Theorem 1.1 can be improved. Sharper estimates in dimensions
$n\geq 7$ for solutions of (1.1) looking like a tower of positive bubbles
(with or without alternating signs) were for instance obtained in Premoselli
[41] (sections $3$ and $4.2$). If one of the $V_{i}$, $1\leq i\leq k$, is
degenerate then the possible component along the kernel in (4.21) seems
difficult to control, and it is not clear whether one can expect better
estimates than in Theorem 1.1. No examples of _degenerate_ solutions of (2.1)
are known, but their existence has not been ruled out yet.

##  Appendix A Technical results

Throughout this appendix we use the notations of Section 3. We let
$(\mu_{i,\alpha})_{\alpha}$ and $(x_{i,\alpha})_{\alpha}$, $1\leq i\leq k$, be
sequences respectively of positive numbers and points of $M$ satisfying (1.4)
and we let $B_{i,\alpha}=B_{\mu_{i,\alpha},x_{i,\alpha}}$ be given by (2.11).

###  A.1. Some estimates

######  Proposition A.1.

Let $n\geq 3$. The following estimates hold true:

  * •

Let $i,j\in\\{1,\dots,k\\}$, $i\neq j$. Then for any sequence
$(x_{\alpha})_{\alpha}$ of points of $M$ we have

$\int_{M}d_{g}(x_{\alpha},\cdot)^{2-n}B_{i,\alpha}dv_{g}\lesssim\Theta_{i,\alpha}(x_{\alpha})B_{i,\alpha}(x_{\alpha})$
(A.1)

where $\Theta_{i,\alpha}$ is as in (3.3), and

$\int_{M}d_{g}(x_{\alpha},\cdot)^{1-n}B_{i,\alpha}dv_{g}\lesssim\Xi_{i,\alpha}(x_{\alpha})B_{i,\alpha}(x_{\alpha})$
(A.2)

where we have let, for $x\in M$,

$\Xi_{i,\alpha}(x)=\left\\{\begin{aligned}
&\theta_{i,\alpha}(x)\big{|}\ln\theta_{i,\alpha}(x)\big{|}&\textrm{ if }n=3\\\
&\theta_{i,\alpha}(x)&\textrm{ if }n\geq 4\end{aligned}\right.,$ (A.3)

and where $\theta_{i,\alpha}$ is as in (3.2).

  * •

Let $i,j\in\\{1,\dots,k\\}$, $i\neq j$. There exists a sequence
$(\sigma_{\alpha})_{\alpha}$ as in (3.7) such that for any sequence
$(x_{\alpha})_{\alpha}$ of points in $M$ we have

$\int_{M}d_{g}(x_{\alpha},\cdot)^{2-n}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}dv_{g}\lesssim\sigma_{\alpha}\big{(}B_{i,\alpha}(x_{\alpha})+B_{j,\alpha}(x_{\alpha})\big{)}$
(A.4)

and

$\int_{M}d_{g}(x_{\alpha},\cdot)^{1-n}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}dv_{g}\lesssim\sigma_{\alpha}\big{(}\mu_{i,\alpha}^{\frac{n-2}{2}}\theta_{i,\alpha}(x_{\alpha})^{1-n}+\mu_{j,\alpha}^{\frac{n-2}{2}}\theta_{j,\alpha}(x_{\alpha})^{1-n}\big{)}.$
(A.5)

  * •

Let $i,j\in\\{1,\dots,k\\}$, $i\neq j$. There exists a sequence
$(\sigma_{\alpha})_{\alpha}$ as in (3.7) such that

$\int_{M}\mu_{i,\alpha}^{\frac{n-2}{2}}\theta_{i,\alpha}^{1-n}\mu_{j,\alpha}^{\frac{n-2}{2}}\theta_{j,\alpha}^{1-n}dv_{g}\lesssim\sigma_{\alpha}.$
(A.6)

  * •

Let $(\delta_{\alpha})_{\alpha}$ be a sequence of positive numbers with
$\delta_{\alpha}\geq\mu_{i,\alpha}$. Then for any sequence
$(x_{\alpha})_{\alpha}$ of points of $M$ we have

$\int_{M\backslash
B_{g}(x_{i,\alpha},\delta_{\alpha})}d_{g}(x_{\alpha},\cdot)^{2-n}B_{i,\alpha}^{2^{*}-1}dv_{g}\lesssim\left(\frac{\mu_{i,\alpha}}{\delta_{\alpha}}\right)^{2}B_{i,\alpha}(x_{\alpha}).$
(A.7)

In particular, if $\delta_{\alpha}\geq r_{i,\alpha}$,

$\int_{M\backslash
B_{g}(x_{i,\alpha},\delta_{\alpha})}d_{g}(x_{\alpha},\cdot)^{2-n}B_{i,\alpha}^{2^{*}-1}dv_{g}\lesssim\sigma_{\alpha}B_{i,\alpha}(x_{\alpha}).$

In (A.1) – (A.7) the sequences $(\sigma_{\alpha})_{\alpha}$ actually only
depend on $(\mu_{i,\alpha})_{\alpha}$ and $(x_{i,\alpha})_{\alpha}$, $1\leq
i\leq k$. The notation “$\;\lesssim\;$” means “$\;\leq C\dots\;$”, where $C$
only depends on $n$ and $(M,g)$.

###### Proof.

These estimates follow from applications of Giraud’s lemma and from (1.4) and
are obtained by straightforward computations. They can be found for instance
in Hebey [25], Chapter $7$. ∎

###  A.2. Proof of (4.4)

The following lemma was used in the proof of Proposition 4.2:

######  Lemma A.2.

Let $0<\tau\leq 2^{*}-1$ be fixed. There exists a sequence
$(\sigma_{\alpha})_{\alpha}$ as in (3.7), but depending only on
$(\mu_{i,\alpha})_{\alpha}$ and $(x_{i,\alpha})_{\alpha}$, $1\leq i\leq k$,
such that for all $x\in M$

$\Big{(}\sum_{i=1}^{k}B_{i,\alpha}(x)\Big{)}^{2^{*}-1-\tau}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{\tau}\leq\sigma_{\alpha}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)+\sum_{i=1}^{k}B_{i,\alpha}(x)^{2^{*}-1}\Big{)}$

holds.

###### Proof.

Let $x\in M$ and let

$S_{\alpha,x}=\big{\\{}i\in\\{1,\dots k\\},x\in
B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})\big{\\}}.$

By (3.3) we have

$\sum_{i\in S_{\alpha,x}}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\leq
C\sqrt{\mu_{1,\alpha}}\sum_{i=1}^{k}B_{i,\alpha}(x)$

for some $C=C(n)>0$. Again by (3.3), we also have

$\sum_{i\in\\{1,\dots,k\\}\backslash
S_{\alpha,x}}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\leq C\sqrt{\mu_{1,\alpha}}.$

In the end, for all $x\in M$,

$\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\leq
C\sqrt{\mu_{1,\alpha}}\sum_{i=1}^{k}B_{i,\alpha}(x)+C\sqrt{\mu_{1,\alpha}},$

so that

$\displaystyle\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)\Big{)}^{2^{*}-1-\tau}$
$\displaystyle\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{\tau}$
(A.8) $\displaystyle\leq
C\mu_{1,\alpha}^{\frac{\tau}{2}}\sum_{i=0}^{k}B_{i,\alpha}(x)^{2^{*}-1}+C\mu_{1,\alpha}^{\frac{\tau}{2}}\sum_{i=0}^{k}B_{i,\alpha}(x)^{2^{*}-1-\tau}.$

Assume first that $0<\tau\leq 2^{*}-2$. Then $2^{*}-1-\tau\in[1,2^{*}-1[$ and
thus, for any $1\leq i\leq k$,

$\displaystyle B_{i,\alpha}(x)^{2^{*}-1-\tau}$ $\displaystyle\leq
C\left\\{\begin{aligned} &B_{i,\alpha}(x)^{2^{*}-1}&\textrm{ if
}d_{g}(x_{i,\alpha},x)\leq\sqrt{\mu_{i,\alpha}}\\\ &B_{i,\alpha}(x)&\textrm{
if }d_{g}(x_{i,\alpha},x)>\sqrt{\mu_{i,\alpha}}\end{aligned}\right.$
$\displaystyle\leq C\big{(}B_{i,\alpha}(x)^{2^{*}-1}+B_{i,\alpha}(x)\big{)}$

for some $C=C(n)>0$. Since either $B_{0,\alpha}\equiv 0$ or
$B_{0,\alpha}\equiv 1$, going back to (A.8) proves the lemma in this case with
$\sigma_{\alpha}=C\mu_{1,\alpha}^{\frac{\tau}{2}}$.

Assume now that $2^{*}-2<\tau\leq 2^{*}-1$. By (3.3) there exists $C=C(n)>0$
such that $\Theta_{i,\alpha}(x)\leq C$ for all $1\leq i\leq k$ and all $x\in
M$. We can then write that

$\displaystyle\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)\Big{)}^{2^{*}-1-\tau}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{\tau}$
$\displaystyle=\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)\Big{)}^{2^{*}-1-\tau}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{\tau-(2^{*}-2)}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{2^{*}-2}$
$\displaystyle\leq
C\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)\Big{)}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{2^{*}-2},$

and the proof follows from the $\tau=2^{*}-2$ case proven above. ∎

## References

  * [1] Antonio Ambrosetti and Andrea Malchiodi, _A multiplicity result for the Yamabe problem on $S^{n}$_, J. Funct. Anal. 168 (1999), no. 2, 529–561. MR 1719213
  * [2] by same author, _Perturbation methods and semilinear elliptic problems on ${\bf R}^{n}$_, Progress in Mathematics, vol. 240, Birkhäuser Verlag, Basel, 2006. MR 2186962 (2007k:35005)
  * [3] Thierry Aubin, _Nonlinear analysis on manifolds. Monge-Ampère equations_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982\. MR 681859 (85j:58002)
  * [4] A. Bahri, _Critical points at infinity in some variational problems_ , Pitman Research Notes in Mathematics Series, vol. 182, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1019828
  * [5] Gabriele Bianchi and Henrik Egnell, _A note on the Sobolev inequality_ , J. Funct. Anal. 100 (1991), no. 1, 18–24. MR 1124290 (92i:46033)
  * [6] Simon Brendle, _Blow-up phenomena for the Yamabe equation_ , J. Amer. Math. Soc. 21 (2008), no. 4, 951–979. MR 2425176 (2009m:53084)
  * [7] Simon Brendle and Fernando C. Marques, _Blow-up phenomena for the Yamabe equation. II_ , J. Differential Geom. 81 (2009), no. 2, 225–250. MR 2472174 (2010k:53050)
  * [8] 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. MR 982351 (90c:35075)
  * [9] Chiun-Chuan Chen and Chang-Shou Lin, _Estimate of the conformal scalar curvature equation via the method of moving planes. II_ , J. Differential Geom. 49 (1998), no. 1, 115–178. MR 1642113
  * [10] by same author, _Topological degree for a mean field equation on Riemann surfaces_ , Comm. Pure Appl. Math. 56 (2003), no. 12, 1667–1727. MR 2001443
  * [11] Wenyi Chen, Juncheng Wei, and Shusen Yan, _Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^{N}$ with critical growth_, J. Differential Equations 252 (2012), no. 3, 2425–2447. MR 2860624
  * [12] 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. MR 2825341 (2012k:35176)
  * [13] by same author, _Torus action on $S^{n}$ and sign-changing solutions for conformally invariant equations_, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 1, 209–237. MR 3088442
  * [14] Bin Deng, Liming Sun, and Juncheng Wei, _Optimal quantitative estimates of struwe’s decomposition_ , (2021), Preprint.
  * [15] Giuseppe Devillanova and Sergio Solimini, _Concentration estimates and multiple solutions to elliptic problems at critical growth_ , Adv. Differential Equations 7 (2002), no. 10, 1257–1280. MR 1919704
  * [16] Wei Yue Ding, _On a conformally invariant elliptic equation on ${\bf R}^{n}$_, Comm. Math. Phys. 107 (1986), no. 2, 331–335. MR 863646
  * [17] Olivier Druet, _From one bubble to several bubbles: the low-dimensional case_ , J. Differential Geom. 63 (2003), no. 3, 399–473. MR 2015469
  * [18] by same author, _Compactness for Yamabe metrics in low dimensions_ , Int. Math. Res. Not. (2004), no. 23, 1143–1191. MR 2041549 (2005b:53056)
  * [19] Olivier Druet and Emmanuel Hebey, _Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium_ , Anal. PDE 2 (2009), no. 3, 305–359. MR 2603801
  * [20] 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. MR 2063399 (2005g:53058)
  * [21] Olivier Druet and Pierre-Damien Thizy, _Multi-bump analysis for Trudinger-Moser nonlinearities. I. Quantification and location of concentration points_ , J. Eur. Math. Soc. (JEMS) 22 (2020), no. 12, 4025–4096. MR 4176786
  * [22] 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. MR 3495446
  * [23] Pierpaolo Esposito, Angela Pistoia, and Jérôme Vétois, _The effect of linear perturbations on the Yamabe problem_ , Math. Ann. 358 (2014), no. 1-2, 511–560. MR 3158007
  * [24] 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., To appear.
  * [25] Emmanuel Hebey, _Compactness and stability for nonlinear elliptic equations_ , Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. MR 3235821
  * [26] M. A. Khuri, F. C. Marques, and R. M. Schoen, _A compactness theorem for the Yamabe problem_ , J. Differential Geom. 81 (2009), no. 1, 143–196. MR 2477893 (2010e:53065)
  * [27] Yan Yan Li and Itai Shafrir, _Blow-up analysis for solutions of $-\Delta u=Ve^{u}$ in dimension two_, Indiana Univ. Math. J. 43 (1994), no. 4, 1255–1270. MR 1322618
  * [28] Yan Yan Li and Lei Zhang, _Compactness of solutions to the Yamabe problem. III_ , J. Funct. Anal. 245 (2007), no. 2, 438–474. MR 2309836 (2008f:53038)
  * [29] Yanyan Li, _Prescribing scalar curvature on $S^{n}$ and related problems. II. Existence and compactness_, Comm. Pure Appl. Math. 49 (1996), no. 6, 541–597. MR 1383201
  * [30] Yanyan Li and Meijun Zhu, _Yamabe type equations on three-dimensional Riemannian manifolds_ , Commun. Contemp. Math. 1 (1999), no. 1, 1–50. MR 1681811 (2000m:53051)
  * [31] 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. MR 2197144 (2006i:53046)
  * [32] 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. MR 4280834
  * [33] 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. MR 3926123
  * [34] Filippo Morabito, Angela Pistoia, and Giusi Vaira, _Towering phenomena for the Yamabe equation on symmetric manifolds_ , Potential Anal. 47 (2017), no. 1, 53–102. MR 3666799
  * [35] Monica Musso and Angela Pistoia, _Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent_ , Indiana Univ. Math. J. 51 (2002), no. 3, 541–579. MR 1911045
  * [36] Monica Musso and Juncheng Wei, _Nondegeneracy of nodal solutions to the critical Yamabe problem_ , Comm. Math. Phys. 340 (2015), no. 3, 1049–1107. MR 3406641
  * [37] Morio Obata, _The conjectures on conformal transformations of Riemannian manifolds_ , J. Differential Geometry 6 (1971/72), 247–258. MR 303464
  * [38] Angela Pistoia and Giusi Vaira, _Clustering phenomena for linear perturbation of the Yamabe equation_ , Partial differential equations arising from physics and geometry, London Math. Soc. Lecture Note Ser., vol. 450, Cambridge Univ. Press, Cambridge, 2019, pp. 311–331. MR 3930853
  * [39] Bruno Premoselli, _A pointwise finite-dimensional reduction method for a fully coupled system of Einstein-Lichnerowicz type_ , Commun. Contemp. Math. 20 (2018), no. 6, 1750076, 72. MR 3848069
  * [40] by same author, _A pointwise finite-dimensional reduction method for Einstein-Lichnerowicz-type systems: the six-dimensional case_ , Nonlinear Anal. 172 (2018), 200–215. MR 3790374
  * [41] by same author, _Towers of bubbles for yamabe-type equations and for the brézis-nirenberg problem in dimensions $n\geq 7$_, (2020), arXiv:2009.01515.
  * [42] Bruno Premoselli and Pierre-Damien Thizy, _Bubbling above the threshold of the scalar curvature in dimensions four and five_ , Calc. Var. Partial Differential Equations 57 (2018), no. 6, Paper No. 147, 39. MR 3858830
  * [43] 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. MR 3926106
  * [44] Olivier Rey, _The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent_ , J. Funct. Anal. 89 (1990), no. 1, 1–52. MR 1040954 (91b:35012)
  * [45] Olivier Rey and Juncheng Wei, _Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity_ , J. Eur. Math. Soc. (JEMS) 7 (2005), no. 4, 449–476. MR 2159223
  * [46] Frédéric Robert, _Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d’ordre deux_ , http://www.iecn.u-nancy.fr/ frobert/ConstrucGreen.pdf.
  * [47] Frédéric Robert and Jérôme Vétois, _Blowing-up solutions for second-order critical elliptic equations: the impact of the scalar curvature_ , Int. Math. Res. Not. IMRN, To appear.
  * [48] by same author, _Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non-locally conformally flat manifolds_ , J. Differential Geom. 98 (2014), no. 2, 349–356. MR 3263521
  * [49] by same author, _A general theorem for the construction of blowing-up solutions to some elliptic nonlinear equations with Lyapunov-Schmidt’s finite-dimensional reduction_ , Concentration Compactness and Profile Decomposition (Bangalore, 2011), Trends in Mathematics, Springer, Basel (2014), 85–116.
  * [50] Michael Struwe, _A global compactness result for elliptic boundary value problems involving limiting nonlinearities_ , Math. Z. 187 (1984), no. 4, 511–517. MR 760051
  * [51] Pierre-Damien Thizy, _The Lin-Ni conjecture in negative geometries_ , J. Differential Equations 260 (2016), no. 4, 3658–3690. MR 3434412
  * [52] Pierre-Damien Thizy and Jérôme Vétois, _Positive clusters for smooth perturbations of a critical elliptic equation in dimensions four and five_ , J. Funct. Anal. 275 (2018), no. 1, 170–195. MR 3799627
  * [53] Neil S. Trudinger, _Remarks concerning the conformal deformation of Riemannian structures on compact manifolds_ , Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274. MR 0240748
  * [54] Jérôme Vétois, _Multiple solutions for nonlinear elliptic equations on compact Riemannian manifolds_ , Internat. J. Math. 18 (2007), no. 9, 1071–1111. MR 2360648
  * [55] Jérôme Vétois and Shaodong Wang, _Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four_ , Adv. Nonlinear Anal. 8 (2019), no. 1, 715–724. MR 3918401
  * [56] Juncheng Wei, _Existence and stability of spikes for the Gierer-Meinhardt system_ , Handbook of differential equations: stationary partial differential equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 487–585. MR 2497911 (2011b:35214)
  * [57] Tobias Weth, _Energy bounds for entire nodal solutions of autonomous superlinear equations_ , Calc. Var. Partial Differential Equations 27 (2006), no. 4, 421–437. MR 2263672