#  InteraRec: Interactive Recommendations Using Multimodal Large Language
Models

Saketh Reddy Karra<EMAIL_ADDRESS>Theja Tulabandhula<EMAIL_ADDRESS>University
of Illinois Chicago, 601 S Morgan St, Chicago, IL 60607, United States

###### Abstract

Weblogs, comprised of records detailing user activities on any website, offer
valuable insights into user preferences, behavior, and interests. Numerous
recommendation algorithms, employing strategies such as collaborative
filtering, content-based filtering, and hybrid methods, leverage the data
mined through these weblogs to provide personalized recommendations to users.
Despite the abundance of information available in these weblogs, identifying
and extracting pertinent information and key features necessitates extensive
engineering endeavors. The intricate nature of the data also poses a challenge
for interpretation, especially for non-experts. In this study, we introduce a
sophisticated and interactive recommendation framework denoted as InteraRec,
which diverges from conventional approaches that exclusively depend on weblogs
for recommendation generation. This framework captures high-frequency
screenshots of web pages as users navigate through a website. Leveraging
state-of-the-art multimodal large language models (MLLMs), it extracts
valuable insights into user preferences from these screenshots by generating a
user behavioral summary based on predefined keywords. Subsequently, this
summary is utilized as input to an LLM-integrated optimization setup to
generate tailored recommendations. Through our experiments, we demonstrate the
effectiveness of InteraRec in providing users with valuable and personalized
offerings.

###### keywords:

Large language models Screenshots User preferences Recommendations

authoryear

##  1 Introduction

In the evolving landscape of e-commerce, the need to understand user
preferences for offering personalized content is increasingly crucial for
online platforms to maximize revenue, build customer loyalty, and maintain
competitive advantage. Typically, the nuanced information that can explain the
user browsing behavior on these platforms is stored as weblogs (Rosenstein,
2000). Numerous state-of-the-art recommendation systems, employing strategies
such as collaborative filtering, content-based filtering, and hybrid methods,
leverage the data mined through these weblogs to provide personalized
recommendations. This data-driven approach ensures that recommendations align
closely with user preferences, contributing to a more seamless and satisfying
shopping journey and ultimately increasing overall platform revenue.

Raw weblogs contain a wealth of information on key browsing session details,
including start and end times, visited pages, and clickstream data. Beyond
capturing fundamental details, these weblogs delve into user-specific data,
encompassing identifiers and IP addresses. However, interpreting the raw web
log data directly can be challenging for non-experts, requiring them to
navigate its complexity and dissect it to build recommendation models. Often,
sophisticated data engineering techniques are necessary to extract the
relevant features and datasets needed for training these models. With recent
advancements in artificial intelligence, particularly within the domain of
LLMs (Zhao et al., 2023), a compelling case emerges for developing a
principled framework that could present a superior solution to the current
dependency on weblogs.

Figure 1:  A comparison of weblog-based framework and InteraRec framework
(refer to Figure 2 for full system description)

.

Inspired by the above, we propose an innovative approach leveraging
information from screenshots of users’ internet browsing activities. Our
approach leverages MLLMs adept at seamless processing and generating content
across diverse modalities. By opting for screenshots as inputs instead of
relying on weblogs, the system benefits from heightened interpretability. The
visual nature of screenshots provides a lucid and transparent representation
of user actions, significantly enhancing the comprehension of the inferences
made by the LLM. Unlike weblogs, which may introduce complexities associated
with various features, the use of screenshots simplifies the inputs, offering
a more straightforward and intuitive approach as shown in Figure 1.
Furthermore, this method facilitates concurrent, real-time recommendations, as
screenshots can be captured almost instantaneously.

In this paper, we introduce InteraRec, a novel screenshot-based user
recommendation system. We capture high-frequency screenshots of web pages as
users navigate through them in real-time. Leveraging the capabilities of
MLLMs, InteraRec processes these screenshots to extract meaningful insights
into user behavior and execute pertinent optimization tools to deliver
personalized recommendations. By seamlessly integrating visual data, language
models, and optimization tools, InteraRec transcends the limitations of
existing systems, promising a more personalized and effective recommendation
system for users.

##  2 Related Work

In this study, we expand upon two key streams of research: (a) LLMs for user
behavior modeling and (b) Tools and their integration with LLMs. We briefly
discuss some of the related works below.

#### LLMs for user behavior modeling

LLMs are known for their capability to dynamically generate diverse facets of
user profiles by analyzing their historical viewing and transaction data.
(Chen, 2023) generated user profiles from TV viewing history to retrieve
candidates from an item pool, subsequently leveraging LLM for item
recommendations. (Liu et al., 2023) employed LLMs to create user profiles,
exploring topics and regions of interest based on user browsing history, and
integrated the inferred user profile to enhance recommendations. Unlike the
approaches above that rely on text data from user history for recommendation
generation, our novel proposal creates dynamic user profiles by leveraging
screenshots of user interactions using MLLMs.

#### Tools and their integration with LLMs

: Researchers have made significant strides in using LLMs to tackle complex
tasks by extending their capabilities to include planning and API selection
for tool utilization. For instance, Schick et al. (2023) introduced the
pioneering work of incorporating external API tags into text sequences,
enabling LLMs to access external tools. TaskMatrix.AI Liang et al. (2023)
utilizes LLMs to generate high-level solution outlines tailored to specific
tasks, matching subtasks with suitable off-the-shelf models or systems.
HuggingGPT Shen et al. (2023) harnesses LLMs as controllers to manage existing
domain models for intricate tasks effectively. Lastly, Qin et al. (2023)
proposed a tool-augmented LLM framework that dynamically adjusts execution
plans, empowering LLMs to complete each subtask using appropriate tools
proficiently. Li et al. (2023) introduced the Optiguide framework, leveraging
LLMs to elucidate supply chain optimization solutions and address what-if
scenarios. In contrast to the aforementioned approaches, InteraRec harnesses
the power of LLMs to generate item recommendations by converting user behavior
summaries into mathematical formulations that can be readily interpreted by an
optimization solver.

##  3 The InteraRec Framework

Addressing the challenge of offering real-time recommendations to users is a
multi-step process. It typically commences with thorough data collection and
analysis, followed by the careful selection and training of appropriate user
behavior models. After a rigorous evaluation, the trained model is deployed to
generate optimal recommendations. The user is then empowered to make a
decision, choosing whether to act upon or disregard the presented
recommendations.

Our framework, InteraRec, depicted in Figure 2, begins by systematically
capturing screenshots of a user’s browsing activity on a designated webpage at
regular intervals. Subsequently, an MLLM, with its inferential capabilities,
translates user interaction behavior into summary information comprising
predefined keywords. This information is then processed by an LLM equipped
with function-calling ability, transforming the summary into specific
constraints and executing optimization tools following a validation check. The
generated solutions are communicated to the user via the LLM through a user
interface. The process outlined above is organized into multiple stages: 1)
Screenshot generation, 2) Behavioral summarization, and 3) Response
generation.

Figure 2: Overview of InteraRec framework delivering real-time personalized
recommendations

###  3.1 Screenshot generation

InteraRec initiates an automated script to systematically capture high-
frequency, high-quality screenshots of the web pages users navigate during a
browsing session. Notably, the screenshots are confined to the user’s current
viewing area on the screen rather than encompassing the full scrollable
content of each webpage. This allows for efficient targeting of the visible
content that users actively engage with. InteraRec then stores these
representative screenshots in a database for further processing and analysis.

###  3.2 Behavioral summarization

In this stage, InteraRec sequentially processes a finite number of screenshots
in real-time using an MLLM to analyze and provide detailed responses.
Specifically, InteraRec instructs the MLLM to succinctly summarize user
interaction behavior across predefined categories such as product
characteristics, lowest and highest prices, brand preferences, product
specifications, user reviews, comparisons, and promotions as shown in Figure
3. This precise instruction is crucial to filter the information and capture
elements directly aligned with user interests. Generic instructions like
“describe the user behavior on screenshots” or “explain the differences in
screenshots” yielded responses that lacked specificity and failed to reflect
user preferences. If the model cannot deduce information for a given category,
InteraRec generates a response indicating ‘does not know’ or ‘not applicable.’
Additionally, InteraRec ensures that the model presents information in a
summarized JSON format, facilitating its use as a constraint or filter for
subsequent processing.

⬇ response = client.chat.completions.create(  model=”gpt-4-vision-preview”
messages=[  {  ”role”: ”user”  ”content”: [  {  ”type”: ”text”  ”text”: ”What
can you infer from the images below \ with regards to a user preference in the
following categories? \ Product Characteristics, \ Lowest Price, \ Highest
Price, \ Brand Preference, \ Product Specifications, \ User Reviews and
Testimonials, \ Comparisons, \ Promotions. \ Write a response that contains
the above information. \ if any of the categorical information is unavailable,
\ mark it as not available ”}…],…}  Figure 3: Guidelines for MLLM to generate
a summary of user interactions using predefined keywords

###  3.3 Response generation

The keyword-based summary generated from the previous stage is rich in
information, forming a valuable resource for generating recommendations. The
relevant information to be extracted from the summary depends on the
capabilities of recommendation methods that can incorporate them. In this
study, InteraRec harnesses the InteraSSort framework (Karra and Tulabandhula,
2023) to deconstruct the user behavior summary into relevant constraints and
uses them to solve an assortment optimization problem to generate optimal
recommendations. Specifically, InteraRec leverages the function-calling
capabilities of the LLM to decompose the user interaction summary into
relevant constraints using appropriate functions as shown in Figure 4.
Subsequently, rigorous validation checks are conducted, encompassing range and
consistency assessments of the decomposed constraints. InteraRec maintains an
extensive database containing parameters for discrete choice models estimated
using historical purchase transactions. By leveraging these choice model
parameters and additional constraints as arguments, InteraRec executes
optimization scripts employing tools like optimization solvers to generate
optimal solutions. Ultimately, InteraRec empowers the LLM to incorporate these
results as input, generating user-friendly personalized product
recommendations.

⬇ function_descriptions = [  {  ”name”: ”get_user_recommendations”,
”description”: ”Generate dynamic recommendations based \ the summary of user
behavior”,  ”parameters”: {  ”type”: ”object”, ”properties”: {
”lowest_price”: {  ”type”: ”integer”,  ”description”: ”get lowest price
preference of user.”  ,},  ”highest_price”: {  ”type”: ”integer”,
”description”: ”get highest price preference of user.”  ,},  ”color”: {
”type”: ”string”,  ”description”: ”get color preference of user”,  }, }, }, },
]  Figure 4: Prospective function for decomposing user interaction summaries.

##  4 Illustration

In this section, we discuss the components needed to run our experiments,
followed by an illustrative example.

###  4.1 Assortment planning

The assortment planning problem involves choosing an assortment among a set of
feasible assortments ($\mathcal{S}$) that maximizes the expected revenue.
Consider a set of products indexed from $1$ to $n$ with their respective
prices being $p_{1},p_{2},\cdots p_{n}$. The revenue of the assortment is
given by $R(S)=\sum_{k\in S}p_{k}\times\mathbb{P}(k|S)$ where
$S\subseteq\\{1,...,n\\}$. The expected revenue maximization problem is
simply: $\max_{S\in\mathcal{S}}R(S)$. Here $\mathbb{P}(k|S)$ represents the
probability that a user chooses product $k$ from an assortment $S$ and is
determined by a choice model.

The complex nature of the assortment planning problem requires the development
of robust optimization methodologies that can work well with different types
of constraints and produce viable solutions within reasonable time frames. In
our experiments, we adopt a series of scalable efficient algorithms
(Tulabandhula et al., 2022) for solving the assortment optimization problem.

###  4.2 Multinomial logit (MNL)

The MNL model (Luce, 2012) is one of the most extensively studied discrete
choice models and is frequently utilized across various marketing
applications. The parameters of the MNL model are represented by a vector
$\mathbf{v}=\left(v_{0},v_{1},\cdots v_{n}\right)$ with $0\leq v_{i}\leq
1\;\;\forall i$. Parameter $v_{i},\ 1\leq i\leq n$, captures the user
preference for purchasing product $i$. Under this model, the probability that
a user chooses product $k$ from an assortment $S$ is given by
$\mathbb{P}(k|S)=v_{l}/(v_{0}+\sum_{k^{\prime}\in S}v_{k^{\prime}})$.

###  4.3 LLM

We employ the gpt-4-vision-preview (GPT-4V) and gpt-3.5-turbo variants from
the GPT model series as our MLLM and LLM, respectively. Both the models are
publicly accessible through the OpenAI API 111https://platform.openai.com/.

###  4.4 Illustrative example

Figure 5: Six screen captures showcasing user interactions with the Asos.com
website, each captured at a 1-second interval. Given the images shown, it’s
important to note that while certain inferences can be made about product
preferences and price sensitivity among other things, these are assumptions
based on a set of images featuring green dresses of different styles and
prices. Here is a possible interpretation in JSON format: “json { “Product
Characteristics”: “User appears to be interested in dresses, specifically in
shades of green, with various styles including mini, midi, and maxi lengths.”,
“Lowest Price”: “$18.00\$$ for the JDY puff sleeve mini smock dress in bright
green”, “Highest Price”: “$144.00\$$ for the Y.A.S Bridesmaid satin cami maxi
dress with ruching detail in deep teal green”, “Brand Preference”: “Variety of
brands shown, such as Y.A.S, Wednesday’s Girl, Nobody’s Child, suggesting no
clear preference for a single brand.”, “Product Specifications”: “Not
Available”, “User Reviews and Testimonials”: “Not Available”, “Comparisons”:
“The user may be comparing dresses based on style and discounts, with several
items showing reduced prices.”, “Promotions”: “Several dresses are listed with
percentage discounts, indicating that promotions may influence user
preferences.” } ” The response indicates an interest in green dresses,
sensitivity to promotions (as indicated by discounted items), a range of price
points indicating a potentially broader budget range or comparison shopping,
and a variety of brands. Figure 6: Extracted summary from $Asos$ website
activities revealing a user preference for green dresses, diverse brand
interest, and price sensitivity. Figure 7: InteraRec Real-time recommendations
incorporating the dynamic user behavior.

Through this illustrative example, we present the dynamic process of
generating personalized, real-time recommendations for users navigating the
$Asos$ fashion apparel website. The recommender system (InteraRec) initiates
by capturing screenshots every 2 seconds, as depicted in Figure 5, and stores
them in the database. Following this, InteraRec employs GPT-V to process these
images in batches of 10 due to its input token size limitation and generate a
summary based on provided instructions, as illustrated in Figure 6(see A for
more examples). In our approach, we focus on extracting the following set of
constraints, specifically the color and price range of products. Subsequently,
the system parses the input, extracting details such as color (‘green’),
lowest price as ‘$18\$$ and highest price as $144\$$’) by leveraging the
function-calling ability of the gpt-3.5-turbo model. Using the parsed
information as arguments, InteraRec executes the MNL optimization script and
communicates outcomes through the interface. We can observe the impact of
incorporating user behavior in the form of updated recommendations limited to
green apparel, as shown in the Figure7. This methodical, step-by-step approach
ensures a consistent refinement of recommendations through continuous analysis
of successive image batches.

##  5 Conclusion

In this paper, we introduce InteraRec, an interactive framework designed to
craft personalized recommendations for users browsing an e-commerce platform.
Overcoming the limitations of extensive weblog engineering, InteraRec utilizes
screenshots to capture and understand user behavior. Harnessing the
capabilities of both LLM and MLLMs, InteraRec leverages insights from the
screenshots and intelligently executes suitable optimization tools,
translating user behavior into concise, easily interpretable recommendations
for users. Overall, InteraRec heralds a new era in recommendation systems
driven by visual data.

## References

  * Rosenstein (2000) M. Rosenstein,  What is actually taking place on web sites: e-commerce lessons from web server logs,  in: Proceedings of the 2nd ACM conference on Electronic commerce, 2000, pp. 38–43.
  * Zhao et al. (2023) W. X. Zhao, K. Zhou, J. Li, T. Tang, X. Wang, Y. Hou, Y. Min, B. Zhang, J. Zhang, Z. Dong, et al.,  A survey of large language models,  arXiv preprint arXiv:2303.18223 (2023).
  * Chen (2023) Z. Chen,  Palr: Personalization aware llms for recommendation,  arXiv preprint arXiv:2305.07622 (2023).
  * Liu et al. (2023) Q. Liu, N. Chen, T. Sakai, X.-M. Wu,  A first look at llm-powered generative news recommendation,  arXiv preprint arXiv:2305.06566 (2023).
  * Schick et al. (2023) T. Schick, J. Dwivedi-Yu, R. Dessì, R. Raileanu, M. Lomeli, L. Zettlemoyer, N. Cancedda, T. Scialom,  Toolformer: Language models can teach themselves to use tools,  arXiv preprint arXiv:2302.04761 (2023).
  * Liang et al. (2023) Y. Liang, C. Wu, T. Song, W. Wu, Y. Xia, Y. Liu, Y. Ou, S. Lu, L. Ji, S. Mao, et al.,  Taskmatrix. ai: Completing tasks by connecting foundation models with millions of apis,  arXiv preprint arXiv:2303.16434 (2023).
  * Shen et al. (2023) Y. Shen, K. Song, X. Tan, D. Li, W. Lu, Y. Zhuang,  Hugginggpt: Solving ai tasks with chatgpt and its friends in huggingface,  arXiv preprint arXiv:2303.17580 (2023).
  * Qin et al. (2023) Y. Qin, S. Hu, Y. Lin, W. Chen, N. Ding, G. Cui, Z. Zeng, Y. Huang, C. Xiao, C. Han, et al.,  Tool learning with foundation models,  arXiv preprint arXiv:2304.08354 (2023).
  * Li et al. (2023) B. Li, K. Mellou, B. Zhang, J. Pathuri, I. Menache,  Large language models for supply chain optimization,  arXiv preprint arXiv:2307.03875 (2023).
  * Karra and Tulabandhula (2023) S. R. Karra, T. Tulabandhula,  Interassort: Interactive assortment planning using large language models,  arXiv preprint arXiv:2311.12241 (2023).
  * Tulabandhula et al. (2022) T. Tulabandhula, D. Sinha, S. Karra,  Optimizing revenue while showing relevant assortments at scale,  European Journal of Operational Research 300 (2022) 561–570.
  * Luce (2012) R. D. Luce, Individual choice behavior: A theoretical analysis, Courier Corporation, 2012.

## Appendix A

Figure 8: Nine screen captures showcasing user interactions with the Nike.com
website, each captured at a 4-second interval. “json “Product
Characteristics”: “The products featured are mostly casual and sports
footwear, with a focus on style and functionality.”, “Lowest Price”: “$63.97,
as seen on the Nike Waffle Debut Womenś Shoes.”, “Highest Price”: “$180, as
seen on the Air Jordan 1 High G Menś Golf Shoes.”, “Brand Preference”: “The
brand showcased is Nike, including its Air Jordan line, indicating a
preference for this brand.”, “Product Specifications”: “Gender“: “Men, Women,
and Unisex options available”, “Color”: “Multiple color options are available
for selection”, “Best For”: “options like Cold Weather, Dry Weather
Conditions, Wet Weather Conditions are available, although specific
preferences are not indicated in the images.” , “User Reviews and
Testimonials”: “Not available”, “Comparisons”: “The images showcase different
models and prices, suggesting that comparisons can be made within the
selection, but specific comparative analysis is not displayed.”, “Promotions”:
“Promotions are visible, such as ẂINTER SALE: UP TO 50% OFFánd discounts on
specific products.”” Figure 9: Extracted summary of user preferences shaped by
activities on the Nike.com website.

Figure 10: Four screen captures showcasing user interactions with the
Zillow.com website, each captured at a 10-second interval. “Based on the
provided images, it appears that the user is searching for real estate
properties for sale on a website, likely with certain preferences. The images
show a website with real estate listings that include price ranges, the number
of bedrooms and bathrooms, square footage, and location. Using the data
visible in the images, I will attempt to create JSON-formatted data that
represents the userś inferred preferences within the limitations of the
images.“json “Product Characteristics”: “Location”: “Millbrae, CA; South San
Francisco, CA; Pacifica, CA; San Jose, CA”, “Property Type”: “House”,
“Bedrooms”: ”3+”, “Bathrooms”: ”2+” , “Lowest Price”: “$799,888”, “Highest
Price”: “$1,350,000”, “Brand Preference”: “not available”, “Product
Specifications”: “Square Footage”: “Low”: “,054 sqft”, “High”: “3,040 sqft” ,
“User Reviews and Testimonials”: “not available”, “Comparisons”: “Available”:
true, “Criteria”: “Price, Location, Size, Bedrooms, Bathrooms” , “Promotions”:
“Price Cut”: “$20,000 (Oct 31) on a property listed for $989,000” ‘’Please
note that the information above is constructed from the context provided by
the images and might lack accuracy in details that are not visible or provide
insufficient context. Some categories like B́rand Preferenceánd Úser Reviews
and Testimonialsd́o not apply in the context of real estate listings and
therefore are marked as “not available.” Also, the promotion information is
based on limited data indicating a price cut on a specific property, as seen
in the images.” Figure 11: Extracted summary of user preferences shaped by
activities on the Zillow.com website.


# On the shape of solutions to elliptic equations in possibly non convex
planar domains

Luca Battaglia, Fabio De Regibus and Massimo Grossi  Dipartimento di
Matematica e Fisica, Università degli Studi Roma Tre, Largo S. Leonardo
Murialdo 1 - 00146 Roma, Italy.
e-mail<EMAIL_ADDRESS> Dipartimento di Matematica, Università
degli Studi di Torino, Via Carlo Alberto 10 - 10123 Torino, Italy.
e-mail<EMAIL_ADDRESS> Dipartimento di Scienze di Base e Applicate
per l’Ingegneria, Università di Roma “La Sapienza”, Via Antonio Scarpa 14 -
00161 Roma, Italy.
e-mail<EMAIL_ADDRESS>
###### Abstract.

In this note we prove uniqueness of the critical point for positive solutions
of elliptic problems in bounded planar domains: we first examine the Poisson
problem $-\Delta u=f(x,y)$ finding a geometric condition involving the
curvature of the boundary and the normal derivative of $f$ on the boundary to
ensure uniqueness of the critical point. In the second part we consider stable
solutions of the nonlinear problem $-\Delta u=f(u)$ in perturbation of convex
domains.

This work was supported by INDAM-GNAMPA

##  1\. Introduction and main results

In this note we are interested in investigating the number of critical points
of solutions of elliptic equations. Let $\Omega\subseteq\mathbb{R}^{2}$ be a
smooth and bounded domain and consider the following problem

$\begin{cases}-\Delta u=f\left((x,y),u\right)&\text{in }\Omega\\\ u>0&\text{in
}\Omega\\\ u=0&\text{on }\partial\Omega,\end{cases}$

where $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a smooth function. It is
known that the number of critical points of solutions of the previous problem
is strongly influenced by the function $f$ and by the geometry of the domain
$\Omega$. In this context the literature is too wide to report all the result,
so we just focus on some of them which are closest to the interest of this
paper (see [17, 13] for a recent survey on the topic in the case $f=f(u)$ and
the monograph [14]).

The first fundamental result has been proved for the solution of the torsion
problem, i.e. $f\equiv 1$ by Makar-Limanov [18]: uniqueness of critical point
holds if $\Omega$ is convex. The same is true for the first eigenfunction,
$f(u)=\lambda u$, see [3, 1].

Under symmetry assumptions the well known results of Gidas, Ni and Nirenberg
[10] gives the uniqueness of the critical point for general $f$.

Uniqueness of critical point can also be proved for solutions of the
Schrödinger eigenvalue problem

$-\Delta u+V(x)u=\lambda u,$

with $V$ convex, $\lambda>0$. It follows from the results in [16, 5], still in
convex domains (their result holds also for more general equations).

If we consider the Poisson problem $-\Delta u=f(x,y)$, as a consequence of a
result by Kennington in [15] one can prove uniqueness of the critical point if
$\Omega$ is convex and if $f$ is $\beta$-concave, i.e. $f^{\beta}$ is concave,
for some $\beta>1$.

The last result we want to recall is due to Cabré and Chanillo [4]. They show
that if the curvature of $\partial\Omega$ is striclty positive and $u$ is a
semi-stable solution of the nonlinear problem $-\Delta u=f(u)$, then $u$ has
exactly one critical point (the result has been extended to the case of non
negative curvature in [8]). Remember that $u$ is said to be a _(semi-)stable_
solution if the linearized operator at $u$ is (nonnegative) positive definite,
i.e. if for all $\varphi\in\mathcal{C}^{\infty}_{0}(\Omega)$ one has

$\int_{\Omega}|\nabla\varphi|^{2}-\int_{\Omega}f^{\prime}(u)|\varphi|^{2}>(\geq)0,$

or equivalently if the first eigenvalue of the linearized operator
$-\Delta-f^{\prime}(u)$ in $\Omega$ is (nonnegative) positive.

All the results mentioned before hold in convex domain and it is known that,
in general, we cannot expect uniqueness of the critical point in non convex
domains, as showed by the case of the torsion problem in a dumbell domain (see
for instance [22]). Then it is natural to ask whether it is possible to
recover the uniqueness in (possibly) non convex domain, under suitable
assumptions.

In the first part of this paper we examine the Poisson problem

(1.1) $\begin{cases}-\Delta u=f(x,y)&\text{in }\Omega\\\ u=0&\text{on
}\partial\Omega,\end{cases}$

where $f:\overline{\Omega}\to\mathbb{R}$ is a smooth function and
$\Omega\subseteq\mathbb{R}^{2}$ is a smooth, bounded (and simply connected)
domain. We have the following result.

######  Theorem 1.1.

Assume $f>0$ in $\overline{\Omega}$ and

(1.2) $\Delta(\log f)=0,\quad\text{in }\Omega,$

and

(1.3) $\frac{1}{2}\frac{\partial f}{\partial\nu}+\mathfrak{K}f\geq
0,\quad\text{on }\partial\Omega,$

where $\nu$ is the outnormal unit vector to $\partial\Omega$ and
$\mathfrak{K}$ is its curvature. If $u$ is the solution of problem 1.1, then
it has a unique nondegenerate critical point $(x_{0},y_{0})\in\Omega$.

######  Remark 1.2.

  1. (1)

Theorem 1.1 holds even if $\Omega$ is not convex. For instance, if we consider
$f(x,y):=e^{2x}$ then equation 1.2 is trivially satisfied while equation 1.3
is satisfied if $\Omega$ is such that

(1.4) $\nu_{x}\geq-\mathfrak{K},\quad\text{on }\partial\Omega,$

where we write $\nu:=(\nu_{x},\nu_{y})$. Hence, we can find a non convex
domain $\Omega$ such that 1.4 is satisfied, see Figure 1.

  2. (2)

Let us point out that we cannot drop assumption 1.2 or 1.3, otherwise we can
lose the uniqueness of the critical point: see Remark 2.4 and Remark 2.5 for
the details.

  3. (3)

The preceding theorem can be seen as a generalization of Makar-Limanov’s
result in [18] for the torsion problem. Indeed, for $f\equiv 1$ and $\Omega$
convex it is easy to see that assumptions 1.2 and 1.3 are trivially satisfied.

$x$$y$$\Omega$ Figure 1. Example of domain which satisfies 1.4.

The idea of the proof is the following: thanks to the assumptions 1.2 and 1.3
we can construct an invertible conformal map that maps $\Omega$ to a bounded
and convex domain where we are able to estimate the number of critical points
of the composition of our solution with the inverse of the conformal map.

In the second part of this paper, we want to investigate the nonlinear problem

(1.5) $\begin{cases}-\Delta u=f(u)&\hbox{in }\Omega\\\ u>0&\hbox{in }\Omega\\\
u=0&\hbox{on }\partial\Omega,\end{cases}$

where $\Omega$ is still a smooth bounded and convex domain in $\mathbb{R}^{2}$
and $f:\mathbb{R}\to\mathbb{R}$ is a smooth nonlinearity.

We are interested in the number of critical points of solutions of the
preceding problem when $\Omega$ is close to be a convex domain.

We recall that if $\Omega$ is convex, we know that a semi-stable solution $u$
of 1.5 admits exactly one critical point which turns out to be nondegenerate,
see [4, 8]. Then we can ask what happens if we consider domains which are
(possibly) non convex, but close to a convex one.

First of all let us recall that for a convex domain
$\Omega\subseteq\mathbb{R}^{N}$, with no empty interior, we can find a
Lipschitz function $\chi_{\Omega}:\mathbb{S}^{N-1}\to(0+\infty)$ such that

$\Omega=\Set{tP(\left(1+\chi_{\Omega}(P)\right)}{P\in\mathbb{S}^{N-1},\quad
t\in[0,1)},$

where up to a translation and a dilatation we assumed without loss of
generality that $B_{1}(\mathbf{0})\subseteq\Omega$. Furthermore, if we assume
$\Omega$ to be of class $\mathcal{C}^{k}$ then
$\chi_{\Omega}\in\mathcal{C}^{k}(\mathbb{S}^{N-1})$.

Hence let us give the following definition of convergence of a family of
smooth sets to a smooth and convex one.

######  Definition 1.1.

Given a bounded and convex set $\Omega\subseteq\mathbb{R}^{N}$ of class
$\mathcal{C}^{k}$ and with no empty interior, we say that the family
$(\Omega_{\varepsilon})_{\varepsilon}\subseteq\mathbb{R}^{N}$ of bounded sets
of class $\mathcal{C}^{k}$ _converges to the convex set_ $\Omega$ for
$\varepsilon\to 0$ \- and we write $\Omega_{\varepsilon}\to\Omega$ for
$\varepsilon\to 0$ \- if there exists a family of functions
$(\chi_{\Omega_{\varepsilon}})_{\varepsilon}\subseteq\mathcal{C}^{k}(\mathbb{S}^{N-1})$
such that

$\Omega_{\varepsilon}=\Set{tP(\left(1+\chi_{\Omega_{\varepsilon}}(P)\right)}{P\in\mathbb{S}^{N-1},\quad
t\in[0,1)},$

and

$\left\|\chi_{\Omega_{\varepsilon}}-\chi_{\Omega}\right\|_{\mathcal{C}^{k}(\mathbb{S}^{N-1})}\to
0,\quad\text{as }\varepsilon\to 0.$

Here again we assumed, without loss of generality, that
$B_{1}(\mathbf{0})\subseteq\Omega$.

Now, let us fix a smooth and convex domain $\Omega\subseteq\mathbb{R}^{2}$,
with $B_{1}(\mathbf{0})\subseteq\Omega$ and consider a family of domains
$\Omega_{\varepsilon}$ that are smooth and such that
$\Omega_{\varepsilon}\to\Omega$ for $\varepsilon\to 0$, say at least in
$\mathcal{C}^{4}$ sense, according to the preceding definition.

The following result holds true.

######  Theorem 1.3.

Let $u_{\varepsilon}$ be a semi-stable solution the following problem

(1.6) $\begin{cases}-\Delta u_{\varepsilon}=f(u_{\varepsilon})&\text{ in }\
\Omega_{\varepsilon}\\\ u_{\varepsilon}>0&\text{ in }\ \Omega_{\varepsilon}\\\
u_{\varepsilon}=0&\text{ on }\ \partial\Omega_{\varepsilon},\end{cases}$

with $f(0)\geq 0$ and assume that
$\left\|u_{\varepsilon}\right\|_{L^{\infty}(\Omega_{\varepsilon})}\leq C$ for
some $C>0$. Then $u_{\varepsilon}$ has a unique nondegenerate critical point
$P_{\varepsilon}\in\Omega_{\varepsilon}$. Moreover $P_{\varepsilon}\to P_{c}$
where $P_{c}\in\Omega$ is the unique critical point of a semi-stable solution
$u$ of problem 1.5 in $\Omega$.

######  Remark 1.4.

As will be explained in details in Remark 3.1, let us point out that the
hypothesis that
$\left\|u_{\varepsilon}\right\|_{L^{\infty}(\Omega_{\varepsilon})}$ is
uniformly bounded is always satisfied if, for instance, we assume that the
nonlinearity has the form $f(u)=\lambda g(u)$, $g$ is smooth and satisfies

(1.7) $\displaystyle g:\mathbb{R}\to\mathbb{R}\,\hbox{ is increasing and
convex},$ (1.8) $\displaystyle g(0)>0,$

and $\lambda\in(0,\lambda^{*}(\Omega))$, with $\lambda^{*}(\Omega)>0$. We
recall that in this setting it is well known that there exists
$\lambda^{*}(\Omega)>0$ such that for all $\lambda\in(0,\lambda^{*}(\Omega))$
the problem

(1.9) $\begin{cases}-\Delta u=\lambda g(u)&\hbox{in }\Omega\\\ u>0&\hbox{in
}\Omega\\\ u=0&\hbox{on }\partial\Omega\end{cases}$

admits a stable solution, see for instance [2, 6, 19] and the references
therein.

######  Remark 1.5.

Observe that if the limit domain is convex, but unbounded, then the result of
Theorem 1.3 does not hold. Indeed, it is possible to build a family of domains
which converges to the strip $\mathcal{S}:=\mathbb{R}\times(-1,1)$ and such
that the corresponding solutions have an arbitrary large (finite) number of
critical points. More precisely for any $\lambda\in(0,\lambda^{*}(-1,1))$ and
for all $k\in\mathbb{N}$ there exists a family of smooth and bounded domains
$\widetilde{\Omega}_{\varepsilon}\subseteq\mathbb{R}^{2}$ such that

  1. (i)

$\widetilde{\Omega}_{\varepsilon}$ is star-shaped with respect to and interior
point;

  2. (ii)

$\widetilde{\Omega}_{\varepsilon}$ locally converges to the strip
$\mathcal{S}$ for $\varepsilon\to 0$, i.e. for all compact sets
$K\subseteq\mathbb{R}^{2}$ it holds $\lvert
K\cap(\mathcal{S}\Delta\widetilde{\Omega}_{\varepsilon})\rvert\to 0$ as
$\varepsilon\to 0$;

  3. (iii)

$\lambda^{*}(\widetilde{\Omega}_{\varepsilon})\geq\lambda^{*}(-1,1)$ for
$\varepsilon$ small enough;

  4. (iv)

if $\widetilde{u}_{\varepsilon}$ is the stable solution of problem 1.9 in
$\widetilde{\Omega}_{\varepsilon}$ then $\widetilde{u}_{\varepsilon}$ has at
least $k$ maximum points.

This fact has been proved in [11] for the torsion problem $f\equiv 1$ and
generalized to any $f=\lambda g$, with $g$ satisfying 1.7 and 1.8 in [7].

To prove Theorem 1.3 we show that $u_{\varepsilon}$ converges to $u$, the
solution of probem 1.5 in $\Omega$, and then the claim can be deduced by the
case of the convex domain.

The paper is organized as follows: in the next section we prove Theorem 1.1,
and we conclude it by showing that if at least one between 1.2 and 1.3 does
not hold, then Theorem 1.1 may fail. In Section 3 we prove Theorem 1.3.

##  2\. The Poisson problem

In this section we prove Theorem 1.1. Until the end of the section we identify
$\mathbb{R}^{2}$ with $\mathbb{C}$ and we write $z:=x+iy$.

Assume $f:\overline{\Omega}\to\mathbb{R}$ be positive and such that
assumptions 1.2 and 1.3 are satisfied. Then the following proposition holds
true.

######  Proposition 2.1.

There exists a holomorphic function $T:\Omega\to\mathbb{C}$ such that

  1. (i)

$\lvert T^{\prime}\rvert^{2}=f$ in $\Omega$,

  2. (ii)

$T(\Omega)$ is bounded and convex,

  3. (iii)

there exists a holomorphic function $\tau:T(\Omega)\to\mathbb{C}$ such that
$\tau=T^{-1}$.

###### Proof.

Without loss of generality, we can assume $\mathbf{0}\in\Omega$. Since $\log
f$ is harmonic in $\Omega$ by 1.2 and $\Omega$ is simply connected, we can
find a holomorphic function $w:\Omega\to\mathbb{R}$ such that

$\Re(w)=\frac{1}{2}\log f.$

Moreover, since $f>0$ in $\overline{\Omega}$, $w$ is holomorphic up to the
boundary of $\Omega$. Hence, also the function $\mathfrak{t}:=e^{w}$ is
holomorphic in $\overline{\Omega}$ and if we decompose it by modulus and
principal argument we have

$\mathfrak{t}=\lvert\mathfrak{t}\rvert e^{i\Theta}.$

Finally

$\frac{1}{2}\log
f=\Re(w)=\Re\left(\log\mathfrak{t}\right)=\Re\left(\log(\lvert\mathfrak{t}\rvert
e^{i\Theta})\right)=\log\lvert\mathfrak{t}\rvert,$

which yields to

(2.1) $\lvert\mathfrak{t}\rvert^{2}=f.$

Then, since $\Omega$ is simply connected and $\mathfrak{t}$ is holomorphic, we
can define $T$ as the primitive of $\mathfrak{t}$ such that
$T(\mathbf{0})=\mathbf{0}$. Clearly $T$ is holomorphic and 2.1 implies $\lvert
T^{\prime}\rvert^{2}=f$, proving $(i)$.

To prove $(ii)$ note that since $T$ is continuous up to the boundary of
$\Omega$ we have that $\lvert T\rvert$ is bounded and then $T(\Omega)$ is. To
show the second claim we recall that the curvature $\widetilde{\mathfrak{K}}$
of the boundary of $T(\Omega)$, in $\zeta\in\partial T(\Omega)$ is given by
(see [20, equation (23) pag. 234])

(2.2) $\widetilde{\mathfrak{K}}(\zeta)=\frac{1}{\lvert
T^{\prime}(z)\rvert}\left(\Im\left(\frac{\textsl{t}(z)T^{\prime\prime}(z)}{T^{\prime}(z)}\right)+\mathfrak{K}(z)\right),$

where $z\in\partial\Omega$ satisfies $T(z)=\zeta$ and $\textsl{t}(z)$ is the
unit tangent vector to $\partial\Omega$ in $z$. Here, since $\Omega$ is simply
connected then its boundary $\partial\Omega$ is a Jordan curve and we assume
it is orientated is such a way that the winding number satisfies

$W_{\partial\Omega}(z):=\frac{1}{2\pi
i}\int_{\partial\Omega}\frac{d\mathbf{z}}{\mathbf{z}-z}=\begin{cases}1&\text{if
}z\in\Omega\\\ 0&\text{if
}z\in\mathbb{C}\setminus\overline{\Omega}.\end{cases}$

Thanks to this convention the unit tangent vector t is uniquely determined by
the orientation of $\partial\Omega$. If we write $T:=h+ig$ we have

$T^{\prime}=h_{x}-ih_{y},\quad\text{and}\quad
T^{\prime\prime}=h_{xx}-ih_{xy},$

and then, writing $\textsl{t}=x_{\textsl{t}}+iy_{\textsl{t}}$, one has

$\displaystyle\Im\left(\frac{\textsl{t}T^{\prime\prime}}{T^{\prime}}\right)$
$\displaystyle=\Im\left(\frac{(x_{\textsl{t}}+iy_{\textsl{t}})(h_{xx}-ih_{xy})}{h_{x}-ih_{y}}\right)$
$\displaystyle=\Im\left(\frac{x_{\textsl{t}}h_{xx}+y_{\textsl{t}}h_{xy}+i(-x_{\textsl{t}}h_{xy}+y_{\textsl{t}}h_{xx})}{h_{x}-ih_{y}}\right)$
$\displaystyle=\frac{x_{\textsl{t}}(h_{y}h_{xx}-h_{x}h_{xy})+y_{\textsl{t}}(h_{y}h_{xy}+h_{x}h_{xx})}{h_{x}^{2}+h_{y}^{2}}$
(2.3)
$\displaystyle=\frac{y_{\textsl{t}}(h_{x}h_{xx}+h_{y}h_{xy})-x_{\textsl{t}}(h_{x}h_{xy}+h_{y}h_{yy})}{h_{x}^{2}+h_{y}^{2}}.$

Taking into account

$\displaystyle\partial_{x}f$
$\displaystyle=\partial_{x}(h_{x}^{2}+h_{y}^{2})=2h_{x}h_{xx}+2h_{y}h_{xy},$
$\displaystyle\partial_{y}f$
$\displaystyle=\partial_{y}(h_{x}^{2}+h_{y}^{2})=2h_{x}h_{xy}+2h_{y}h_{yy},$

equation 2.3 becomes

$\Im\left(\frac{\textsl{t}T^{\prime\prime}}{T^{\prime}}\right)=\frac{y_{\textsl{t}}f_{x}-x_{\textsl{t}}f_{y}}{2f}=\frac{\nu\cdot\nabla
f}{2f},$

where $\nu=(y_{\textsl{t}},-x_{\textsl{t}})$. Finally, the previous equation
and 2.2 imply

$\lvert
T^{\prime}\rvert\widetilde{\mathfrak{K}}=\Im\left(\frac{\textsl{t}T^{\prime\prime}}{T^{\prime}}\right)+\mathfrak{K}=\frac{f_{\nu}+2\mathfrak{K}f}{2f}\geq
0,$

where the last inequality holds true by 1.3. Then
$\widetilde{\mathfrak{K}}\geq 0$ and $T(\Omega)$ is convex.

Since $T$ is proper, $T^{\prime}\not=0$ and $T(\Omega)$ is simply connected,
[12, Theorem B] tells us that $T$ is invertible. Finally, the inverse is
holomorphic by the Open Mapping Theorem and $(iii)$ follows. ∎

######  Remark 2.2.

In particular $T$ is a conformal map, indeed $\lvert T^{\prime}\rvert^{2}=f>0$
in $\overline{\Omega}$ and then $\Omega$ and $T(\Omega)$ are conformally
equivalent.

######  Remark 2.3.

The function $T$ can be written in a more explicit way by setting

$T^{\prime}(z)=\mathfrak{t}(x+iy):=\frac{1}{\overline{\mathfrak{t}(\mathbf{0})}}f\left(\frac{x+iy}{2},\frac{y-ix}{2}\right).$

See [21, equation (4.3)].

We can now prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let us denote $\Lambda:=T(\Omega)$ with coordinates $\zeta:=\xi+i\eta$ and set

$v(\xi,\eta):=u(\tau(\xi,\eta)),$

where we recall that $\tau=T^{-1}$. If we write $\tau:=\varphi+i\psi$ we have

$\displaystyle\partial_{\xi}v$
$\displaystyle=u_{x}\varphi_{\xi}+u_{y}\psi_{\xi},$
$\displaystyle\partial_{\eta}v$
$\displaystyle=u_{x}\varphi_{\eta}+u_{y}\psi_{\eta},$

and

$\displaystyle\partial_{\xi\xi}v$
$\displaystyle=u_{xx}\varphi_{\xi}^{2}+2u_{xy}\varphi_{\xi}\psi_{\xi}+u_{yy}\psi_{\xi}^{2}+u_{x}\varphi_{\xi\xi}+u_{y}\psi_{\xi\xi},$
$\displaystyle\partial_{\eta\eta}v$
$\displaystyle=u_{xx}\varphi_{\eta}^{2}+2u_{xy}\varphi_{\eta}\psi_{\eta}+u_{yy}\psi_{\eta}^{2}+u_{x}\varphi_{\eta\eta}+u_{y}\psi_{\eta\eta}.$

Hence by the Cauchy-Riemann equations one has

$\displaystyle\Delta v$
$\displaystyle=u_{xx}\lvert\nabla\varphi\rvert^{2}+2u_{xy}\nabla\varphi\nabla\psi+u_{yy}\lvert\nabla\psi\rvert^{2}+u_{x}\Delta\varphi+u_{y}\Delta\psi=\Delta
u\lvert\tau^{\prime}\rvert^{2}=-f\lvert\tau^{\prime}\rvert^{2},$

and then by $(i)$ of Proposition 2.1 we get

$-\Delta v=f\lvert T^{\prime}\rvert^{-2}=1,$

that is $v$ is the solution of the torsion problem in $\Lambda$, i.e.

$\begin{cases}-\Delta v=1&\text{in }\Lambda\\\ v=0&\text{on
}\partial\Lambda.\end{cases}$

By [18, Theorem 1], $v$ has a unique nondegenerate critical point
$(\xi_{0},\eta_{0})\in\Lambda$ and then
$(x_{0},y_{0}):=\tau(\xi_{0},\eta_{0})\in\Omega$ is the unique critical point
of $u$. To show the nondegeneracy of $(x_{0},y_{0})$, since
$u_{x}(x_{0},y_{0})=u_{y}(x_{0},y_{0})=0$ one has

$\displaystyle\partial_{\xi\xi}v(\tau(\xi_{0},\eta_{0}))$
$\displaystyle=u_{xx}\varphi_{\xi}^{2}+2u_{xy}\varphi_{\xi}\psi_{\xi}+u_{yy}\psi_{\xi}^{2},$
$\displaystyle\partial_{\xi\eta}v(\tau(\xi_{0},\eta_{0}))$
$\displaystyle=u_{xx}\varphi_{\xi}\varphi_{\eta}+u_{xy}(\varphi_{\xi}\psi_{\eta}+\varphi_{\eta}\psi_{\xi})+u_{yy}\psi_{\xi}\psi_{\eta},$
$\displaystyle\partial_{\eta\eta}v(\tau(\xi_{0},\eta_{0}))$
$\displaystyle=u_{xx}\varphi_{\eta}^{2}+2u_{xy}\varphi_{\eta}\psi_{\eta}+u_{yy}\psi_{\eta}^{2}.$

Then

$\displaystyle\partial_{\xi\xi}v\partial_{\eta\eta}v$
$\displaystyle=u_{xx}^{2}\varphi_{\xi}^{2}\varphi_{\eta}^{2}+2u_{xx}u_{xy}\varphi_{\xi}^{2}\varphi_{\eta}\psi_{\eta}+u_{xx}u_{yy}\varphi_{\xi}^{2}\psi_{\eta}^{2}+$
$\displaystyle+2u_{xx}u_{xy}\varphi_{\xi}\varphi_{\eta}^{2}\psi_{\xi}+4u_{xy}^{2}\varphi_{\xi}\varphi_{\eta}\psi_{\xi}\psi_{\eta}+2u_{xy}u_{yy}\varphi_{\xi}\psi_{\xi}\psi_{\eta}^{2}+$
$\displaystyle+u_{xx}u_{yy}\varphi_{\eta}^{2}\psi_{\xi}^{2}+2u_{xy}u_{yy}\varphi_{\eta}\psi_{\xi}^{2}\psi_{\eta}+u_{yy}^{2}\psi_{\xi}^{2}\psi_{\eta}^{2},$

and

$\displaystyle\left(\partial_{\xi\eta}v\right)^{2}$
$\displaystyle=u_{xx}^{2}\varphi_{\xi}^{2}\varphi_{\eta}^{2}+u_{xy}^{2}(\varphi_{\xi}\psi_{\eta}+\varphi_{\eta}\psi_{\xi})^{2}+u_{yy}^{2}\psi_{\xi}^{2}\psi_{\eta}^{2}+2u_{xx}u_{yy}\varphi_{\xi}\varphi_{\eta}\psi_{\xi}\psi_{\eta}+$
$\displaystyle+2u_{xx}u_{xy}\varphi_{\xi}^{2}\varphi_{\eta}\psi_{\eta}+2u_{xx}u_{xy}\varphi_{\xi}\varphi_{\eta}^{2}\psi_{\xi}+2u_{xy}u_{yy}\varphi_{\xi}\psi_{\xi}\psi_{\eta}^{2}+2u_{xy}u_{yy}\varphi_{\eta}\psi_{\xi}^{2}\psi_{\eta}.$

Finally we have that

$v_{\xi\xi}v_{\eta\eta}-v_{\xi\eta}^{2}=(u_{xx}u_{yy}-u_{xy}^{2})(\varphi_{\xi}\psi_{\eta}-\varphi_{\eta}\psi_{\xi})^{2},$

and since $(\xi_{0},\eta_{0})$ is nondegenerate, the same holds true for
$(x_{0},y_{0})$. ∎

###  2.1. Final remarks

We conclude this section by showing that if at least one between 1.2 and 1.3
does not hold, then Theorem 1.1 may fail.

######  Remark 2.4.

If $f>0$ in $\overline{\Omega}$, satisfies 1.2, but does not satisfy 1.3 then
the solution of the Poisson problem 1.1 can have more than one critical point.

Indeed in [11] it is shown that for any $\delta>0$, there exists a star-shaped
domain $\Omega:=\Omega(\delta)$ such that the solution of the torsion problem,
i.e. $f\equiv 1$, admits at least two critical points. Moreover one has
$\mathfrak{K}_{|\partial\Omega}\geq-\delta$ and it is negative somewhere. Then
$f>0$ in $\overline{\Omega}$, 1.2 is satisfied but 1.3 is not.

######  Remark 2.5.

If $f>0$ in $\overline{\Omega}$, satisfies 1.3, but does not satisfy 1.2 then
the solution of the Poisson problem 1.1 can have more than one critical point.

Indeed, as a consequence of [9, Theorem 1.1], one has that if $p>1$ is large
enough there exists a solution $u$ of the following Hénon problem

$\begin{cases}-\Delta u=(x^{2}+y^{2})^{\alpha}\lvert u\rvert^{p}&\text{in
}B\\\ u>0&\text{in }B\\\ u=0&\text{on }\partial B,\end{cases}$

with $\alpha>0$, $B:=B_{1}(\mathbf{0})$, and there exist $Q_{1},Q_{2}\in B$
such that

$\max_{B\setminus\bigcup_{i=1}^{2}B_{2\delta}(Q_{i})}u\leq\frac{\sqrt{e}}{4},\quad\text{and}\quad\sup_{B_{\delta}(Q_{i})}u\geq\frac{\sqrt{e}}{2},\quad
i=1,2,$

for some $0<\delta<\frac{\operatorname{dist}(Q_{1},Q_{2})}{4}$. Then let $v$
be the solution of the torsion problem in $B$ with Dirichlet boundary
conditions and set

$u_{\varepsilon}:=u+\varepsilon v,\quad
0<\varepsilon<\frac{\sqrt{e}}{4\left\|v\right\|_{\infty}},$

which solves

$\begin{cases}-\Delta u_{\varepsilon}=f_{\varepsilon}(x,y)&\text{in }B\\\
u_{\varepsilon}>0&\text{in }B\\\ u_{\varepsilon}=0&\text{on }\partial
B,\end{cases}$

with $f_{\varepsilon}(x,y):=(x^{2}+y^{2})^{\alpha}\lvert
u\rvert^{p}+\varepsilon>0$ in $\overline{B}$. Then $u=0$ on $\partial B$
implies

$\frac{1}{2}\frac{\partial
f_{\varepsilon}}{\partial\nu}+\mathfrak{K}f_{\varepsilon}=0+\varepsilon>0,\quad\text{on
}\partial\Omega,$

that is 1.3 is satisfied. Moreover, it is easy to see that $u_{\varepsilon}$
admits at least one critical point in each $B_{\delta}(Q_{i})$ for $i=1,2$.

Now let $Q\in B_{\delta}(Q_{1})\setminus\\{\mathbf{0}\\}$ be a critical point
for $u$ such that $u(Q)\geq\frac{\sqrt{e}}{2}$. Then one has

$\displaystyle f_{\varepsilon}(Q$ $\displaystyle)^{2}\Delta\log
f_{\varepsilon}(Q)$ $\displaystyle={f_{\varepsilon}(Q)\Delta
f_{\varepsilon}(Q)-\lvert\nabla f_{\varepsilon}(Q)\rvert^{2}}$
$\displaystyle=-p\lvert Q\rvert^{6\alpha}\lvert
u(Q)\rvert^{3p-1}+4\varepsilon\alpha^{2}\lvert Q\rvert^{2(\alpha-1)}\lvert
u(Q)\rvert^{p}-\varepsilon p\lvert Q\rvert^{4\alpha}\lvert
u(Q)\rvert^{2p-1}<0,$

for $\varepsilon$ small enough and then 1.2 is not satisfied.

##  3\. The nonlinear problem

In this section we prove Theorem 1.3. Hence, let $\Omega$ be a fixed bounded
and convex domain in $\mathbb{R}^{2}$ and assume
$B_{1}(\mathbf{0})\subseteq\Omega$. Let $\Omega_{\varepsilon}$ be the family
of smooth domains converging to $\Omega$ as $\varepsilon\to 0$ and
$u_{\varepsilon}$ be the solution of problem 1.6, as in Theorem 1.3.

###### Proof of Theorem 1.3.

Since $\Omega$ is convex we know that if $u$ is a semi-stable solution of
problem 1.5 in $\Omega$, then it admits a unique nondegenerate critical point,
which we denote by $P_{c}$. Hence, it is enough to show that for all multi-
indices $\alpha$ with $\lvert\alpha\rvert\leq 2$ it holds

(3.1)
$\sup_{\overline{\Omega_{\varepsilon}\cap\Omega}}\left|D^{\alpha}\big{(}u_{\varepsilon}-u\big{)}\right|\to
0,\quad\text{for }\varepsilon\to 0.$

Indeed, if $P_{\varepsilon}\in\Omega_{\varepsilon}$ is a critical point for
$u_{\varepsilon}$, then $P_{\varepsilon}\to P_{c}$ as $\varepsilon\to 0$, and
it is a nondenerate maximum thanks to 3.1. Then uniqueness follows from the
convergence to $P_{c}$ and the nondegeneracy.

We prove 3.1 through several steps.

_Step 1: there exists $C>0$ such that
$\left\|u_{\varepsilon}\right\|_{H^{2}(\Omega_{\varepsilon})}\leq C$._
From classical regularity theory one has

$\left\|u_{\varepsilon}\right\|_{H^{m+2}(\Omega_{\varepsilon})}\leq
C(\Omega_{\varepsilon})\left(\left\|f(u_{\varepsilon})\right\|_{H^{m}(\Omega_{\varepsilon})}+\left\|u_{\varepsilon}\right\|_{L^{2}(\Omega_{\varepsilon})}\right),$

but from the convergence of $\Omega_{\varepsilon}$ to $\Omega$ one can see
that $C(\Omega_{\varepsilon})$ does not really depend on $\varepsilon$. Then
for $m=0$ using the assumption
$\left\|u_{\varepsilon}\right\|_{L^{\infty}(\Omega_{\varepsilon})}\leq C$ we
get the desired claim.

_Step 2: $u_{\varepsilon}\rightharpoonup u$ in $H^{1}(\Omega^{\rho})$, where
$u$ is a semi-stable solution of problem 1.5 in $\Omega$ and
$\Omega^{\rho}:=\set{(x,y)\in\mathbb{R}^{2}}{(\frac{x}{\rho},\frac{y}{\rho})\in\Omega}$,
for some $\rho>1$._
First of all from the convergence of $\Omega_{\varepsilon}$ to $\Omega$ we can
find $\rho>1$ such that $\Omega_{\varepsilon}\subseteq\Omega^{\rho}$. Since
$\Omega_{\varepsilon}$ are smooth we can consider $u_{\varepsilon}$ defined in
$\Omega^{\rho}$ by means of zero extension outside $\Omega_{\varepsilon}$ and
with a little abuse of notation we still denote such an extension by
$u_{\varepsilon}$. Then from the previous step we have
$u_{\varepsilon}\rightharpoonup u$ in $H^{1}(\Omega^{\rho})$. Then it is easy
to see that by means of the dominated convergence theorem it holds

$\int_{\Omega}\nabla u\nabla\varphi=\int_{\Omega}f(u)\varphi,\quad\text{for
all }\varphi\in\mathcal{C}^{\infty}_{0}(\Omega),$

and $u=0$ on $\partial\Omega$ in trace sense. Finally for any
$\xi\in\mathcal{C}^{\infty}_{0}(\Omega)$ again the dominated convergence
theorem gives

$\int_{\Omega}\lvert\nabla\xi\rvert^{2}-\int_{\Omega}f^{\prime}(u)\xi^{2}=\lim_{\varepsilon\to
0}\left(\int_{\Omega}\lvert\nabla\xi\rvert^{2}-\int_{\Omega}f^{\prime}(u_{\varepsilon})\xi^{2}\right)\geq
0.$

Hence we proved that $u$ is a semi-stable solution of problem 1.5 in $\Omega$.

_Step 3: end of the proof_
From the convergence of $\Omega_{\varepsilon}$ to $\Omega$ we can find $r>0$
and $Q_{1},\dots,Q_{k}\in\partial\Omega$ such that

$\displaystyle\partial\Omega\subseteq\bigcup_{i=1}^{k}B_{r}(Q_{i}),\quad\text{and}\quad\partial\Omega_{\varepsilon}\subseteq\bigcup_{i=1}^{k}B_{r}(Q_{i}),$
$\displaystyle\Omega\cap B_{2r}(Q_{i})=\set{(x,y)\in
B_{2r}(Q_{i})}{y>\Gamma^{i}(x)},$ $\displaystyle\Omega_{\varepsilon}\cap
B_{2r}(Q_{i})=\set{(x,y)\in
B_{2r}(Q_{i})}{y>\Gamma^{i}(x)+\gamma_{\varepsilon}^{i}(x)},$

where the last two relations hold up to a rotation and where
$\Gamma^{1},\dots,\Gamma^{k},\gamma_{\varepsilon}^{1},\dots,\gamma_{\varepsilon}^{k}$
are smooth functions such that

$\gamma_{\varepsilon}^{i}\to 0\quad\text{in }\mathcal{C}^{4},\quad\text{for
all }i=1,\dots,k.$

Now, let us fix $i=1$: up to a translation we can assume $Q_{1}=\mathbf{0}$
and consider

$\bar{u}_{\varepsilon}(x,y):=u_{\varepsilon}(x,y+\gamma_{\varepsilon}(x)),\quad\text{for
all }(x,y)\in\Omega\cap B_{2r},$

where we omitted the index $i$ and $B_{2r}:=B_{2r}(\mathbf{0})$. Then one has
$\bar{u}_{\varepsilon}=0$ on $\partial\Omega\cap B_{2r}$ and since
$\gamma_{\varepsilon}\to 0$ in $\mathcal{C}^{2}$ it holds

$\sup_{\overline{\Omega_{\varepsilon}\cap\Omega\cap
B_{2r}}}\left|D^{\alpha}\big{(}u_{\varepsilon}-\bar{u}_{\varepsilon}\big{)}\right|\to
0,\quad\text{for }\varepsilon\to 0,$

for all multi-indices $\alpha$, with $\lvert\alpha\rvert\leq 2$. Moreover, we
have

$\begin{cases}-\Delta(u-\bar{u}_{\varepsilon})=h_{\varepsilon}&\text{in
}\Omega\cap B_{2r}\\\ u-\bar{u}_{\varepsilon}=0&\text{on }\partial\Omega\cap
B_{2r},\end{cases}$

where

$h_{\varepsilon}:=f(u)-f(\bar{u}_{\varepsilon})-2\partial_{xy}u_{\varepsilon|(x,y+\gamma_{\varepsilon}(x))}\dot{\gamma}_{\varepsilon}-\partial_{yy}u_{\varepsilon|(x,y+\gamma_{\varepsilon}(x))}\dot{\gamma}_{\varepsilon}^{2}-\partial_{y}u_{\varepsilon|(x,y+\gamma_{\varepsilon}(x))}\ddot{\gamma}_{\varepsilon}.$

For $m\in\mathbb{N}$, by means of the mean value theorem and taking into
account that $u$, $u_{\varepsilon}$ and in turn $\bar{u}_{\varepsilon}$ are
uniformly bounded, we have $\lvert f(u)-f(\bar{u}_{\varepsilon})\rvert\leq
C\lvert u-\bar{u}_{\varepsilon}\rvert$, then

$\left\|h_{\varepsilon}\right\|_{H^{m}(\Omega\cap B_{2r})}\leq
C\left(\left\|u-\bar{u}_{\varepsilon}\right\|_{H^{m}(\Omega\cap
B_{2r})}+\left\|u_{\varepsilon}\right\|_{H^{m+2}(\Omega_{\varepsilon})}\left\|\gamma_{\varepsilon}\right\|_{\mathcal{C}^{m+2}}\right).$

Iterating the argument in _Step 1_ , we can find $C>0$ such that
$\left\|u_{\varepsilon}\right\|_{H^{m+2}(\Omega_{\varepsilon})}\leq C$ and

$\left\|u-\bar{u}_{\varepsilon}\right\|_{L^{2}(\Omega\cap
B_{2r})}\leq\left\|u-u_{\varepsilon}\right\|_{L^{2}(\Omega\cap
B_{2r})}+C\left\|\bar{u}_{\varepsilon}-u_{\varepsilon}\right\|_{L^{\infty}(\Omega\cap
B_{2r})}\to 0,\quad\text{for }\varepsilon\to 0,$

thanks to the compact embedding of $H^{1}(\Omega^{\rho})$ in
$L^{2}(\Omega^{\rho})$. Then classical boundary regularity theory gives
$\left\|u-\bar{u}_{\varepsilon}\right\|_{\mathcal{C}^{2}(\Omega\cap B_{r})}\to
0$ for $\varepsilon\to 0$ and in turn

$\sup_{\overline{\Omega_{\varepsilon}\cap\Omega\cap
B_{r}}}\left|D^{\alpha}\big{(}u_{\varepsilon}-u\big{)}\right|\to
0,\quad\text{for }\varepsilon\to 0,$

for all multi-indices $\alpha$, with $\lvert\alpha\rvert\leq 2$. To complete
the proof of 3.1 it is enough to repeat the argument for all $i=1,\dots,k$ and
use interior regularity estimates taking into account that
$\lvert-\Delta(u-u_{\varepsilon})\rvert\leq C\lvert u-u_{\varepsilon}\rvert$,
for some $C>0$. ∎

######  Remark 3.1.

Here we show that if we assume that the nonlinearity has the form
$f(u)=\lambda g(u)$, $g$ is smooth and satisfies 1.7 and 1.8, that are

$\displaystyle g:\mathbb{R}\to\mathbb{R}\,\hbox{ is increasing and convex},$
$\displaystyle g(0)>0,$

and $\lambda\in(0,\lambda^{*}(\Omega))$, then there exists $C>0$ such that
$\left\|u_{\varepsilon}\right\|_{L^{\infty}(\Omega_{\varepsilon})}\leq C$.

First of all note that, since $\Omega_{\varepsilon}\to\Omega$ then
$\lambda^{*}(\Omega_{\varepsilon})\to\lambda^{*}(\Omega)$ and then
$\lambda<\lambda^{*}(\Omega_{\varepsilon})$ for $\varepsilon$ small enough.

Remember that from the convergence of $\Omega_{\varepsilon}$ to $\Omega$ we
can find $\rho>1$ such that
$\Omega_{\varepsilon}\subseteq\Omega^{\rho}=\set{(x,y)\in\mathbb{R}^{2}}{(x/\rho,y/\rho)\in\Omega}$.
Moreover under this set of assumptions one has
$\lambda^{*}(\Omega^{\rho})>\lambda$ for $\varepsilon$ small enough. Hence if
we consider the stable solution $u^{\rho}$ of 1.5 in $\Omega^{\rho}$ \- using
the convexity of $g$ \- we have

$\begin{cases}-\Delta(u_{\varepsilon}-u^{\rho})=\lambda\left(g(u_{\varepsilon})-g(u^{\rho})\right)\leq\lambda
g^{\prime}(u_{\varepsilon})(u_{\varepsilon}-u^{\rho})&\hbox{in}\
\Omega_{\varepsilon}\\\ u_{\varepsilon}-u^{\rho}\leq 0&\hbox{on}\
\partial\Omega_{\varepsilon}\end{cases}$

and then from the stability of $u_{\varepsilon}$ we can apply the maximum
principle to deduce $u_{\varepsilon}\leq
u^{\rho}\leq\left\|u^{\rho}\right\|_{L^{\infty}(\Omega^{\rho})}$ in
$\Omega_{\varepsilon}$.

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