# Automated Molecular Concept Generation and Labeling with Large Language
Models

Shichang Zhang1 Botao Xia111footnotemark: 1 Zimin Zhang111footnotemark: 1
Qianli Wu111footnotemark: 1 Fang Sun1 Ziniu Hu2 Yizhou Sun1 1University of
California, Los Angeles Equal contribution 2California Institute of Technology

###### Abstract

Artificial intelligence (AI) is significantly transforming scientific
research. Explainable AI methods, such as concept-based models (CMs), are
promising for driving new scientific discoveries because they make predictions
based on meaningful concepts and offer insights into the prediction process.
In molecular science, however, explainable CMs are not as common compared to
black-box models like Graph Neural Networks (GNNs), primarily due to their
requirement for predefined concepts and manual label for each instance, which
demand domain knowledge and can be labor-intensive. This paper introduces a
novel framework for Automated Molecular Concept (AutoMolCo) generation and
labeling. AutoMolCo leverages the knowledge in Large Language Models (LLMs) to
automatically generate predictive molecular concepts and label them for each
molecule. Such procedures are repeated through iterative interactions with
LLMs to refine concepts, enabling simple linear models on the refined concepts
to outperform GNNs and LLM in-context learning on several benchmarks. The
whole AutoMolCo framework is automated without any human knowledge inputs in
either concept generation, labeling, or refinement, thereby surpassing the
limitations of extant CMs while maintaining their explainability and allowing
easy intervention. Through systematic experiments on MoleculeNet and High-
Throughput Experimentation (HTE) datasets, we demonstrate that the AutoMolCo-
induced explainable CMs are beneficial and promising for molecular science
research.

##  1 Introduction

Artificial intelligence (AI) has been a driving force behind several
groundbreaking scientific discoveries, particularly in the domain of molecular
science. A prime example is the utilization of deep learning by the MIT Jameel
Clinic, leading to the identification of halicin – the first antibiotic
discovered in three decades that is effective against a broad spectrum of 35
bacteria [33]. In molecular science, deep learning models like Graph Neural
Networks (GNNs) have shown promising capabilities for learning complex atomic
structures and making accurate predictions of molecular properties [36].
However, a major challenge with such deep-learning-based models like GNNs is
their “black boxes” nature and lack of explainability [42]. Despite their high
predictive performance, black-box models fail to provide insights into the
underlying reasoning behind their predictions, making it difficult for
scientists to interpret and intervene in the model’s decision-making process,
which hinders scientific understanding and limits the potential for knowledge
discovery.

Figure 1: The prediction process of molecule properties is greatly illuminated
with AutoMolCo. First, concepts are generated and labeled with an LLM.
Afterward, a simple prediction model (e.g., linear regression) can be fitted
to achieve explainable predictions. LLM-relevant pieces are highlighted in
green, and specifically, green arrows indicate prompting.

In contrast, concept-based models (CMs) [20, 19, 38, 35] have emerged as a
promising explainable AI (XAI) solution for scientific problems, whose
explainability can provide valuable insights and increase the chance of new
scientific discoveries. Unlike other black-box models, CMs first predict
labels of human-interpretable concepts from the data and then use these
concept labels as features to predict the actual task label. For example, in
computer vision, CMs are applied for predicting the species of birds from
their images by identifying the relevant concepts like “wing color”, which
greatly increases the prediction explainability [19]. In molecular science,
CMs can be especially beneficial as they can break down the prediction of
complex molecular properties into more interpretable molecular concepts like
functional groups and molecular descriptors. For instance, Figure 1
demonstrates the prediction process of a CM for molecules A and B. The CM
first elucidates the relevant molecular descriptors as concepts, such as the #
of nitrogen atoms and the topological polar surface area (TPSA), which are
further employed to predict the molecule’s solubility. This process not only
yields a prediction but also provides a rationale that can be scrutinized and
built upon. Like molecule B is predicted to be soluble based on its high TPSA,
researchers can explore similar molecules with slight variations in these key
concepts to hypothesize and test solubility-enhancing modifications. In
contrast, black-box models, while potentially accurate, offer no such
explanatory power.

Despite being a promising XAI approach, CMs have not been widely applied to
molecular science problems, primarily due to their limitations in concept
generation and labeling. Extant CM approaches either require predefined
concepts and manual labels generated by domain experts [19] or can only
utilize simple and qualitative concepts that are insufficient for molecular
problems [27]. Collecting such concepts and their labels is feasible and
adequate for some computer vision tasks [19, 27]. However, the desired
concepts for molecules are sophisticated and nuanced, which require a depth of
chemical knowledge and accurate quantitative labels. For example, TPSA in
Figure 1 is a molecule descriptor relevant to the solubility prediction. It
provides insight into a molecule’s absorption and permeability
characteristics, representing a quantitative concept pivotal for understanding
the pharmacokinetics of potential drug compounds. Identifying and quantifying
such a concept often demands domain knowledge and computational methods that
extant CM approaches cannot achieve, thereby highlighting the challenge in
effective CMs for molecular sciences.

In response to the effectiveness of CMs and the challenges of applying them to
molecular science, we propose Automated Molecular Concept (AutoMolCo)
generation and labeling. AutoMolCo leverages Large Language Models (LLMs) to
generate molecular concepts that are predictive for the task and label these
concepts for each molecule instance. AutoMolCo also repeats these procedures
through iterative interactions with LLMs to refine concepts, enabling simple
linear models on the refined concepts to outperform GNNs and LLM in-context
learning (ICL) on several molecular benchmarks. The whole AutoMolCo framework
is automated and does not require human knowledge inputs in either concept
generation, labeling, or refinement, thus surpassing the limitations of extant
CMs.

The motivation of AutoMolCo is founded on the idea that LLMs can be treated as
extensive and integrated knowledge bases [30, 4], and their effectiveness for
solving molecular science problems has been benchmarked in [13] through ICL.
In our work, we exploit the potential of LLMs for XAI through CMs. For concept
generation, we prompt LLMs with the task description and ask LLMs to suggest
relevant concepts. For concept labeling, since it is particularly challenging,
we explore three different approaches: direct LLM prompting, function code
generation, and external tool calling. The concept generation and labeling
process form the foundation of our framework. Afterward, we fit simple
prediction models on the concepts to achieve an explainable CM. Moreover, we
also do an iterative refinement of the concepts by running feature selection
algorithms of the prediction model and prompting LLMs again with the feature
selection results. This allows LLMs to generate new concepts to replace the
less useful ones from the previous iteration, which ensures the CM remains up-
to-date with the most relevant concepts thus boosting its performance.

In this work, we first show that AutoMolCo can produce meaningful concepts and
accurate labels, which lead to CMs with simple prediction models to achieve
surprisingly good performance for molecular science problems. Then we perform
a systematic study via experiments on MoleculeNet [36] and High-Throughput
Experimentation (HTE) [1, 31] datasets to show the strengths and weaknesses of
AutoMolCoand answer five research questions. In summary, our contribution
includes:

  1. •

Automated framework: We propose AutoMolCo, which leverages LLMs for automated
concept generation and labeling, eliminating the need for human domain
knowledge and labor-intensive data collection, thereby streamlining the
development of CMs.

  2. •

Accuracy and explainability: AutoMolCo produces meaningful molecular concepts
that, when combined with simple prediction models for CMs, can achieve
superior or comparable accuracy to powerful black-box models while providing
greater explainability.

  3. •

LLM-driven XAI for science: Our work highlights the potential of LLMs in
addressing complex problems in molecular science, introduces a novel
perspective on CMs with LLMs, and paves the way for future research to further
exploit the capabilities of LLMs in molecular science and beyond.

##  2 Related work

Concept-based Models. A popular example of CMs is the Concept Bottleneck Model
(CBM) [19]. CBMs make predictions through an intermediate layer of human-
specified concepts, such as “wing color” in specious classification from bird
images. While CBMs offer more transparent predictions, they are often
constrained by the predefined concepts and label demands. Several follow-up
CBMs with predefined concepts targeting different tasks [10, 39, 7, 23, 8].
One particular follow-up that is more relevant to this work, label-free CBM
[27], tries to bypass the need for predefined concepts and labels by
generation and similarity matching. Unlike the other CBMs, this model
identifies concepts using GPT-3 for concept set creation and CLIP-Dissect for
matching text concepts with images. However, label-free CBM only focuses on
vision tasks, and the generated concepts are often simple and qualitative,
e.g., “yellow” is a concept, with similarity as the label, e.g., how similar a
“lemon” image is to the text “yellow”. Such concepts and labels are
insufficient for molecules, where a depth of chemical knowledge is required
for meaningful concepts and accurate qualitative labels.

Concept Learning in Graphs/Molecule. Several lines of research study concept
learning from graph data, particularly for molecules. One line identifies
graph motifs as concepts through counting or sampling [26, 34] and builds GNNs
on top of them [44, 41]. However, motif identification cannot be comprehensive
as it is NP-complete. Another line tries to use concept-based explanations for
GNNs with human-in-the-loop, e.g., GCExplainer [24]. Subsequent works have
refined this idea with k-means clustering and similarity scoring algorithms to
neuron-level grouping within activation layers [25, 37]. These methods
exemplify the attempt to extract and interpret salient features in graph data,
yet they often face challenges in fully capturing the nuanced complexity of
molecular structures.

LLMs for Molecular Science. GPT4Graph [12] prompts LLMs to explain the format
or to summarize a raw molecule graph input, where the graph is represented by
the Graph Modelling Language (GML) [16] or Graph Markup Language (GraphML)
[6]. Graph-ToolFormer [43] lets LLMs generate API calls to use external graph
reasoning tools, which can be applied to molecule function reasoning problems.
[13] studies solving molecular problems with LLMs ICL. We show our AutoMolCo
outperforms ICL and enjoys better explainability. Some other survey papers
discussing the potential of LLMs for molecular science include: [45] from a
scientific research perspective and [18] from an LLM for graph perspective,
and [40] for fine-tuning LLMs.

##  3 AutoMolCo: automated molecular concept generation and labeling

In this section, we describe AutoMolCo for concept generation, labeling, and
refinement, where the concepts are used to build an explainable CM. Figure 2
depicts the three major steps of AutoMolCo: 1) concept generation, 2) concept
labeling, and 3) CM fitting and concept selection.

Figure 2: The AutoMolCo framework. Step 1: concept generation. Step 2: concept
labeling with three different strategies. Step 3: fitting a prediction model
and perform concept selection. These three steps are repeated for multiple
iterations to achieve a refined list of meaningful concepts, where the
selected concepts in each iteration are feedback to the LLM through prompting.
LLM outputs are highlighted as green boxes.

Step 1: Concept Generation Given a particular task on molecules, e.g.,
predicting the hydration-free energy of small molecules in water, the first
step is to prompt LLMs to propose a diverse list of concepts that are
potentially relevant to the task. This step is analogous to a brainstorming
process. Concepts range from counting-based ones, like # nitrogen atoms, to
more complicated ones that require precise calculation, like TPSA. Without
LLMs, coming up with meaningful concepts requires domain experts. The
underlying intuition for concept generation is founded on the idea that LLMs
can be treated as extensive and integrated knowledge bases. Their capacity to
comprehend and output meaningful concepts is pivotal in this phase, yielding a
wide spectrum of potentially relevant concepts for our analysis. The prompt
for this step is shown in Figure 3 Step 1. The LLM-suggested concepts might be
less relevant initially, but they will be refined later.

Step 2: Concept Labeling Following the concept generation step, we then label
the generated concepts for each data instance. Compared to human labeling,
which requires domain knowledge and can be labor-intensive. Labeling with LLMs
is streamlined to a process of interaction with a single LLM interface, which
can be easily scaled and minimizes human error. This automation with LLMs is
crucial for efficiently processing large volumes of data encountered in
molecular studies. In this step, we consider three different labeling
strategies to enhance labeling quality.

Labeling Strategy 1: Direct LLM prompting We prompt LLMs directly to assign
each data instance numerical or categorical labels for the generated concepts
from Step 1. Similar to concept generation, this strategy builds up on the
idea that LLMs can be treated as integrated knowledge bases for retrieving
useful information. For each data instance, we provide LLMs with the molecule
names or SMILES strings. The prompt is shown in Figure 3 Step 2.

Labeling Strategy 2: Function code generation with LLMs Since LLMs are
particularly skilled in code generation, we explore a second approach for
concept labeling: generate functions in Python code for computing the concept
labels. The function generation approach has two advantages. Firstly, it
greatly reduces the need for repeated LLM API calls. Only a single API call is
required for each concept to obtain the function code, as opposed to making a
separate call for each data instance in the direct label prompting case.
Secondly, the generated functions can utilize preprocessed dataset features as
function arguments, such as atom types in terms of node features and molecule
structures in terms of adjacency matrix. These features provide more direct
information beyond molecule names or SMILES strings. Leveraging these
features, the LLM-generated functions can offer more nuanced and accurate
concept labels, enhancing the effectiveness of AutoMolCo. The prompt is shown
in Figure 7 in Appendix A.

Labeling Strategy 3: External tool calling with LLMs We also utilize LLMs to
call external tools like RDKit [21] for labeling, combining the generation
ability of LLMs with the reliability of specialized tools. This strategy
enjoys the same efficiency advantage as the function generation approach,
meaning it requires only a single API call of the LLM per concept to get the
API code for calling the tool. Moreover, the use of labeling tools ensures
that labels for all the tool-calculable concepts are accurate and reliable.
One disadvantage of this strategy is that not all generated concepts are
calculable by the external tool, in which case we can only turn to the first
two strategies. The prompt is shown in Figure 8 in Appendix A.

Figure 3: Prompts for concept generation and labeling on the FreeSolv dataset.
Hyperparameters, molecule instance information, and re-used LLM responses from
a previous step are in blue.

Step 3: CM Fitting and Concept Selection After getting the generated concepts
and their labels, we utilize them to fit prediction models for the molecular
task. Since the concept labels can be treated as tabular data, any model from
the off-the-shelf ones in Scikit-learn [29] to sophisticated deep learning
models can be applied. However, we found that explainable models like linear
models and decision trees or simple two-layer multi-layer perceptions (MLPs)
are often sufficient for achieving competitive performance. We attribute the
credit to the high-quality concepts and their labels. We will discuss our
model choice and perform a systematic study of different prediction models in
Section 4. While fitting the model, we also run feature selection methods like
Akaike Information Criterion (AIC) [2, 3] and Recursive Feature Elimination
(RFE) [15] to determine the useful concepts. Feature selection not only boosts
the model performance but also leads to automated iterative refinement for
identifying a list of the most useful concepts.

Iterative Concepts Refinement After all three steps. We do an iterative
refinement of the generated concepts by prompting LLMs again with the
empirical performance of our prediction model and the concept selection
results from Step 3. We include such information in an updated prompt to make
LLMs generate new concepts to replace the less useful ones from the previous
iteration. Using the empirical results as feedback, we ensure that our CM
remains adaptable and up-to-date with the most relevant molecular concepts.
Through this iterative refinement process, we guarantee that the model
performance improves over iterations and prune the irrelevant concepts
generated in previous iterations. The prompt for this step is shown in Figure
3.

##  4 Experiments

###  4.1 Experiment settings

Datasets  We include four datasets from MoleculeNet [36]. Two regression
datasets, e.g., FreeSolv and ESOL, and two classification datasets, e.g., BBBP
and BACE. FreeSolv provides hydration-free energy (which induces solubility)
for 642 molecules, while ESOL contains water solubility data for 1128 organic
small molecules. BBBP contains 2,039 molecules and assesses compounds’ blood-
brain barrier penetration. BACE has 1,513 molecules and predicts
$\beta$-secretase 1 inhibitors, relevant for Alzheimer’s research. For all
datasets, we employed the same scaffold splits as the Open Graph Benchmark
(OGB) [17]. We also include two HTE datasets, e.g., Buchwald-Hartwig (BH) [1]
and Suzuki-Miyaura (SM) [31] for chemical reaction yield prediction. BH
provides the yields of the Buchwald-Hartwig reaction for 3957 molecules, and
SM provides the yield of the Suzuki reaction for 5650 molecules. We employed
the same data splits as used in [13].

Metrics  We follow the standard evaluation metrics for these datasets. For
FreeSolv and ESOL, results are measured with Root Mean Square Error (RMSE).
For BBBP and BACE, we mainly evaluate these datasets using AUC-ROC and report
results in our main Table 1. Since [13] evaluates BBBP and BACE with accuracy,
we also report a comparison in accuracy in Table 7 in Appendix F. For BH and
SM, we evaluate with accuracy.

Baselines  Our baselines include GNNs, LLM ICL, and GNN + CBM. Specifically,
we use the GIN and GCN. For LLM ICL, we reference the findings from [13] and
use their prompts. For GNN + CBM, we use GIN and use GPT-3.5 Turbo for
generating concept labels for CBM. For HTE datasets, we only consider LLM ICL
baselines as graphs are not provided for the test set.

Models  We employ GPT-3.5 Turbo as our primary LLMs for generating concepts
and directly labeling. Additionally, we utilize GPT-4 for labeling strategy 2:
function code generation, and strategy 3: external tool calling. For strategy
3, the LLM will create code snippets for invoking RDKit [21]. We don’t use
GPT-4 for direct labeling due to the high cost of per-instance labeling. After
collecting the concept labels, we explore four types of prediction models to
cover a broad spectrum of tasks and performance levels. As a basic setting, we
use linear models like linear regression and logistic regression, we also
consider more advanced models including decision trees and 2-layer MLPs. We
use off-the-shelf prediction models from sklearn [29]. We do ablation on LLMs
with Claude-2 in Appendix E. Since we call LLMs through their APIs and the
prediction models are light and off-the-shelf, there is no specially
requirements, like GPUs, for our framework.

Concept Selection  We employ AIC [2, 3] for regression and RFE [14] for
classification. These selection methods are specifically applied to linear
models. For multi-iteration performance with decision trees and MLPs, we use
the selection results from the linear models.

###  4.2 AutoMolCo-induced CM performance

In Table 1, we compare the performance of the AutoMolCo-induced CM to
baselines. Compared to GNNs, our CM achieves better results on MoleculeNet
regression tasks and HTE tasks and competitive results on MoleculeNet
classification tasks. In comparison to the results presented by ICL, our
models have demonstrated a substantial performance advantage on all tasks. Our
best-performing model is the culmination of multiple iterations of refinement
and a combination of labeling strategies. Specifically, the results presented
in Table 1 are achieved using the following approaches: 1. A combination of
all three labeling strategies for concept labeling, with further details
provided in Appendix E.3 2. The optimal CMs from linear models, decision
trees, and MLPs 3. Concepts refinement over three iterations, as discussed in
Section 4.6. An in-depth exposition of these techniques is discussed in detail
in the RQs below through experiments on MoleculeNet datasets. More details on
experiment results for HTE datasets are shown in Appendix H.

Table 1: Performance comparison of the AutoMolCo-induced CM with baselines.
MoleculeNet regression tasks (FreeSolv and ESOL) are measured in RMSE
($\downarrow$). MoleculeNet classification tasks (BBBP and BACE) are measured
in AUC-ROC ($\uparrow$). HTE datasets (BH and SM) are measured in accuracy
($\uparrow$). Ours achieve better results on MoleculeNet regression and HTE
tasks and competitive results on MoleculeNet classification tasks.

| FreeSolv ($\downarrow$) | ESOL ($\downarrow$) | BBBP ($\uparrow$) | BACE ($\uparrow$) | BH ($\uparrow$) | SM ($\uparrow$)
---|---|---|---|---|---|---
GIN | 2.151 | 0.998 | 69.710 | 73.460 | - | -
GCN | 2.186 | 1.015 | 67.800 | 68.930 | - | -
GIN + CBM | 2.412 | 1.373 | 54.500 | 68.457 | - | -
GPT-3.5 Turbo (zero-shot) | 5.450 | 2.039 | 49.256 | 48.765 | 0.320 | 0.473
GPT-3.5 Turbo (4-shot) | 4.852 | 1.161 | 51.580 | 41.871 | 0.640 | 0.630
GPT-3.5 Turbo (8-shot) | 4.491 | 1.128 | 56.632 | 47.757 | 0.706 | 0.693
AutoMolCo-CM (ours) | 2.065 | 0.843 | 65.278 | 70.744 | 0.810 | 0.800

###  4.3 RQ1: Can AutoMolCo generate meaningful molecular concepts?

Figure 4: RQ1: Concepts selected by AutoMolCo in three refinement iterations
on FreeSolv. A detailed version in Appendix B Figure 9.

The effectiveness of the CM is built on meaningful concepts, which used to
rely on domain experts [19]. In this RQ1, we examine the concepts generated by
AutoMolCo in each refinement iteration and consult with domain experts for
their rationality. Figure 4 shows the concepts selected by a linear regression
model for predicting solubility (FreeSolv), evolving from an initial broad set
to a more focused and chemically relevant set. According to our consultation
with domain experts, the eliminated concepts such as the counts of specific
atoms and rotatable bonds, while informative, may not have significantly
contributed to the model’s predictive power or may have been correlated with
other concepts. The retained concepts, like molecular weight, # hydrogen bond
donors, and TPSA, are fundamental properties that influence molecular
interaction and behavior in a solvent. For instance, hydrogen bond donors
directly relate to a molecule’s ability to engage in hydrogen bonding, a key
interaction for solubility. The TPSA measures the molecule’s surface that can
engage in polar interactions, which is crucial for solubility in polar
solvents [28]. Thus, the final selected concepts are indeed more chemically
meaningful for the task and align with domain knowledge.

###  4.4 RQ2: Can AutoMolCo assign molecules reasonable concept labels using
each strategy?

Accurate concept labels are another critical component of CM performance. In
this RQ2, we evaluate AutoMolCo labeling results. we collect ground truth
labels for concepts where labels are available, either through calculation
(e.g., for molecular weights), or manual lookup (e.g., for melting points). We
evaluate labels produced by our direct prompting strategy and function
generation labeling strategy using the Pearson correlation coefficient ($r$)
with the ground truth, due to the scale-invariant nature of the metric. The
external tool calling strategy is excluded from this evaluation as tools will
always provide correct labels, and the downside of this strategy is that not
all the concepts are tool-calculable (e.g., melting points). Results in Table
2 show strong correlations can be achieved on most datasets with AutoMolCo
labeling. Nonetheless, variations in correlation underscore the potential for
method improvement. In addition to this benchmarking effort, we also discuss
several challenges we encountered and overcame in the labeling step, including
imputation for missing values, dictionary for unit inconsistency, and Chain-
of-Thoughts (CoT) prompts for syntax errors in function code. A more detailed
discussion of these studies are in Appendix C.

Table 2: RQ2: Percentage of concepts with a high correlation ($r$ score $\geq$ 0.7) with the ground-truth. Labeling Strategy | LLM |  Molecule Format | FreeSolv | ESOL | BBBP | BACE
---|---|---|---|---|---|---
Str-1 Direct Prompt | GPT-3.5 | Name | 0.82 | 0.63 | 0.06 | -
Str-1 Direct Prompt | GPT-3.5 | SMILES | 0.82 | 0.75 | 0.69 | 0.22
Str-2 Function | GPT-4 | - | 1.00 | 0.79 | 0.69 | 0.67

###  4.5 RQ3: Can AutoMolCo produced concepts and labels be utilized to build
an effective CM?

In RQ1, we have verified that the generated concepts are meaningful according
to domain experts. In RQ2, we have shown that concept labels are relatively
accurately assigned after properly handling potential issues like missing
labels and unit inconsistency. In this RQ3, we compare the performance of the
AutoMolCo-induced CMs when different predictions models and labeling
strategies are adopted. Results in Table 3 show that AutoMolCo can give
reasonable performance even with the most basic direct prompting labeling
strategy and the simplest linear model. The good performance of different
prediction models demonstrates the quality of the concepts and the
effectiveness of AutoMolCo.

Table 3: RQ3: Model performance with different labeling strategies and prediction models. Labeling Strategy | Prediction Model | FreeSolv($\downarrow$) | ESOL ($\downarrow$) | BBBP ($\uparrow$) | BACE ($\uparrow$)
---|---|---|---|---|---
Str-1 Direct Prompt | Linear/Logistic | 2.685 | 1.250 | 52.836 | 56.894
Str-1 Direct Prompt | Decision Tree | 2.791 | 1.272 | 56.887 | 68.632
Str-1 Direct Prompt | MLP | 2.338 | 1.194 | 51.794 | 60.059
Str-2 Function | Linear/Logistic | 3.284 | 1.254 | 55.671 | 56.624
Str-2 Function | Decision Tree | 2.569 | 1.238 | 54.167 | 55.573
Str-2 Function | MLP | 2.805 | 1.034 | 58.738 | 56.894
Str-3 Tool | Linear/Logistic | 3.142 | 1.011 | 57.350 | 63.154
Str-3 Tool | Decision Tree | 3.750 | 1.027 | 55.903 | 65.658
Str-3 Tool | MLP | 1.981 | 0.911 | 58.449 | 60.772

###  4.6 RQ4: Does iterative refinement boost the performance of AutoMolCo-
induced CM?

As one of the most important designs of our AutoMolCo framework, concept
refinement helps to identify meaningful important concepts through iterative
interactions with LLMs. The concept relevance has been shown to improve in
RQ1, but that does not necessarily mean CM performance will also improve. In
this RQ4, we run AutoMolCo with three iterative concept refinements on the
MoleculeNet datasets with linear prediction models. We show the results in
Figure 5, and we observe that the CM prediction performance indeed improved
through concept refinement, especially for classification tasks. The
improvement for regression tasks is marginal, partially because the
performance is already good for regression.

Figure 5: RQ4: Iterative refinement improves CM performance for classification
tasks.

###  4.7 RQ5: Does the AutoMolCo-induced CM facilitates explainable molecular
science?

One of the key advantages of CMs over black-box models is their
explainability. In this section, we evaluate this aspect of AutoMolCo-induced
CMs through three experiments on all three types of the prediction models:
coefficient interpretation of linear models, split interpretation of decision
trees, and concept label intervention of MLPs.

Coefficient Interpretation of Linear Models Using linears model in the
AutoMolCo-induced CM offers excellent explainability through direct
interpretation of the model coefficients. We plot the coefficients of the
linear model on FreeSolv for predicting hydration free energy in Figure 6 (a),
highlighting three significant concepts: # hydrogen bond donors, TPSA, and #
rotatable bonds. According to domain experts, the # hydrogen bond donors
relates to a molecule’s ability in hydrogen bonding, reflecting its potential
to interact with solvents and other molecules. Therefore, an its increment
typically leads to a more favorable (more negative) hydration free energy [9].
TPSA quantifies the surface area of a molecule that can engage in polar
interactions, providing insights into a molecule’s permeability
characteristics. Thus, higher TPSA also leads to more favorable (more
negative) hydration free energy [28]. Conversely, the # rotatable bonds
positively correlated with hydration free energy. More rotatable bonds
increase molecular flexibility, allowing the molecule to adopt conformations
that enhance interactions with water molecules. This increased flexibility can
lead to less favorable hydration free energy (less negative), as it reduces
the stability of the solvation shell around the molecule [11]. Our linear
model interpretation aligns with domain knowledge without requiring any human
knowledge input into the model. We show results on BBBP in Appendix D.

Figure 6: RQ5: Coefficients of the linear regression model and the decision
tree on FreeSolv. Both with concepts refined by AutoMolCo after three
iterations

Splits Interpretation of Decision Trees Complementing to the coefficients of
the linear model, decision tree enhances the understanding of model’s decision
process. In Figure 6 (b), we show the 3-layer decision tree for BBBP dataset.
In the first two layers, the model uses TPSA to categorize the molecules into
four categories, where molecules with TPSA less than 26.99 are likely to
penetrate the BBB while molecules with TPSA greater than 176.22 are rarely
penetrative. The decision tree further differentiates molecules with TPSA
between 26.99 and 107.1 by whether or not it contains a hydrogen bond in its
ring structure, where molecules without this property are more likely to
penetrate the BBB. On the other hand, the model splits molecules with TPSA
between 107.1 and 176.22 using the number of carbon atoms, illustrating that
molecules containing more than 23 carbon atoms are very likely to be
penetrative. Figure 13 shows the impurity details of the decision tree.

Concept Label Intervention Besides analyzing interpretable prediction models
like linear models and decision trees, we also conduct a case study of concept
label interventions with MLPs. Our goal is to identify molecules with similar
concept labels except for the one we intend to intervene on (e.g., similar
molecular weights, # aromatic rings, etc., except logP) but different task
labels (e.g., soluble vs. insoluble). Two examples we identify from the ESOL
dataset are: Diphenylamine (N(c1ccccc1)c2ccccc2) and RTI 17
(CCN2c1ccccc1N(C)C(=S)c3cccnc23). After three iterations of refinement these
two molecules have the same labels for three out of the four remaining
concepts, except for their logP labels, which differ by 0.275 (standardized to
have mean 0 and standard deviation 1). Diphenylamine is predicted to be
insoluble (-3.648), whereas RTI 17 is predicted to be soluble (-4.079), based
on a conventional solubility threshold of -4 [32]. These predictions proved to
be quite accurate, with the ground truth solubility of these two molecules
being -3.857 and -4.227, respectively. By intervening on diphenylamine’s logP
value (0.209) to match RTI 17’s logP value (0.484) through interpolation, we
observe a linear change in solubility. This study highlights the significant
impact of logP on solubility predictions, which is consistent with expert
conclusions [22, 5], providing insights beyond traditional black-box models.
We show the intervention plot in 11 in Appendix D.

###  4.8 Ablation studies and limitations

We conduct ablation studies of AutoMolCo. We found that AutoMolCo can perform
consistently with different LLMs and is robust to molecule input formats.
Also, properly combining the labeling strategies can enhance model
performance. These results are in Appendix E.

Our framework also has limitations that can inspire future work. One
limitation is that LLMs can sometimes hallucinate, generating inaccurate
concepts and labels. This affects the accuracy and reliability of the outputs.
Using more powerful LLMs could improve this issue. Another limitation is that
the evaluation of the generated concepts and labels often requires validation
by human experts, introducing subjectivity and dependency on domain knowledge.
Developing automated evaluation methods is another potential direction for
improvement.

##  5 Conclusion

We propose the AutoMolCo framework that automates the generation and labeling
of molecular concepts, overcoming challenges of existing CMs and enhancing
explainability through iterative refinement of useful concepts. We demonstrate
that, for molecular property prediction tasks, simple linear prediction model
on our generated concepts can perform competitively or even better than GNNs
and LLM ICL. Our work paves the way for future research to further exploit the
capabilities of LLMs for XAI in molecular science and beyond.

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## Appendices

##  Appendix A Example prompts

We show an example prompt for generating the labeling functions in Python code
in Figure 7 and an example prompt for generating code snippet to call external
tools in Figure 8.

Figure 7: Prompts for generating concept labeling functions in Python code for
the FreeSolv dataset. Figure 8: Prompts for calling the external tool RDKit to
label concepts on the FreeSolv dataset.

##  Appendix B RQ1 supplement: the full version of concept refinement

In Figure 9, we show the full list of concepts selected by AutoMolCoon
FreeSolv in three refinement iterations. The result corresponds to the RQ1.

Figure 9: Concepts selected by AutoMolCo in three refinement iterations on
FreeSolv. Full version.

##  Appendix C RQ2 supplement: challenges and solutions for concept labeling

#### Direct LLM prompting

For labeling with direct LLM prompting, we encountered two key issues: missing
labels and unit inconsistency.

Missing labels One issue we found for concept labeling with direct LLM
prompting is that it is challenging to have LLMs generate some concept labels
for certain molecules. For instance, LLMs identified acid dissociation
constant (pKa) as a crucial concept for predicting water solubility. However,
pKa is a quantity only apply to acids, and thus the model will output
“Unknown” for the label. For the ESOL dataset, this results in a $13.03\%$
missing rate for this concept. The missing-label issue underscores AutoMolCo’s
limitation in recognizing concept applicability across molecules. To mitigate
this, we apply various imputation methods, including mean value imputation and
domain-knowledge-driven imputation. For the latter, we set missing pKa labels
to $100$—significantly above water’s pKa of $14$—to denote weak or non-
acidity, which enhances the CM’s performance.

Unit inconsistency We observe that for concepts with multiple possible units,
labels generated by LLMs for these concepts may exhibit inconsistent units
across different molecules. Take our experiments on the ESOL dataset as an
example. The LLM suggest melting point as the relevant concept for predicting
water solubility. When we call LLM API to generate values for melting point,
it randomly chooses from Celsius (${}^{\circ}C$), Fahrenheit (F), or Kelvin
(K) as melting point’s unit, which resulted in inconsistent scales for melting
point’s labels. Similar issues also show up in other concepts, e.g. molecular
Volume and molecular surface area. The inconsistency is due to the randomness
in LLM context generation as we plug in different data instances into the
prompt template. We first tried to fix the problem by specifying with LLM what
unit should be used for each concept in the prompt, it was not every
effective. Therefore, we added an intermediate step between Step 1 and Step 2,
which lets LLM generate a concept-to-unit dictionary for each concept it
proposed in Step 1. Then, we merged the resulting dictionary into Step 2’s
prompt, so LLM can generate labels with consistent units based on the unit
specified in the dictionary.

#### Function code generation

When generating labeling functions in Python code, we find it is non-trivial
to prompt LLMs for executable functions with no errors. We made two efforts to
increase the likelihood of producing executable functions with LLMs. We first
perform prompt engineering to clearly specify atom types, adjacency matrices,
and node and edge features, which enhances the function quality. Through
careful prompt engineering, most generated functions for simpler concepts
become executable. However, functions for labeling complex concepts like
“number of rings” are still unlikely to be error-free due to their intricate
nature. We thus adopt a chain of thought (CoT) approach to generate functions.
For the CoT prompt, we first ask the LLM to describe the function in natural
language, which can best leverage the LLM’s strength in generating natural
language. Then, the CoT prompt asks the LLM to turn the natural language
description of the function into Python code, which we found increases the
likelihood of generating accurate and executable functions. An example of the
CoT function-generation prompt is shown in Figure 7.

#### External tool calling

Given there are external tools for molecular science with API access, we
prompt LLMs to generate code snippets for calling the tool API. We observe
that LLMs are adept at obtaining callable APIs for a majority of our generated
concepts from step 1, which we successfully employed to calculate the concept
labels for each molecule in our dataset. The example prompts and generated API
calls can be found in Figure 8. The drawback of this strategy is that the
external tool cannot cover all the concepts generated by the LLM, especially
for those measured concepts like melting point. For these cases, we turn to
the first two strategies for labeling.

##  Appendix D RQ5 supplement: BBBP linear model coefficients and ESOL
intervention visualization

Coefficient Interpretation of Linear Models Additional to the results in
Section 4.7, we evaluating the linear model on the BBBP dataset. We focused on
the top positive coefficient lipophilicity (logP) and the top negative
coefficient hydrogen bond acceptors shown in Figure 10. Notably, logP has a
coefficient of 1.97 and hydrogen bond acceptors has a coefficient -4.36. These
finding aligns with domain knowledge, as higher logP enhances a molecule’s
ability to cross lipid-rich biological membranes. Conversely, a lower number
of hydrogen bond acceptors generally enhances a molecule’s permeability
through the BBB. These findings validate the CM’s alignment with established
biochemical principles, demonstrating its potential utility in predictive
modeling for molecular properties.

Figure 10: RQ5: Coefficients of the logistic regression model on BBBP with
concepts refined by AutoMolCo after three iterations

Concept Label Intervention Complementing to the intervention study in Section
4.7, we plot the results in Figure 11.

Figure 11: Intervention on logP of diphenylamine for predicting solubility
with MLP

##  Appendix E Ablation Studies

###  E.1 Different LLMs

We study the performance of AutoMolCo with different LLMs. Table 4 compares
the performance of GPT-3.5 Turbo and Claude-2 using the direct LLM prompting
labeling strategy with linear prediction models. While both GPT-3.5 Turbo and
Claude-2 exhibit slightly inferior performance compared to GNNs across four
datasets, they maintain competitive results, emphasizing simplicity and
interpretability. Specifically, Claude-2 underperforms GPT-3.5 Turbo after
first iteration, potentially due to its less consistent and accurate response.
This inconsistency, partly attributed to more frequent issues with missing
values and unit inconsistencies observed in Claude-2, suggests GPT-3.5 Turbo’s
superior ability to generate reliable ground truth knowledge. Additionally,
GPT-3.5 Turbo’s better prompt comprehension and domain knowledge in chemistry
might contribute to its enhanced performance in predicting target concepts.

Table 4: Ablation on LLMs (GPT vs. Claude-2). LLM | FreeSolv($\downarrow$) | ESOL ($\downarrow$) | BBBP ($\uparrow$) | BACE ($\uparrow$)
---|---|---|---|---
GPT-3.5 | 2.685 | 1.250 | 52.84 | 56.89
Claude-2 | 2.804 | 1.327 | 52.78 | 56.11

###  E.2 Direct LLM prompting with molecule names vs. with SMILES strings

Building on insights from [13] regarding LLMs’ challenges with long molecular
representations, we examined LLM’s capability in labeling concepts using
SMILES strings and molecule names across our datasets. Our findings indicate
that LLMs perform reasonably well in identifying basic concepts like molecular
weight and atom counts using either molecule names or SMILES strings, with a
strong correlation to ground truth labels ($r>0.9$). However, LLMs struggle
with complex concepts requiring detailed structural knowledge, such as the
number of chiral centers. Moreover, our analysis reveals a notable decline in
LLM’s performance with molecule names in the larger datasets like BBBP,
suggesting LLM’s familiarity with common molecular names improves its
performance on smaller datasets, but this advantage diminishes with less
familiar names in larger datasets. In contrast, the structural specificity of
SMILES strings maintains more consistent performance across dataset sizes,
highlighting their utility in representing unique molecular concepts.
Furthermore, we compared the performance difference between the two
representations over 3 iterations. As demonstrated in Table 5, the model
performance using SMILES strings matched the model performance using molecule
names on most datasets, but a notable improvement in performance with SMILES
strings is observed on the BBBP dataset.

Table 5: Ablation on input formats (SMILES vs. molecule names). | FreeSolv ($\downarrow$) | ESOL ($\downarrow$) | BBBP ($\uparrow$) | BACE ($\uparrow$)
---|---|---|---|---
Input Format | SMILES | Names | SMILES | Names | SMILES | Names | SMILES
GPT-3.5 iter 1 | 2.854 | 2.685 | 1.401 | 1.250 | 53.88 | 52.84 | 56.89
GPT-3.5 iter 2 | 2.662 | 2.520 | 1.262 | 1.250 | 56.08 | 54.05 | 57.60
GPT-3.5 iter 3 | 2.763 | 2.520 | 1.262 | 1.255 | 60.41 | 56.08 | 58.38

We also present a comparative analysis of the quality of concept labels
generated using molecular names vs. SMILES strings. The comparison is
visualized through a series of heatmaps, as illustrated in Figure 12.

Figure 12: Correlation ($r$) between the ground truth labels and concept
labels generated using molecule names or SMILES strings. Red indicates a
higher correlation.

###  E.3 Combine different labeling strategies

The AutoMolCo framework includes three labeling strategies and allows easy
extension to new ones. We consider combinations of the labeling strategies and
study their impact on model performance, where we adopt a simple priority
heuristic where strategy 3 $>$ strategy 2 = strategy 1. Specifically, whenever
the external tool is available for a suggested concept, we get the accurate
concept labels from calling it. Otherwise, two concept labels are derived from
both direct prompting and function code generation, and they are both
considered in step 3 for the selection. This combined-strategy labeling turns
out to outperform most of the standalone strategies as shown in Table 6. These
findings demonstrate that the three labeling strategies have their own
strengths and weaknesses for different concepts, and they can be complement to
each other to maximize the model performance. We leave exploration of new
strategies and more sophisticated strategy combinations as future work.

Table 6: Combine labeling strategies.

Labeling Strategy | Model | FreeSolv($\downarrow$) | ESOL ($\downarrow$) | BBBP ($\uparrow$) | BACE ($\uparrow$)
---|---|---|---|---|---
Direct Prompt | Linear/Logistic | 2.685 | 1.250 | 52.836 | 56.894
Direct Prompt | Tree Model | 2.791 | 1.272 | 56.887 | 68.632
Direct Prompt | MLP | 2.338 | 1.194 | 51.794 | 60.059
Direct Prompt + Function | Linear/Logistic | 2.697 | 1.254 | 56.134 | 56.712
Direct Prompt + Function | Tree Model | 2.540 | 1.364 | 55.150 | 59.998
Direct Prompt + Function | MLP | 2.211 | 0.971 | 57.697 | 65.797
Direct Prompt + Function + Tool | Linear/Logistic | 3.002 | 1.136 | 55.845 | 64.032
Direct Prompt + Function + Tool | Tree Model | 3.752 | 1.107 | 56.250 | 65.658
Direct Prompt + Function + Tool | MLP | 2.122 | 0.791 | 58.391 | 62.624

##  Appendix F Accuracy comparison with LLM ICL on BBBP and BACE

In our experiments, we follow the standard and widely-used evaluation metrics
for all datasets. For classification tasks on BBBP and BACE, we mainly
evaluate these datasets using the AUC-ROC metric and report results in our
main Table 1. Since [13] provides the ICL prompts but evaluates BBBP and BACE
with accuracy, we also report a comparison in accuracy in Table 7 for a fair
comparison.

Table 7:  Performance comparison of the AutoMolCo-induced CM vs. LLM ICL (results are taken from [13]). Results are evaluated with Accuracy ($\uparrow$). | BBBP ($\uparrow$) | BACE ($\uparrow$)
---|---|---
GPT-4 (zero-shot) | 0.476 | 0.499
GPT-4 (Scaffold, k= 8) | 0.614 | 0.679
GPT-3.5 (Scaffold, k= 8) | 0.463 | 0.496
Ours | 0.657 | 0.704

##  Appendix G Decision trees visualization

As discussed in 4.7, we visualize the decision tree which makes the prediction
process explainable. Figure 13 shows the impurity details of the decision tree
shown in Figure 6 (b) and Figure 14 shows a sample decision tree for the BACE
dataset.

Figure 13: The decision trees for classification on BBBP. Figure 14: The
decision trees for classification on BACE.

##  Appendix H More results on Buchwald-Hartwig and Suzuki-Miyaura

The GPT ICL performance from [13] are measured on 100 data samples. To compare
AutoMolCo’s performance with their numbers, for each dataset we picked the
best model performance from the logistic regression models or MLP models
trained on either 200 or 500 sampled training data. The performance details
are presented in Table 8 with best performance reported in Table 1.

Table 8:  Performance comparison of the best AutoMolCo-induced CM vs. LLM ICL in accuracy ($\uparrow$) (GPT results are taken from [13]). | BH ($\uparrow$) | SM ($\uparrow$)
---|---|---
GPT-4 (random, k = 8) | 0.800 | 0.764
GPT-3.5 + logistic (200 samples) | 0.800 | 0.770
GPT-3.5 + MLP (200 samples) | 0.800 | 0.780
GPT-3.5 + logistic (500 samples) | 0.790 | 0.780
GPT-3.5 + MLP (500 samples) | 0.810 | 0.800


# A Trudinger-Moser inequality for conical metric in the unit ball

Yunyan Yang<EMAIL_ADDRESS>Xiaobao Zhu<EMAIL_ADDRESS>School
of Mathematics, Renmin University of China, Beijing 100872, P. R. China

###### Abstract

In this note, we prove a Trudinger-Moser inequality for conical metric in the
unit ball. Precisely, let $\mathbb{B}$ be the unit ball in $\mathbb{R}^{N}$
$(N\geq 2)$, $p>1$, $g=|x|^{\frac{2p}{N}\beta}(dx_{1}^{2}+\cdots+dx_{N}^{2})$
be a conical metric on $\mathbb{B}$, and
$\lambda_{p}(\mathbb{B})=\inf\left\\{\int_{\mathbb{B}}|\nabla u|^{N}dx:u\in
W_{0}^{1,N}(\mathbb{B}),\,\int_{\mathbb{B}}|u|^{p}dx=1\right\\}$. We prove
that for any $\beta\geq 0$ and
$\alpha<(1+\frac{p}{N}\beta)^{N-1+\frac{N}{p}}\lambda_{p}(\mathbb{B})$, there
exists a constant $C$ such that for all radially symmetric functions $u\in
W_{0}^{1,N}(\mathbb{B})$ with $\int_{\mathbb{B}}|\nabla
u|^{N}dx-\alpha(\int_{\mathbb{B}}|u|^{p}|x|^{p\beta}dx)^{N/p}\leq 1$, there
holds

$\int_{\mathbb{B}}e^{\alpha_{N}(1+\frac{p}{N}\beta)|u|^{\frac{N}{N-1}}}|x|^{p\beta}dx\leq
C,$

where $|x|^{p\beta}dx=dv_{g}$, $\alpha_{N}=N\omega_{N-1}^{1/(N-1)}$,
$\omega_{N-1}$ is the area of the unit sphere in $\mathbb{R}^{N}$; moreover,
extremal functions for such inequalities exist. The case $p=N$, $-1<\beta<0$
and $\alpha=0$ was considered by Adimurthi-Sandeep [1], while the case
$p=N=2$, $\beta\geq 0$ and $\alpha=0$ was studied by de Figueiredo-do Ó-dos
Santos [8].

###### keywords:

Trudinger-Moser inequality , blow-up analysis , conical metric

###### MSC:

[2010] 35J15; 46E35.

††journal: ***

##  1 Introduction

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ $(N\geq 2)$,
$W_{0}^{1,N}(\Omega)$ be the completion of $C_{0}^{\infty}(\Omega)$ under the
Sobolev norm

$\|u\|_{W_{0}^{1,N}(\Omega)}=\left(\int_{\Omega}|\nabla
u|^{N}dx\right)^{1/N},$

where $\nabla$ denotes the gradient operator. Write
$\alpha_{N}=N\omega_{N-1}^{1/(N-1)}$, where $\omega_{N-1}$ stands for the area
of the unit sphere in $\mathbb{R}^{N}$. Then the classical Trudinger-Moser
inequality [39, 29, 28, 33, 24] says

$\sup_{u\in W_{0}^{1,N}(\Omega),\,\|u\|_{W_{0}^{1,N}(\Omega)}\leq
1}\int_{\Omega}e^{\alpha|u|^{\frac{N}{N-1}}}dx<\infty,\quad\forall\alpha\leq\alpha_{N}.$
(1)

This inequality is sharp in the sense that if $\alpha>\alpha_{N}$, all
integrals in (1) are still finite, but the supremum is infinite. While the
existence of extremal functions for it was solved by Carleson-Chang [5],
Flucher [12] and Lin [21].

Using a symmetrization argument and a change of variables, Adimurthi-Sandeep
[1] generalized (1) to the following singular version:

$\sup_{u\in W_{0}^{1,N}(\Omega),\,\|u\|_{W_{0}^{1,N}(\Omega)}\leq
1}\int_{\Omega}\frac{e^{\alpha_{N}\gamma|u|^{\frac{N}{N-1}}}}{|x|^{N\beta}}dx<\infty,\quad\forall\,\,0\leq\beta<1,\,\,\,0<\gamma\leq
1-\beta.$ (2)

Also, it is sharp in the sense that if $\gamma>1-\beta$, integrals are still
finite, but the above supremum is infinite. The inequality (2) was extended to
the whole Euclidean space $\mathbb{R}^{N}$ by Adimurthi-Yang [2]. The
existence of extremal functions for (2) in the case $N=2$ was due to Casto-Roy
[6], Yang-Zhu [38] and Iula-Mancini [13]. An interesting question is whether
or not (2) still holds for $\beta<0$. Generally, the answer is negative. To
see this, we choose $x_{0}\not=0$, $r_{0}>0$ such that
$B_{2r_{0}}(x_{0})\subset\Omega\setminus\\{0\\}$. For any $0<\epsilon<r_{0}$,
we write the Moser function

$m_{\epsilon}(x)=\left\\{\begin{array}[]{lll}\frac{1}{\omega_{N-1}^{1/N}}\left(\log\frac{r_{0}}{\epsilon}\right)^{\frac{N-1}{N}},&{\rm
when}&|x-x_{0}|\leq\epsilon\\\\[6.45831pt]
\frac{1}{\omega_{N-1}^{1/N}}\frac{\log\frac{r_{0}}{|x-x_{0}|}}{\left(\log\frac{r_{0}}{\epsilon}\right)^{1/N}},&{\rm
when}&\epsilon<|x-x_{0}|\leq r_{0}\\\\[6.45831pt] 0,&{\rm
when}&|x-x_{0}|>r_{0}.\end{array}\right.$

An easy computation shows $\|m_{\epsilon}\|_{W_{0}^{1,N}(\Omega)}=1$. Since
$\beta<0$, we have

$\int_{\Omega}e^{\alpha_{N}(1-\beta)m_{\epsilon}^{\frac{N}{N-1}}}|x|^{-N\beta}dx\geq\int_{B_{\epsilon}(x_{0})}e^{\alpha_{N}(1-\beta)m_{\epsilon}^{\frac{N}{N-1}}}|x|^{-N\beta}dx\rightarrow\infty\quad{\rm
as}\quad\epsilon\rightarrow 0.$

Even worse, the above estimate still holds if $\alpha_{N}(1-\beta)$ is
replaced by any $\alpha>\alpha_{N}$. In conclusion, the singular Trudinger-
Moser inequality (2) does not hold for $\beta<0$.

Let us consider the unit ball $\mathbb{B}\subset\mathbb{R}^{N}$, which is
centered at the origin. Let $\mathscr{S}$ be a set of all radially symmetric
functions. With a slight abuse of notations, we say that $u$ is radially
symmetric if $u(x)=u(|x|)$ for almost every $x\in\mathbb{B}$. It was proved by
Ni [27] that $W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S}$ can be imbedded in
$L^{p}(\mathbb{B},|x|^{\alpha})$ with $\alpha>0$ and $p=2(N+\alpha)/(N-2)$
greater than $2^{\ast}=2N/(N-2)$. Motivated by results of Bonheure-Serra-
Tarallo [3], Calanchi-Terraneo [4] and de Figueiredo-dos Santos-Miyagaki [7],
de Figueiredo-do Ó-dos Santos [8] observed that in the case $N=2$,

$\sup_{u\in
W_{0}^{1,2}(\mathbb{B})\cap\mathscr{S},\,\|u\|_{W_{0}^{1,2}(\mathbb{B})}\leq
1}\int_{\mathbb{B}}e^{\alpha
u^{2}}|x|^{2\gamma}dx<\infty,\quad\forall\,\alpha\leq
4\pi(1+\gamma),\,\,\gamma\geq 0;$ (3)

moreover, extremal function for the above supremum exists. Of course they
discussed more general weight $h(|x|)$ and fast growth $F(u)$ instead of
$|x|^{\alpha}$ and $e^{4\pi(1+\alpha)u^{2}}$ respectively.

Our aim is to generalize (3) to higher dimensional case and to stronger
versions. We first have the following:

######  Theorem 1.

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^{N}$ $(N\geq 2)$,
$W_{0}^{1,N}(\mathbb{B})$ and $\mathscr{S}$ be as above. Then there holds for
any $\beta\geq 0$,

$\sup_{u\in
W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S},\,\|u\|_{W_{0}^{1,N}(\mathbb{B})}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}|x|^{N\beta}dx<\infty,\quad\forall\,\gamma\leq\alpha_{N}(1+\beta).$
(4)

Here $\alpha_{N}(1+\beta)$ is the best constant in the sense that if
$\gamma>\alpha_{N}(1+\beta)$, all integrals are finite but the supremum is
infinity. Moreover, for any $\beta\geq 0$ and any
$\gamma\leq\alpha_{N}(1+\beta)$, the supremum in (4) can be attained.

By a rearrangement argument, for any $\gamma\leq\alpha_{N}$, there holds

$\sup_{u\in W_{0}^{1,N}(\mathbb{B}),\,\|u\|_{W_{0}^{1,N}(\mathbb{B})}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}dx=\sup_{u\in
W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S},\,\|u\|_{W_{0}^{1,N}(\mathbb{B})}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}dx.$ (5)

Therefore, when $\Omega=\mathbb{B}$, Theorem 1 includes the classical
Trudinger-Moser inequality (1) as a special case and complements Adimurthi-
Sandeep’s inequality (2).

Motivated by [34, 41, 40], we would generalize Theorem 1 to a version
involving eigenvalue of the $N$-Laplace. For $p>1$, define

$\lambda_{p}(\mathbb{B})=\inf_{u\in W_{0}^{1,N}(\mathbb{B}),\,u\not\equiv
0}\frac{\int_{\mathbb{B}}|\nabla
u|^{N}dx}{(\int_{\mathbb{B}}|u|^{p}dx)^{N/p}}.$ (6)

For $\alpha<\lambda_{p}(\mathbb{B})$, we write for simplicity

$\|u\|_{1,\alpha}=\left(\int_{\mathbb{B}}|\nabla
u|^{N}dx-\alpha(\int_{\mathbb{B}}|u|^{p}dx)^{N/p}\right)^{1/N}.$ (7)

######  Theorem 2.

Given $p>1$. In addition to the assumptions of Theorem 1, let
$\lambda_{p}(\mathbb{B})$ and $\|\cdot\|_{1,\alpha}$ be defined as in (6) and
(7) respectively. Then if $\alpha<\lambda_{p}(\mathbb{B})$, there holds

$\sup_{u\in W_{0}^{1,N}(\mathbb{B}),\,\|u\|_{1,\alpha}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}dx<\infty,\quad\forall\,\gamma\leq\alpha_{N}.$

Moreover, the above supremum can be attained.

When $p=N$, Theorem 2 was proved by Nguyen [26] for a smooth bounded domain.
As a consequence of Theorem 2, we improve Theorem 1 as follows:

######  Theorem 3.

Given $p>1$. Under the same assumptions of Theorem 2, for any $\beta\geq 0$
and any $\alpha<(1+\frac{p}{N}\beta)^{N-1+N/p}\lambda_{p}(\mathbb{B})$ , there
holds

$\sup_{u\in W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S},\,\int_{\mathbb{B}}|\nabla
u|^{N}dx-\alpha(\int_{\mathbb{B}}|u|^{p}|x|^{p\beta}dx)^{N/p}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}|x|^{p\beta}dx<\infty,\quad\forall\,\gamma\leq\alpha_{N}(1+\frac{p}{N}\beta),$
(8)

where $\alpha_{N}(1+\frac{p}{N}\beta)$ is the best constant in the same sense
as in Theorem 1. Furthermore, the supremum in (8) can be attained.

We now explain the geometric meaning of the term $|x|^{p\beta}dx$. Let $g_{0}$
be the standard Euclidean metric, namely
$g_{0}(x)=d{x_{1}}^{2}+\cdots+d{x_{N}}^{2}$. Define a metric
$g(x)=|x|^{\frac{2p}{N}\beta}g_{0}(x)$ for $x\in\mathbb{B}$. Then
$(\mathbb{B},g)$ is a conical manifold with the volume element
$dv_{g}=|x|^{p\beta}dx$. Moreover, $|\nabla u|^{N}dx=|\nabla_{g}u|^{N}dv_{g}$.

The proof of Theorems 1 and 3 is based on a change of variables. While the
proof of Theorem 2 is based on blow-up analysis. In the remaining part of this
note, we shall prove Theorems 1-3 respectively.

##  2 Proof of Theorem 1

Let $\beta\geq 0$ and $\gamma\leq\alpha_{N}(1+\beta)$. Write for simplicity
$u(x)=u(r)$ with $r=|x|$. Following Smets-Willem-Su [31] and Adimurthi-Sandeep
[1], we make a change of variables. Define a function

$v(r)=(1+\beta)^{1-1/N}u(r^{1/(1+\beta)}).$

A straightforward calculation shows

$\displaystyle\int_{\mathbb{B}}|\nabla v|^{N}dx$ $\displaystyle=$
$\displaystyle\omega_{N-1}\int_{0}^{1}|v^{\prime}(r)|^{N}r^{N-1}dr$ (9)
$\displaystyle=$
$\displaystyle\frac{\omega_{N-1}}{1+\beta}\int_{0}^{1}|u^{\prime}(r^{1/(1+\beta)})|^{N}r^{{(N-1-\beta)/(1+\beta)}}dr$
$\displaystyle=$
$\displaystyle\omega_{N-1}\int_{0}^{1}|u^{\prime}(t)|^{N}t^{N-1}dt=\int_{\mathbb{B}}|\nabla
u|^{N}dx$

and

$\displaystyle\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}|x|^{N\beta}dx$
$\displaystyle=$
$\displaystyle\omega_{N-1}\int_{0}^{1}e^{\gamma|u(r)|^{\frac{N}{N-1}}}r^{N-1+N\beta}dr$
$\displaystyle=$
$\displaystyle\omega_{N-1}\int_{0}^{1}e^{\frac{\gamma}{1+\beta}|v(r^{1+\beta})|^{\frac{N}{N-1}}}r^{N-1+N\beta}dr$
$\displaystyle=$
$\displaystyle\frac{\omega_{N-1}}{1+\beta}\int_{0}^{1}e^{\frac{\gamma}{1+\beta}|v(t)|^{\frac{N}{N-1}}}t^{N-1}dt=\frac{1}{1+\beta}\int_{\mathbb{B}}e^{\frac{\gamma}{1+\beta}|v|^{\frac{N}{N-1}}}dx.$
(10)

Then it follows from (9), (10) and (5) that

$\sup_{u\in
W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S},\,\|u\|_{W_{0}^{1,N}(\mathbb{B})}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}|x|^{N\beta}dx=\frac{1}{1+\beta}\sup_{v\in
W_{0}^{1,N}(\mathbb{B}),\,\|v\|_{W_{0}^{1,N}(\mathbb{B})}\leq
1}\int_{\mathbb{B}}e^{\frac{\gamma}{1+\beta}|v|^{\frac{N}{N-1}}}dx.$ (11)

According to Carleson-Chang [5], the supremum on the right-hand side of (11)
can be attained, so does the supremum on the left-hand side. This concludes
Theorem 1. $\hfill\Box$

##  3 Proof of Theorem 2

In this section, we use the standard blow-up analysis to prove Theorem 2. This
method was originally introduced by Ding-Jost-Li-Wang [9] and Li [17, 18], and
extensively employed by Yang [34, 35, 36, 37], Lu-Yang [23], Li-Ruf [19], Zhu
[41], do Ó-de Souza [10, 11], Li-Yang [16], Li [15], Nguyen [25, 26] and
others. Comparing with the case $p\leq N$ [41, 26], we need more analysis to
deal with the general case $p>1$.

###  3.1 The existence of maximizers for subcritical functionals

Let $\alpha<\lambda_{p}(\mathbb{B})$. Denote

$\Lambda_{\gamma,\alpha}=\sup_{u\in
W_{0}^{1,N}(\mathbb{B}),\,\|u\|_{1,\alpha}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}dx.$

######  Lemma 4.

For any positive integer $k$, there exists a decreasing radially symmetric
function $u_{k}\in W_{0}^{1,N}(\mathbb{B})\cap C^{1}(\overline{\mathbb{B}})$
with $\|u_{k}\|_{1,\alpha}=1$ such that
$\int_{\mathbb{B}}e^{\gamma_{k}|u_{k}|^{\frac{N}{N-1}}}dx=\Lambda_{\gamma_{k},\alpha}$,
where $\gamma_{k}=\alpha_{N}-1/k$. Moreover, $u_{k}$ satisfies the Euler-
Lagrange equation

$\left\\{\begin{array}[]{lll}-\Delta_{N}u_{k}-\alpha\left(\int_{\mathbb{B}}u_{k}^{p}dx\right)^{\frac{N}{p}-1}u_{k}^{p-1}=\frac{1}{\lambda_{k}}u_{k}^{\frac{1}{N-1}}e^{\gamma_{k}u_{k}^{\frac{N}{N-1}}}\\\\[5.16663pt]
u_{k}\geq 0\quad{\rm in}\quad\mathbb{B}\\\\[5.16663pt] u_{k}=0\quad{\rm
on}\quad\partial\mathbb{B}\\\\[5.16663pt]
\lambda_{k}=\int_{\mathbb{B}}u_{k}^{\frac{N}{N-1}}e^{\gamma_{k}u_{k}^{\frac{N}{N-1}}}dx,\end{array}\right.$
(12)

where $\Delta_{N}u_{k}={\rm div}(|\nabla u_{k}|^{N-2}\nabla u_{k})$.

###### Proof.

Let $k$ be a positive integer. By a rearrangement argument, there exists a
sequence of decreasing radially symmetric functions $u_{j}\in
W_{0}^{1,N}(\mathbb{B})$ with $\|u_{j}\|_{1,\alpha}\leq 1$ and
$\int_{\mathbb{B}}e^{\gamma_{k}|u_{j}|^{\frac{N}{N-1}}}dx\rightarrow\Lambda_{\gamma_{k},\alpha}$
as $j\rightarrow\infty$. Since $\alpha<\lambda_{p}(\mathbb{B})$, $u_{j}$ is
bounded in $W_{0}^{1,N}(\mathbb{B})$. Without loss of generality, we assume
$u_{j}$ converges to some function $u_{k}$ weakly in
$W_{0}^{1,N}(\mathbb{B})$, strongly in $L^{s}(\mathbb{B})$ for any $s>1$ and
almost everywhere in $\mathbb{B}$. If $u_{k}\equiv 0$, then
$\|u_{j}\|_{W_{0}^{1,N}(\mathbb{B})}\leq 1+o_{j}(1)$. Thus
$e^{\gamma_{k}u_{j}^{{N}/{(N-1)}}}$ is bounded in $L^{q}(\mathbb{B})$ for some
$q>1$. It follows that $e^{\gamma_{k}u_{j}^{{N}/{(N-1)}}}$ converges to $1$ in
$L^{1}(\mathbb{B})$. This implies that
$\Lambda_{\gamma_{k},\alpha}=|\mathbb{B}|$, the volume of $\mathbb{B}$, which
is impossible. Therefore $u_{k}\not\equiv 0$. Clearly $u_{k}$ is also
decreasingly radially symmetric and $\|u_{k}\|_{1,\alpha}\leq 1$. Define a
function sequence

$v_{j}=\frac{u_{j}}{\left(1+\alpha(\int_{\mathbb{B}}u_{j}^{p}dx)^{N/p}\right)^{1/N}}.$

It follows that $\|v_{j}\|_{W_{0}^{1,N}(\mathbb{B})}\leq 1$, $v_{j}$ converges
to $v_{k}=u_{k}/(1+\alpha(\int_{\mathbb{B}}u_{k}^{p}dx)^{N/p})^{1/N}$ weakly
in $W_{0}^{1,N}(\mathbb{B})$. By a result of Lions ([22], Theorem I.6), for
any $q<1/(1-\|v_{k}\|_{W_{0}^{1,N}(\mathbb{B})}^{N})^{1/(N-1)}$, there holds

$\lim_{j\rightarrow\infty}\int_{\mathbb{B}}e^{q\alpha_{N}v_{j}^{\frac{N}{N-1}}}dx<\infty.$
(13)

One can easily check that

$\left(1+\alpha(\int_{\mathbb{B}}u_{k}^{p}dx)^{N/p}\right)\left(1-\|v_{k}\|_{W_{0}^{1,N}(\mathbb{B})}^{N}\right)=1-\|u_{k}\|_{1,\alpha}^{N}<1.$
(14)

It follows from (13) and (14) that $e^{\gamma_{k}u_{j}^{N/(N-1)}}$ is bounded
in $L^{r}(\mathbb{B})$ for some $r>1$, and thus
$e^{\gamma_{k}u_{j}^{N/(N-1)}}\rightarrow e^{\gamma_{k}u_{k}^{N/(N-1)}}$ in
$L^{1}(\mathbb{B})$ as $j\rightarrow\infty$. Hence
$\int_{\mathbb{B}}e^{\gamma_{k}u_{k}^{N/(N-1)}}dx=\Lambda_{\gamma_{k},\alpha}$
and $u_{k}$ is the desired extremal function. Clearly
$\|u_{k}\|_{1,\alpha}=1$. Moreover, the Euler-Lagrange equation of $u_{k}$ is
(12). According to the regularity theory for degenerate elliptic equations,
see Serrin [30], Tolksdorf [32] and Lieberman [20], we have $u_{k}\in
C^{1}(\overline{\mathbb{B}})$. $\hfill\Box$

It is indicated by Lemma 4 that for any $\gamma<\alpha_{N}$ and
$\alpha<\lambda_{p}(\mathbb{B})$, the supremum $\Lambda_{\gamma,\alpha}$ can
be attained. In particular, for any $\gamma_{k}=\alpha_{N}-1/k$, there exists
a maximizer $u_{k}\geq 0$ satisfies (12). It is not difficult to see that

$\lim_{k\rightarrow\infty}\int_{\mathbb{B}}e^{\gamma_{k}u_{k}^{\frac{N}{N-1}}}dx=\Lambda_{\alpha_{N},\alpha}=\sup_{u\in
W_{0}^{1,N}(\mathbb{B}),\,\|u\|_{1,\alpha}\leq
1}\int_{\mathbb{B}}e^{\alpha_{N}|u|^{\frac{N}{N-1}}}dx.$ (15)

Since $\|u_{k}\|_{1,\alpha}=1$, without loss of generality, we can assume that
$u_{k}$ converges to $u_{0}$ weakly in $W_{0}^{1,N}(\mathbb{B})$, strongly in
$L^{s}(\mathbb{B})$ for any $s>1$, and almost everywhere in $\mathbb{B}$. Let
$c_{k}=u_{k}(0)=\max_{\mathbb{B}}u_{k}$. If $c_{k}$ is bounded, then applying
the Lebesgue dominated convergence theorem to (15), we know that $u_{0}$ is
the desired extremal function for the supremum $\Lambda_{\alpha_{N},\alpha}$.
Hereafter we assume

$c_{k}\rightarrow\infty\quad{\rm as}\quad k\rightarrow\infty.$ (16)

######  Lemma 5.

Let $u_{0}$ be the limit of $u_{k}$ as above. Then $u_{0}\equiv 0$ and
$|\nabla u_{k}|^{N}dx\rightharpoonup\delta_{0}$ weakly in the sense of
measure, where $\delta_{0}$ stands for the Dirac measure centered at the
origin.

###### Proof.

We first prove $u_{0}\equiv 0$. Suppose not. It follows from Lions’ lemma that
$e^{\gamma_{k}u_{k}^{N/(N-1)}}$ is bounded in $L^{q}(\mathbb{B})$ for some
$q>1$. Then applying elliptic estimates to (12), we conclude $u_{k}$ is
uniformly bounded in $\mathbb{B}$, which contradicts our assumption (16).
Therefore $u_{0}\equiv 0$.

Next we prove $|\nabla u_{k}|^{N}dx\rightharpoonup\delta_{0}$. Suppose not.
Since $\|u_{k}\|_{1,\alpha}=1$ and $u_{0}\equiv 0$, there would hold
$\|u_{k}\|_{W_{0}^{1,N}(\mathbb{B})}=1+o_{k}(1)$. Thus there exists some
$0<r_{0}<1$ such that

$\limsup_{k\rightarrow\infty}\int_{|x|\leq r_{0}}|\nabla u_{k}|^{N}dx<1.$

It follows from the classical Trudinger-Moser inequality (1) that
$e^{\gamma_{k}(u_{k}-u_{k}(r_{0}))^{N/(N-1)}}$ is bounded in
$L^{q}({B}_{r_{0}})$ for some $q>1$. Since $u_{k}$ is decreasing radially
symmetric and $\|u_{k}\|_{1,\alpha}=1$ with $\alpha<\lambda_{p}(\mathbb{B})$,
we have

$u_{k}^{N}(r_{0})\leq\left(\frac{1}{|B_{r_{0}}|}\int_{|x|\leq
r_{0}}u_{k}^{p}dx\right)^{N/p}\leq\frac{1}{(\lambda_{p}(\mathbb{B})-\alpha)|B_{r_{0}}|^{N/p}}.$

Hence $e^{\gamma_{k}u_{k}^{N/(N-1)}}$ is also bounded in
$L^{q_{1}}({B}_{r_{0}})$ for some $q_{1}>1$. Then applying elliptic estimates
to (12), we conclude that $u_{k}$ is uniformly bounded in $\mathbb{B}$. This
contradicts (16) and ends the proof of the lemma. $\hfill\Box$

###  3.2 Blow-up analysis

Let
$r_{k}=\lambda_{k}^{\frac{1}{N}}c_{k}^{-\frac{1}{N-1}}e^{-\frac{\gamma_{k}}{N}c_{k}^{{N}/{(N-1)}}}$.
Using the same argument as in the proof of ([34], Lemma 4.3), one has by Lemma
5 and the classical Trudinger-Moser inequality (1) that

$\displaystyle r_{k}e^{au_{k}^{N/(N-1)}}\rightarrow
0~{}~{}~{}~{}\mbox{as}~{}~{}k\rightarrow\infty,~{}~{}\forall a<\alpha_{N}/N.$
(17)

For $x\in B_{r_{k}^{-1}}$, we define $\psi_{k}(x)=c_{k}^{-1}u_{k}(r_{k}x)$ and
$\varphi_{k}(x)=c_{k}^{\frac{1}{N-1}}(u_{k}(r_{k}x)-c_{k})$.

######  Lemma 6.

Up to a subsequence, there holds $\psi_{k}\rightarrow 1$ in $C^{1}_{\rm
loc}(\mathbb{R}^{N})$ and $\varphi_{k}\rightarrow\varphi$ in $C^{1}_{\rm
loc}(\mathbb{R}^{N})$ as $k\rightarrow\infty$, where

$\varphi(x)=-\frac{N-1}{\alpha_{N}}\log\left(1+\frac{\alpha_{N}}{N^{N/(N-1)}}|x|^{\frac{N}{N-1}}\right).$
(18)

###### Proof.

A simple calculation gives

$\displaystyle-\Delta_{N}\psi_{k}=\alpha
c_{k}^{p-N}r_{k}^{N}||u_{k}||_{p}^{N-p}\psi_{k}^{p-1}+c_{k}^{-N}e^{\gamma_{k}(u_{k}^{N/(N-1)}-c_{k}^{N/(N-1)})}\psi_{k}^{1/(N-1)}$
(19)

and

$\displaystyle-\Delta_{N}\varphi_{k}=\alpha
c_{k}^{p}r_{k}^{N}||u_{k}||_{p}^{N-p}\psi_{k}^{p-1}+e^{\gamma_{k}(u_{k}^{N/(N-1)}-c_{k}^{N/(N-1)})}\psi_{k}^{1/(N-1)}.$
(20)

Since $u_{k}$ is bounded in $L^{p}(\mathbb{B})$, one has by (16) and (17) that

$\displaystyle\left(\int_{B_{r_{k}^{-1}}}\left(c_{k}^{p-N}r_{k}^{N}||u_{k}||_{p}^{N-p}\psi_{k}^{p-1}\right)^{p/(p-1)}dx\right)^{(p-1)/p}=c_{k}^{1-N}r_{k}^{N/p}||u_{k}||_{p}^{N-1}\rightarrow
0~{}~{}\mbox{as}~{}~{}k\rightarrow\infty.$ (21)

Since $0\leq\psi_{k}\leq 1$, there holds

$\displaystyle
c_{k}^{-N}e^{\gamma_{k}(u_{k}^{N/(N-1)}-c_{k}^{N/(N-1)})}\psi_{k}^{1/(N-1)}\rightarrow
0~{}~{}\mbox{as}~{}~{}k\rightarrow\infty.$ (22)

It follows from (21) and (22) that $\Delta_{N}\psi_{k}$ is bounded in
$L^{{p}/{(p-1)}}(B_{r_{k}^{-1}})$. Applying the regularity theory [32] to
(19), one obtains

$\displaystyle\psi_{k}\rightarrow\psi~{}~{}\mbox{in}~{}~{}C^{0}_{\rm
loc}(\mathbb{R}^{N})~{}~{}\mbox{as}~{}~{}k\rightarrow\infty,$ (23)

where

$\psi(0)=1,\quad 0\leq\psi(x)\leq 1,\,\,\forall x\in\mathbb{R}^{N}.$ (24)

When $1<p\leq N$, one can easily see that

$\displaystyle c_{k}^{p}r_{k}^{N}||u_{k}||_{p}^{N-p}\psi_{k}^{p-1}\rightarrow
0$ (25)

uniformly in $x\in B_{r_{k}^{-1}}$ as $k\rightarrow\infty$. When $p>N$, we
have for any $R>0$ and sufficiently large $k$

$\displaystyle||u_{k}||_{p}^{N-p}=\left(\int_{\mathbb{B}}u_{k}^{p}dx\right)^{N/p-1}\leq\left(\int_{B_{Rr_{k}}}u_{k}^{p}dx\right)^{N/p-1}=c_{k}^{N-p}r_{k}^{N^{2}/p-N}\left(\int_{B_{R}}\psi_{k}^{p}dx\right)^{N/p-1}.$
(26)

In view of (24), we conclude $\int_{B_{R}}\psi^{p}dx>0$, which together with
(23) and (26) leads to

$\displaystyle||u_{k}||_{p}^{N-p}\leq
2\left(\int_{B_{R}}\psi^{p}dx\right)^{N/p-1}c_{k}^{N-p}r_{k}^{N^{2}/p-N}$

for sufficiently large $k$. This together with (17) gives

$\displaystyle c_{k}^{p}r_{k}^{N}||u_{k}||_{p}^{N-p}\psi_{k}^{p-1}\leq
2\left(\int_{B_{R}}\psi^{p}dx\right)^{N/p-1}c_{k}^{N}r_{k}^{N^{2}/p}\rightarrow
0~{}~{}\mbox{as}~{}~{}k\rightarrow\infty.$ (27)

It then follows from (25) and (27) that $\Delta_{N}\psi_{k}$ is bounded in
$L^{\infty}(B_{R})$. Applying again the regularity theory [32] to (19), we
conclude that $\psi_{k}\rightarrow\psi$ in $C^{1}(B_{R/2})$. Since $R$ is
arbitrary, up to a subsequence, there holds

$\psi_{k}\rightarrow\psi~{}~{}\mbox{in}~{}~{}C^{1}_{\rm
loc}(\mathbb{R}^{N})~{}~{}\mbox{as}~{}~{}k\rightarrow\infty,$

where $\psi$ is a solution of

$\displaystyle-\Delta_{N}\psi=0~{}~{}\mbox{in}~{}~{}\mathbb{R}^{N},~{}~{}~{}~{}0\leq\psi\leq\psi(0)=1.$

The Liouville theorem implies that $\psi\equiv 1$ in $\mathbb{R}^{N}$.

Recalling (25) and (27),
$c_{k}^{p}r_{k}^{N}||u_{k}||_{p}^{N-p}\psi_{k}^{p-1}\rightarrow 0$ in
$L^{\infty}_{\rm loc}(\mathbb{R}^{N})$. Then using the same argument as in
([19], Section 3) or ([16], Lemma 17), we have by applying elliptic estimates
to (20),

$\varphi_{k}\rightarrow\varphi\,\,{\rm in}\,\,C^{1}_{\rm
loc}(\mathbb{R}^{N})\,\,{\rm as}\,\,k\rightarrow\infty,$

where $\varphi$ satisfies

$\left\\{\begin{array}[]{lll}-\Delta_{N}\varphi=e^{\alpha_{N}\frac{N}{N-1}\varphi}\quad{\rm
in}\quad\mathbb{R}^{N}\\\\[5.16663pt]
\sup_{\mathbb{R}^{N}}\varphi=\varphi(0)=0.\end{array}\right.$

Observing that $\varphi$ is radially symmetric, one gets (18) by solving the
corresponding ordinary differential equation. $\hfill\Box$

Lemma 6 describes the asymptotic behavior of $u_{k}$ near the blow-up point
$0$. To know $u_{k}$’s behavior away from $0$, by the same argument as in the
proof of ([34], Lemma 4.11), we have that

$\left\\{\begin{array}[]{lll}c_{k}^{{1}/{(N-1)}}u_{k}\rightharpoonup G&{\rm
weakly\,\,in}&W_{0}^{1,q}(\mathbb{B}),\,\,\,\forall 1<q<N\\\\[5.16663pt]
c_{k}^{{1}/{(N-1)}}u_{k}\rightarrow G&{\rm
strongly\,\,in}&L^{s}(\mathbb{B}),\,\,\,\forall 1<s\leq
Nq/(N-q)\\\\[5.16663pt] c_{k}^{{1}/{(N-1)}}u_{k}\rightarrow G&\quad\quad{\rm
in}&C^{1}_{\rm loc}(\overline{\mathbb{B}}\setminus\\{0\\}),\end{array}\right.$

where $G$ is a distributional solution of

$-\Delta_{N}G-\alpha||G||_{p}^{N-p}G^{p-1}=\delta_{0}\quad{\rm
in}\quad\mathbb{B}.$ (28)

According to Kichenassamy-Veron [14], $G$ can be represented by

$G(x)=-\frac{N}{\alpha_{N}}\log|x|+A_{0}+w(x),$

where $A_{0}$ is a constant, $w\in C^{\nu}(\mathbb{B})$ for some $0<\nu<1$ and
$w(0)=0$. In view of (15), we also have an analog of ([34], Proposition 5.2),
namely

$\Lambda_{\alpha_{N},\alpha}\leq|\mathbb{B}|+\frac{\omega_{N-1}}{N}e^{\alpha_{N}A_{0}+\sum_{j=1}^{N-1}\frac{1}{j}}.$
(29)

For its proof, since no new idea comes out, we omit the details but refer the
readers to [34] (see also [41, 15, 25]).

###  3.3 Test function computation

In this subsection, we construct a sequence of functions to show that

$\Lambda_{\alpha_{N},\alpha}>|\mathbb{B}|+\frac{\omega_{N-1}}{N}e^{\alpha_{N}A_{0}+\sum_{j=1}^{N-1}\frac{1}{j}}.$
(30)

The contradiction between (30) and (29) indicates that $c_{k}$ is a bounded
sequence, and whence the desired extremal function exists. This completes the
proof of Theorem 2.

For any positive integer $k$, we set

$\phi_{k}(x)=\left\\{\begin{array}[]{lll}c+\frac{1}{c^{1/(N-1)}}\left(-\frac{N-1}{\alpha_{N}}\log(1+c_{N}|kx|^{\frac{N}{N-1}})+b\right),&|x|<\frac{\log
k}{k}\\\\[5.16663pt] \frac{G}{c^{1/(N-1)}},&\frac{\log k}{k}\leq|x|\leq
1,\end{array}\right.$ (31)

where $c$ and $b$ are constants, depending only on $k$, to be determined
later. To ensure $\phi_{k}\in W_{0}^{1,N}(\mathbb{B})$, we need

$c+\frac{1}{c^{1/(N-1)}}\left(-\frac{N-1}{\alpha_{N}}\log(1+c_{N}|\log
k|^{\frac{N}{N-1}})+b\right)=\frac{G(\frac{\log k}{k})}{c^{1/(N-1)}},$

which implies that

$c^{\frac{N}{N-1}}=G(\frac{\log k}{k})+\frac{N-1}{\alpha_{N}}\log(1+c_{N}(\log
k)^{\frac{N}{N-1}})-b.$ (32)

We now calculate the energy of $\phi_{k}$. In view of (31), a straightforward
calculation gives

$\int_{|x|<\frac{\log
k}{k}}|\nabla\phi_{k}|^{N}dx=\frac{1}{c^{{N}/{(N-1)}}}\frac{N-1}{\alpha_{N}}\left\\{-\sum_{j=1}^{N-1}\frac{1}{j}+\log(1+c_{N}(\log
k)^{\frac{N}{N-1}})+O((\log k)^{-\frac{N}{N-1}})\right\\}.$

By (28) and the divergence theorem

$\displaystyle\int_{\frac{\log k}{k}\leq|x|\leq 1}|\nabla G|^{N}dx$
$\displaystyle=$ $\displaystyle\int_{\frac{\log k}{k}\leq|x|\leq
1}G(-\Delta_{N}G)dx+\int_{|x|=\frac{\log k}{k}}G|\nabla G|^{N-1}ds$
$\displaystyle=$ $\displaystyle\alpha(\int_{\frac{\log k}{k}\leq|x|\leq
1}G^{p}dx)^{N/p}+G(\frac{\log k}{k})\int_{|x|=\frac{\log k}{k}}|\nabla
G|^{N-1}ds$ $\displaystyle=$ $\displaystyle G(\frac{\log
k}{k})+\alpha(\int_{\mathbb{B}}G^{p}dx)^{N/p}+O\left(\frac{(\log
k)^{N+p-1}}{k^{N}}\right).$

As a consequence

$\displaystyle\int_{\mathbb{B}}|\nabla\phi_{k}(x)|^{N}dx$ $\displaystyle=$
$\displaystyle\frac{1}{c^{N/(N-1)}}\left\\{-\frac{N-1}{\alpha_{N}}\sum_{j=1}^{N-1}\frac{1}{j}+\frac{N-1}{\alpha_{N}}\log(1+c_{N}(\log
k)^{\frac{N}{N-1}})\right.$ $\displaystyle\quad\quad\left.+G(\frac{\log
k}{k})+\alpha(\int_{\mathbb{B}}G^{p}dx)^{N/p}+O((\log
k)^{-\frac{N}{N-1}})\right\\}.$

Noting also that

$\left(\int_{\mathbb{B}}\phi_{k}^{p}(x)dx\right)^{N/p}=\frac{1}{c^{N/(N-1)}}\left(\int_{\mathbb{B}}G^{p}dx+O(\frac{(\log
k)^{2N}}{k^{N}})\right)^{N/p},$

we obtain

$\|\phi_{k}\|_{1,\alpha}^{N}=\frac{1}{c^{N/(N-1)}}\left(-\frac{N-1}{\alpha_{N}}\sum_{j=1}^{N-1}\frac{1}{j}+\frac{N-1}{\alpha_{N}}\log(1+c_{N}(\log
k)^{\frac{N}{N-1}})+G(\frac{\log k}{k})+O((\log k)^{-\frac{N}{N-1}})\right).$

Set $\|\phi_{k}\|_{1,\alpha}=1$. It then follows that

$c^{\frac{N}{N-1}}=\frac{N}{\alpha_{N}}\log
k-\frac{N-1}{\alpha_{N}}\sum_{j=1}^{N-1}\frac{1}{j}+\frac{N-1}{\alpha_{N}}\log
c_{N}+A_{0}+O((\log k)^{-\frac{N}{N-1}}).$ (33)

This together with (32) leads to

$b=\frac{N-1}{\alpha_{N}}\sum_{j=1}^{N-1}\frac{1}{j}+O((\log
k)^{-\frac{N}{N-1}}).$ (34)

When $|x|<\frac{\log k}{k}$, we calculate

$\displaystyle\alpha_{N}\phi_{k}^{\frac{N}{N-1}}(x)$ $\displaystyle=$
$\displaystyle\alpha_{N}c^{\frac{N}{N-1}}\left(1+\frac{1}{c^{N/(N-1)}}\left(-\frac{N-1}{\alpha_{N}}\log(1+c_{N}|kx|^{\frac{N}{N-1}})+b\right)\right)^{\frac{N}{N-1}}$
(35) $\displaystyle\geq$
$\displaystyle\alpha_{N}c^{\frac{N}{N-1}}\left(1+\frac{N}{N-1}\frac{1}{c^{N/(N-1)}}\left(-\frac{N-1}{\alpha_{N}}\log(1+c_{N}|kx|^{\frac{N}{N-1}})+b\right)\right)$
$\displaystyle=$
$\displaystyle\alpha_{N}c^{\frac{N}{N-1}}+\frac{N\alpha_{N}b}{N-1}-{N}\log(1+c_{N}|kx|^{\frac{N}{N-1}}).$

In view of (33) and (34),

$\alpha_{N}c^{\frac{N}{N-1}}+\frac{N\alpha_{N}b}{N-1}=N\log
k+\sum_{j=1}^{N-1}\frac{1}{j}+(N-1)\log c_{N}+\alpha_{N}A_{0}+O((\log
k)^{-\frac{N}{N-1}}).$ (36)

Moreover, integration by parts leads to

$\displaystyle\int_{|x|<\frac{\log
k}{k}}e^{-{N}\log(1+c_{N}|kx|^{\frac{N}{N-1}})}dx$ $\displaystyle=$
$\displaystyle k^{-N}\int_{|y|<\log
k}\frac{dy}{(1+c_{N}|y|^{\frac{N}{N-1}})^{N}}$ $\displaystyle=$ $\displaystyle
k^{-N}\int_{0}^{(\log
k)^{\frac{N}{N-1}}}\frac{N-1}{N}\frac{t^{N-2}dt}{(1+c_{N}t)^{N}}$
$\displaystyle=$ $\displaystyle k^{-N}(1+O((\log k)^{-\frac{N}{N-1}})).$ (37)

Combining (35), (36) and (37), we obtain

$\int_{|x|<\frac{\log
k}{k}}e^{\alpha_{N}\phi_{k}^{\frac{N}{N-1}}(x)}dx\geq\frac{\omega_{N-1}}{N}e^{\alpha_{N}A_{0}+\sum_{j=1}^{N-1}\frac{1}{j}}+O((\log
k)^{-\frac{N}{N-1}}).$ (38)

Using an inequality $e^{t}\geq 1+t$, we have

$\int_{\frac{\log k}{k}\leq|x|\leq
1}e^{\alpha_{N}\phi_{k}^{\frac{N}{N-1}}(x)}dx\geq|\mathbb{B}|+\frac{\alpha_{N}\int_{\mathbb{B}}{G^{\frac{N}{N-1}}}dx}{c^{{N}/{(N-1)^{2}}}}+O((\log
k)^{-\frac{N}{N-1}}).$ (39)

Then (30) follows from (38) and (39) immediately.

##  4 Proof of Theorem 3

As in the proof of Theorem 1, we set
$v(r)=(1+\frac{p}{N}\beta)^{1-1/N}u(r^{1/(1+\frac{p}{N}\beta)})$. Note that

$\displaystyle\int_{\mathbb{B}}|u|^{p}|x|^{p\beta}dx$ $\displaystyle=$
$\displaystyle\omega_{N-1}\int_{0}^{1}|u(r)|^{p}r^{N-1+p\beta}dr$
$\displaystyle=$
$\displaystyle\frac{\omega_{N-1}}{(1+\frac{p}{N}\beta)^{p-p/N}}\int_{0}^{1}|v(r^{1+\beta})|^{N}r^{N-1+N\beta}dr$
$\displaystyle=$
$\displaystyle\frac{\omega_{N-1}}{(1+\frac{p}{N}\beta)^{p+1-p/N}}\int_{0}^{1}|v(t)|^{N}t^{N-1}dt=\frac{1}{(1+\frac{p}{N}\beta)^{p+1-p/N}}\int_{\mathbb{B}}|v|^{p}dx.$

Similar calculations as in (9) and (10) tell us that

$\displaystyle\sup_{u\in
W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S},\,\int_{\mathbb{B}}|\nabla
u|^{N}dx-\alpha(\int_{\mathbb{B}}|u|^{p}|x|^{p\beta}dx)^{N/p}\leq
1}\int_{\mathbb{B}}e^{\gamma|u|^{\frac{N}{N-1}}}|x|^{p\beta}dx$ (40)
$\displaystyle=$ $\displaystyle\frac{1}{1+\frac{p}{N}\beta}\sup_{v\in
W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S},\,\int_{\mathbb{B}}|\nabla
v|^{N}dx-\frac{\alpha}{(1+\frac{p}{N}\beta)^{N-1+N/p}}(\int_{\mathbb{B}}|v|^{p}dx)^{N/p}\leq
1}\int_{\mathbb{B}}e^{\frac{\gamma}{1+\frac{p}{N}\beta}|v|^{\frac{N}{N-1}}}dx.$

By a rearrangement argument, we have

$\displaystyle\sup_{v\in
W_{0}^{1,N}(\mathbb{B})\cap\mathscr{S},\,\int_{\mathbb{B}}|\nabla
v|^{N}dx-\frac{\alpha}{(1+\frac{p}{N}\beta)^{N-1+N/p}}(\int_{\mathbb{B}}|v|^{p}dx)^{N/p}\leq
1}\int_{\mathbb{B}}e^{\frac{\gamma}{1+\frac{p}{N}\beta}|v|^{\frac{N}{N-1}}}dx$
(41) $\displaystyle=$ $\displaystyle\sup_{v\in
W_{0}^{1,N}(\mathbb{B}),\,\int_{\mathbb{B}}|\nabla
v|^{N}dx-\frac{\alpha}{(1+\frac{p}{N}\beta)^{N-1+N/p}}(\int_{\mathbb{B}}|v|^{p}dx)^{N/p}\leq
1}\int_{\mathbb{B}}e^{\frac{\gamma}{1+\frac{p}{N}\beta}|v|^{\frac{N}{N-1}}}dx.$

Since $\alpha<(1+\frac{p}{N}\beta)^{N-1+N/p}\lambda_{p}(\mathbb{B})$ and
$\gamma\leq\alpha_{N}(1+\frac{p}{N}\beta)$, in view of (40) and (41), Theorem
3 follows from Theorem 2 immediately. $\hfill\Box$

Acknowledgements. This work is partly supported by the National Science
Foundation of China (Grant Nos. 11471014, 11401575 and 11761131002).

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