#  Search engines post-ChatGPT: How generative artificial intelligence could
make search less reliable
What happens when inherently hallucinating language models are employed within
search engines without proper guardrails in place?

Shahan Ali Memon  Center for an Informed Public, Information School,
University of Washington, Seattle, USA
Email<EMAIL_ADDRESS>Jevin D. West  Center for an Informed Public,
Information School, University of Washington, Seattle, USA
Email<EMAIL_ADDRESS>
###### Abstract

In this commentary, we discuss the evolving nature of search engines, as they
begin to generate, index, and distribute content created by generative
artificial intelligence (GenAI). Our discussion highlights challenges in the
early stages of GenAI integration, particularly around factual inconsistencies
and biases. We discuss how output from GenAI carries an unwarranted sense of
credibility, while decreasing transparency and sourcing ability. Furthermore,
search engines are already answering queries with error-laden, generated
content, further blurring the provenance of information and impacting the
integrity of the information ecosystem. We argue how all these factors could
reduce the reliability of search engines. Finally, we summarize some of the
active research directions and open questions.

###### keywords:

Generative AI $|$ Search Engines $|$ Large Language Models $|$ Misinformation
$|$ Hallucinations

With the rise of generative AI (GenAI), question answering systems or
chatbots, such as ChatGPT and Perplexity AI, have rapidly become alternate
sources of information retrieval. Leading search engines have begun
experimenting and incorporating GenAI into their platforms, likely as a move
to remain relevant and competitive in the AI race. This includes You.com with
its YouChat feature, Bing’s introduction of BingChat, and Google’s launch of
the Search Generative Experience (SGE).

Figure 1: Searching controversial topics, such as abortion. (a) shows the
search engine response to the query “problems with abortion” correctly citing
the source. (b) shows the response to the query “problems with abo rt” mis-
citing the source. Search results are from November 22, 2023.

We explored some of these _generative search engines_ across divisive topics
such as vaccination, COVID-19, elections, and abortions. Notably, many of them
did not generate results for such sensitive subjects. Exploring the topic of
abortion on Google’s SGE platform, with slightly varied but semantically
similar search queries, revealed a mix of outcomes. For example, the query
“problems with aborting pregnancy” yielded no results. The query “problems
with abortion” did generate scientifically credible results citing appropriate
sources (see image 1a). Our exploration took an unexpected turn when we
introduced typos in our searches. A query “problems with abo rt”, displayed
the following erroneous claim (also shown in image 1b):

> “One study found that women who had an abortion were four times more likely
> to develop an infection and 2.4 times more likely to experience a postpartum
> hemorrhage. In some cases, infections can become severe or life-
> threatening.”

In an effort to present a coherent response, the search engine integrated two
semantically relevant but contextually erroneous texts, with some portions of
the text (shown in green) quoted verbatim from the cited article and other
segments (shown in red) generated based on our search query. The generated
response (mis)cited an article from The Texas Tribune that was in fact
discussing the risks of the expectant management after the premature rupture
of membranes in cases where the abortion was not pursued. The search results,
however, presented these statistics as risks of abortion instead. Quoting the
article directly, it stated:

> “For the women, expectant management after premature rupture of membranes
> comes with its own health risks. One study showed they were four times as
> likely to develop an infection and 2.4 times as likely to experience a
> postpartum hemorrhage, compared with women who terminated the pregnancy.”

The article went on to make a case for pro-choice.

Figure 2: Searching for benefits of nicotine. Search results are from December
4, 2023

In another example, searching for “beneficial effects of nicotine” generated a
list of benefits, including improved mood, improved concentration, and so on
(see image 2). However, the listed source actually pointed to an article
discussing why smoking is addictive. The article referenced stated:

> “When a person smokes, nicotine reaches the brain within about ten seconds.
> At first, nicotine improves mood and concentration, decreases anger and
> stress, relaxes muscles and reduces appetite. Regular doses of nicotine lead
> to changes in the brain, which then lead to nicotine withdrawal symptoms
> when the supply of nicotine decreases.”

In the two examples above, the search engine manufactured information that did
not exist in the first place. While the search responses are coherent and
linguistically relevant to the search query, the information being presented
is contextually erroneous and without clear source attribution to mistakes. In
certain cases, the information is often supported by incorrect citations.

## Generative search can make stuff up

Generative search systems are powered by a special type of GenAI known as
large language models (LLMs). Generative LLMs are statistical models of
natural language and function as sophisticated “next-word predictors”, a
capability they hone by learning from terabytes of web data and one that
allows them to produce large chunks of coherent text. Instead of storing and
retrieving information directly like in a database, LLMs internalize
information in a less direct, and more _abstract_ manner. Some even argue that
LLMs are “merely a lossy compression of all the text they are trained on”,
implying a reduction in information fidelity. However, unlike lossy
compression, these models can and often produce seemingly new content. This
does not mean that these models can _think_ or _reason_ like humans; they are
only able to combine contextually relevant words to produce seemingly coherent
and arguably original text. Quoting from the seminal paper, _On the Dangers of
Stochastic Parrots: Can Language Models Be Too Big?_ by Bender et al. [1],

> “Contrary to how it may seem when we observe its output, an LM is a system
> for haphazardly stitching together sequences of linguistic forms it has
> observed in its vast training data, according to probabilistic information
> about how they combine, but without any reference to meaning: a stochastic
> parrot.”

LLMs’ ability to stitch together sequences of words without regard to meaning
naturally leads them to _make stuff up_ , sometimes leading to undesirable or
unpredictable outputs, a phenomenon technically referred to as a
“hallucination”. Hallucination is a commonly acknowledged issue in LLMs, yet
there lacks a universally accepted definition for what constitutes a
hallucination. It is usually referred to a _factual inconsistency_ or a
_factual fabrication_ in the content generated by LLMs. Many researchers have
recently argued that hallucination is in fact an expected and desired
_feature_ of LLMs, and that in fact LLMs are always hallucinating. We concur
with this perspective and argue that the problem of hallucination cannot be
meaningfully defined on a model level. Language models are statistical models
that generate responses based on the patterns they learn from the training
data. This process inherently includes a degree of unpredictability. As such,
with the exception of rare cases of regurgitation, “the phenomenon where
generative AI models spit out training data verbatim (or near-verbatim)”, an
LLM is always hallucinating. In the words of Andrej Karpathy, a leading voice
in this space,

> _“An LLM is 100% dreaming and has the hallucination problem. A search engine
> is 0% dreaming and has the creativity problem.”_

Traditional search engines rank relevant pages. Their modus operandi is not to
generate new information. If the query were a direct question, the search
engine would display an answer box or a featured snippet at the top of the
search results, containing a direct quote from the _relevant_ information
source. The important question is, what happens when the inherently
hallucinating language models are employed within search engines without
proper guardrails in place?

In the examples above, the generative search engine _retrieved_ the pages in
the closest match to the user query, extracted the relevant information,
_augmented_ the query with the retrieved information, and used the LLM to
_generate_ a coherent response. This is a simplified explanation for what is
known as _Retrieval Augmented Generation (or RAG)_[2]. The RAG framework was
in fact proposed to minimize hallucinations in the generated content, by
anchoring the content in factual knowledge. However, evidently, it is not
impervious to hallucinations. From the example above, we observe that the
search engine decontextualizes the information from a reliable source. To
label this _mix-up_ of factual inconsistency as a _hallucination_ is to
overstate the capabilities of LLMs. The LLM-based search is not really
creating new information. It is simply missing the context. This type of error
is considered a factuality hallucination by others in the literature [3].

## Generative search systems don’t like to say ‘No’

LLMs are also known to exhibit a reluctance to indicate uncertainty[4]. In
other words, without appropriate guardrails, LLMs are more likely to _guess_
an answer instead of acknowledging their lack of knowledge by stating “I don’t
know.” In traditional search engines, users would typically encounter a _no
match found_ response to a query without relevant results. However, in the
generative search engine, the inherent nature of LLMs to guess rather than
refuse could lead to a hallucinated answer. An example of this is shown in
image 3 where a question regarding a fictitious concept, “Jevin’s theory of
social echoes,” resulted in a hallucinated yet confident response from both
the Perplexity AI and the Arc search engine, both responses supported by fake
citations – or _hallu-citations_.

Figure 3: Searching for “Jevin’s theory of social echoes” on (a) Perplexity AI
and (b) Arc Search. Almost none of this is true. Search results are from (a)
January 18, 2024 and (b) February 15, 2024.

## Generative search engines obscure the provenance of information

This brings us to one of the defining characteristics of a search engine: the
source or the _provenance_ of information. A search engine, as we know it,
optimizes for _relevance_ , and _efficiency_. These metrics have sufficed
because the primary goal of a search engine is to assist users in finding
relevant information by guiding them to the relevant web pages. In doing so,
it inherently takes into account the provenance of information. However, the
generative search engine takes on the burden of responsibility to be accurate
or verifiable as well. In not doing so or doing so haphazardly as shown in the
examples above, it puts the search user at risk. We argue that this is the
fundamental issue with the integration of GenAI in search. In the examples
above, we highlight the issue with Google’s SGE. However, verifiability is a
problem across all major generative search engines. In a recent research [5]
by Liu et al. on evaluating verfiability across four major generative search
engines (BingChat, NeevaAI, Perplexity AI, and YouChat) found that results
from these systems “frequently contain unsupported statements and inaccurate
citations: on average, a mere 51.5% of generated sentences are fully supported
by citations and only 74.5% of citations support their associated sentence.”
The issue of provenance around generative search engines has also been raised
by other researchers in the past, including Khattab et al. (2021) [6], and
Shah and Bender (2022) [7].

To portray an even grimmer reality, with the increasing prominence of GenAI,
the search engines will likely start indexing the generated content. This may
seem like a far-fetched claim, but it is not. In fact, a recent example on X
(also reproduced in image 4) demonstrated how Google search is already
indexing content generated by Quora’s AI chatbot Poe. It has also been found
that AI-generated content can occasionally appear in Google News. Google has
also helped drive massive traffic to an AI spam blogging site. More recently,
Google Scholar has been found indexing papers generated entirely by ChatGPT
[8]. For generative search engines, this is akin to _a dream within a dream_ ,
where eventually the generated content will make its way to the generative
search engine to be indexed to provide answers to the users, further obscuring
the provenance of online information.

Figure 4: An example of how generative content online may affect search engine
results. Courtesy: Delip Rao who shared this example on X. Search results are
from January 3, 2024.

## Generative search engines can reinforce biases

Figure 5: Search results from Perplexity AI for the query “gift ideas for a 7
year old girl”(a) versus “gift ideas for a 7 year old boy”(b). Search results
are from January 8, 2024.

Another major issue with LLMs, also emphasized by Bender et al., is that the
vast data they are trained on can often present a narrow and hegemonic view of
the world. These training data, sourced primarily from the Web, are filled
with biases, stereotypes, and messy societal narratives, which are then
learned and mirrored inadvertently by these language models. Research on the
evaluation of language models has repeatedly shown that these models can
exhibit gender bias [9, 10, 11, 12], racism [13], antimuslim stereotypes[14],
antisemitism [15], etc. When these models are integrated into widely used
user-facing systems, such as search engines, they unsurprisingly risk
amplifying and reinforcing these biases. In Birhane and Prabhu’s [16] words:

> “Feeding AI systems on the world’s beauty, ugliness, and cruelty, but
> expecting it to reflect only the beauty is a fantasy.”

Prior work has shown how the integration of artificial intelligence in search
has amplified biases in search results. In this regard, Noble’s documentation
of racism in search results is of notable mention [17], in which search for
“black girls”, for example, have been shown to result in pornography web
pages. The issue becomes particularly concerning when the biases are embedded
within the search results in a subtle and insidious manner. This is
problematic because they are not immediately apparent to the search users, and
are hidden within the seemingly neutral responses.

Consider, for example, a seemingly innocuous query such as “gift ideas for a 7
year old girl”. Users might not instinctively compare it to a similar search
for a boy, thus overlooking potential biases in the results. However, such a
search on Perplexity AI (shown in image 5) reveals markedly different
suggestions: educational, STEM-related items, outdoor toys, and cooperative
games or puzzles for boys, whereas arts and crafts, or make your own soap
jellies for girls111Similar results were obtained on Google search as well.

Such results, while not overtly harmful, contribute to the reinforcement of
gender stereotypes. Furthermore, while these biases existed in search before
the integration of GenAI, LLMs have the tendency to produce confident and
authoritative sounding statements, which can mislead users into actually
trusting and internalizing bias of all kinds. We expand on this issue of
_misplaced trust_ in generative search engines in a later section of this
discussion.

## The efficiency of generative search comes with the trade-off of its
reliability

Considering the aforementioned challenges, it prompts us to question the
fundamental role of a generative search engine: What key advantages does the
generative approach offer to the conventional search engine? Google’s SGE blog
gives us some perspective:

> “With new generative AI capabilities in Search, we’re now taking more of the
> work out of searching, so you’ll be able to understand a topic faster,
> uncover new viewpoints and insights, and get things done more easily.”

However, this raises the question noted by Shah and Bender (2022)[7], _“is
getting the user to a piece of relevant information as fast as possible the
only or the most important goal of a search system?”_

We argue that in their pursuit of speed and convenience, generative search
engines are, in fact, compromising the depth, diversity, and accuracy of
information — a phenomenon we refer to as the _efficiency-reliability trade-
off_. When traditional search engines organize and rank information, they
prioritize relevance and efficiency but not reliability because they are not
arbiters of truth. This is in fact a desired property of search engines, as it
makes it feasible to locate the relevant information while also offering an
array of relevant web pages for exploration and cross-verification. In the
generative age, however, when a search engine transitions from helping find
the information to answering the query, it moves from presenting the list of
web pages to synthesizing information from them. This process of synthesizing
information entails _selection_ of certain data over others, inevitably
limiting the depth and diversity of information due to constraints of
expression. Moreover, the system’s preferential treatment of certain sources
over others increases the chances of further bias. This reduction in
diversity, depth, and increased bias is unavoidable. Additionally, issues of
hallucination and lack of provenance in language models, as discussed earlier,
may compound these issues even further, at least until they are resolved.
Ultimately, these issues make the search response much less reliable.

Perplexity AI CEO Aravind Srinivas says: “If you can directly answer
somebody’s question, nobody needs those 10 blue links.” We disagree. We need
those 10+ links to assess where the information is coming from. BBC is a much
more reliable source than some random blog post.

With GenAI integration, while the reliability of the search engine results
decreases, paradoxically, its _perceived reliability_ may increase. This is
because LLMs often produce confident and authoritative text, leading users to
perceive the information as more credible. Moreover, research has also shown
that search users prioritize information at the top of search results,
especially if it provides a direct answer[18]. Users also often overestimate
the credibility of the information in the featured snippets[19]. The trust
users place in search engines also stems from the understanding that, unlike
humans with limited personal knowledge, experience, and biases, these
platforms draw from an expansive repository of diverse, structured, and
constantly updated information sources. This ability to present an extensive
range of information sources is what separates them from the more subjective
and limited scope of human knowledge. Unfortunately, with generative search,
the user’s trust in the search engine is _misplaced_ as it offers a less
diverse, more biased, hallucinated, and unverifiable, but confident-sounding
answer.

## Outlook

In Jorge Luis Borge’s Library of Babel, he describes an enormous library, with
nearly infinite collection of books that have and can ever be written. The
library contains the universe of all the information that ever existed or will
ever exist. People initially rejoiced at the thought of unlimited knowledge.
But their joy soon fizzled into frustration when they realized that despite -
or perhaps due to - the excessive abundance of information, finding relevant
and reliable information was an impossible task; the library was not just a
repository of knowledge, but also a minefield of half-truths, falsehoods, and
gibberish.

Generative AI has the potential to transform our current information ecosystem
into a contemporary sibling of the Library of Babel. In this new version,
there would be an infinite array of texts, blending truth and fabrication, but
worse, they would be stripped of their covers, thereby obscuring the
provenance and sources of information. Our door to the internet, the search
engine – akin to a digital librarian – tends to _hallucinate_ and _generate_
fabricated tales. Finding reliable information in this world is like a wild
goose chase, except that the geese are on roller skates.

Yet, search engines are swiftly advancing their efforts to incorporate GenAI.
There are multiple models evolving. One type of model (e.g., Perplexity) is a
fully question-answering system without any traditional search elements. A
second model is a hybrid of traditional search engines and question-answering
systems (e.g., Bing and Google); however, these hybrid models come in many
forms, where some, for example, prioritize the question and answer element
(e.g., You.com). Those who choose not to embrace this form of search likely
risk falling behind. The question-answering-based search engine Perplexity AI,
for example, has emerged as a viable competitor to Google, with a current
valuation of $520 million by The Wall Street Journal as of January 2024.
Although the integration of LLMs into search has many technical and ethical
challenges, an encouraging aspect is that many AI researchers are actively
working towards addressing these issues. For example, a recent survey [20]
identified 32 mitigation techniques for the problem of hallucination, all
developed in the past few years. One of these methods is _Refusal-Aware
Instruction Tuning_[21] aimed at teaching LLMs when to refrain from
responding. Bias and fairness in LLMs [22] is also an active area of research.
A notable initiative in this regard is Latimer, also known as Black GPT, which
is intended to prioritize diversity and inclusion in its training.

The changing landscape of search is also going to cause a major behavioral
shift in how users access and trust online information. Recent studies
indicate a trend toward biased querying practices in generative search engines
[23]. More research is required to understand how users’ information-seeking
behaviors change, including the types of queries. Are users’ perceptions of
the reliability of search systems changing? It is also important to understand
if these new search systems are introducing new information asymmetries; does
the LLM alignment research, predominantly focused on the Global North, affect
the equitability of search systems? Evaluation of these systems is another
critical area of research, especially in terms of the reliability and
provenance of information presented by AI-based search systems compared to
traditional search. As discussed above, LLMs often exhibit a reluctance to
indicate uncertainty[4]. In this regard, an important question is how this
alters the reliability of search responses for queries that typically return
no results on traditional search engines. Furthermore, it is likely that many
small domain-specific search engines will emerge as a result of this
evolution; examples include Consensus222Full disclosure, one of the authors of
this piece, Jevin West, is on the board of this startup. and SciSpace, which
are GenAI-powered search engines for conducting research. Exploring the role
and impact of these domain-specific search engines on the broader information
ecosystem would also be a valuable line of study.

It is important to recognize Google, and many other tech companies’ efforts in
identifying many of the raised challenges as inherent limitations of their
systems. Search engines are also increasingly refraining from displaying
generated content for sensitive queries, although there are methods to
jailbreak these safeguards as shown in the case of our abortion example. We
believe that as the technology around GenAI matures, many of these problems
will be resolved. But until that happens, we are in a _technological
liminality_ , a state of in-between, where there is great potential for
innovation but also a lot of uncertainty as the risks are not clearly
understood. The competitive landscape around GenAI has driven tech companies
to hastily deploy underdeveloped systems for public use. Maintaining vigilance
is the key to staying informed.

## Further reading

  * •

Shah and Bender, Situating search – an extremely insightful paper discussing
many of the issues we have raised in much detail, and a vision for the goals
the future search engines must accomplish.

  * •

Noble, Algorithms of oppression – an account of negative biases embedded in
search results and algorithms, especially against women of color.

  * •

Bender et al., On the Dangers of Stochastic Parrots: Can Language Models Be
Too Big? – a classic paper on risks associated with language models.

  * •

Abercrombie et al., Mirages. On Anthropomorphism in Dialogue Systems – a
recent paper discussing linguistic factors that contribute to the
anthropomorphism of dialogue systems, as well as the associated downstream
harms.

  * •

Miller, Generative Search Engines: Beware the Facade of Trustworthiness – blog
post discussing the paper on Evaluating Verifiability in Generative Search
Engines that quantifies the issue of provenance in LLMs.

  * •

Sharma et al., Generative Echo Chamber? Effects of LLM-Powered Search Systems
on Diverse Information Seeking – a recent paper on how information seeking
behaviors are changing as a result of generative search.

  * •

Khattab et al., A Moderate Proposal for Radically Better AI-powered Web Search
– blog post highlighting the issue of provenance in question-answer based web
search and potential solutions.

## About the authors

Shahan Ali Memon is a first-year doctoral student at the University of
Washington Information School. Jevin West is an associate professor at the
University of Washington Information School and co-founder of the Center for
an Informed Public.

## Acknowledgment

We would like to thank Soham De, Nic Weber, Damian Hodel, Asad Memon, Apala
Chaturvedi, Abdulaziz Yafai, and Hisham Abdus-Salam for helpful discussions
and feedback.

## References

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# Interpolation inequalities on the sphere: rigidity, branches of solutions,
and symmetry breaking

Esther Bou Dagher  E. Bou Dagher: Department of Mathematics, Imperial College
London, 180 Queen’s Gate,
London SW7 2AZ, UK<EMAIL_ADDRESS>and  Jean Dolbeault  J.
Dolbeault: CEREMADE (CNRS UMR n∘ 7534), PSL university, Université Paris-
Dauphine,
Place de Lattre de Tassigny, 75775 Paris 16, France
<EMAIL_ADDRESS>
(Date: . File: SBI.tex)

###### Abstract.

This paper is devoted to three Gagliardo-Nirenberg-Sobolev interpolation
inequalities on the sphere. We are interested in branches of optimal functions
when a scale parameter varies and investigate whether optimal functions are
constant, or not. In the latter case, a symmetry breaking phenomenon occurs
and our goal is to decide whether the threshold between symmetry and symmetry
breaking is determined by a spectral criterion or not, that is, whether it
appears as a perturbation of the constants or not. The first inequality is
classical while the two other inequalities are variants which reproduce
patterns similar to those observed in Caffarelli-Kohn-Nirenberg inequalities,
for weighted inequalities on the Euclidean space. In the simpler setting of
the sphere, it is possible to implement a parabolic version of the entropy
methods associated to nonlinear diffusion equations, which is so far an open
question on weighted Euclidean spaces.

###### Key words and phrases:

Interpolation, Gagliardo-Nirenberg-Sobolev inequalities, best constants, heat
equation, nonlinear diffusion, entropy methods, carré du champ, bifurcations,
symmetry, symmetry breaking, uniqueness, rigidity

###### 2020 Mathematics Subject Classification:

Primary: 58J35,35B06 Secondary: 26D10, 35K55, 53C21.

##  1\. Introduction

In this paper we consider some interpolation inequalities on the sphere which
can be written in generic form as

$F\left(\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2},\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)\geq\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}\quad\forall\,u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$
(1)

where $p>2$ has to be specified. Here the function
$F:({\mathbb{R}}^{+})^{2}\to{\mathbb{R}}$ is a two-homogeneous function,
_i.e._ ,

$F(h\,\mathsf{a},h\,\mathsf{b})=h^{2}\,F(\mathsf{a},\mathsf{b})\quad\forall\,(h,\mathsf{a},\mathsf{b})\in({\mathbb{R}}^{+})^{3}\,,$

and we assume that $F(0,1)=1$. We will not give general assumptions of
regularity on $F$ as we shall rather focus on three particular cases that
illustrate the variety of possible cases. With standard notations,
$d\sigma=\big{|}\mathbb{S}^{d}\big{|}^{-1}\,dv_{g}$ denotes the uniform
probability measure on the unit sphere
${\mathbb{S}}^{d}\subset{\mathbb{R}}^{d+1}$ induced by Lebesgue’s measure on
${\mathbb{R}}^{d+1}$. The space $\mathrm{L}^{q}({\mathbb{S}}^{d},d\sigma)$
with $q\in[1,+\infty)$ is the standard Lebesgue space with norm
$\|{\cdot}\|_{\mathrm{L}^{q}({\mathbb{S}}^{d})}$ defined by

$\|{u}\|_{\mathrm{L}^{q}({\mathbb{S}}^{d})}:=\left(\int_{{\mathbb{S}}^{d}}{|u|^{q}}\,d\sigma\right)^{1/q}.$

The fact that $d\sigma$ is a probability measure means that
$\|{u}\|_{\mathrm{L}^{q}({\mathbb{S}}^{d})}=1$ if $u=1$ a.e. on
${\mathbb{S}}^{d}$ and we also have
$\|{u}\|_{\mathrm{L}^{q_{1}}({\mathbb{S}}^{d})}\leq\|{u}\|_{\mathrm{L}^{q_{2}}({\mathbb{S}}^{d})}$
for any $u\in\mathrm{L}^{q_{2}}({\mathbb{S}}^{d},d\sigma)$ as soon as $1\leq
q_{1}\leq q_{2}$. The space $\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$ is
obtained by completion of smooth functions on ${\mathbb{S}}^{d}$ with respect
to the norm $u\mapsto\big{(}\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\big{)}^{1/2}$.

If (1) is an optimal inequality, and if there is some
$u_{\star}\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$ such that equality
holds in (1), the question that we address is the _symmetry_ versus _symmetry
breaking_ issue: is $u_{\star}$ necessarily a constant function (symmetry) or
not (symmetry breaking). In this last case, there is a manifold of non-
constant optimal functions generated by all possible rotations. We also
consider limit cases for which compactness is lost and there is no optimal
function (or a sequence of functions approximating the equality case is not
compact even after normalization): there is _symmetry breaking_ if the
optimality case is not achieved by constant functions.

As a side question, let us mention the question of the _rigidity_ in related
elliptic equations: up to a normalization by a multiplicative constant, is
there at most one nonnegative solution to the Euler-Lagrange equation

$-\,\partial_{\mathsf{a}}F(\mathsf{a},\mathsf{b})\,\Delta
u+\partial_{\mathsf{b}}F(\mathsf{a},\mathsf{b})\,u=\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2-p}\,u^{p-1}\quad\mbox{with}\quad\mathsf{a}=\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\quad\mbox{and}\quad\mathsf{b}=\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}$

or not. Here $\Delta$ denotes the Laplace-Beltrami operator on
${\mathbb{S}}^{d}$ with the convention that $-\Delta$ is a nonnegative
operator. In case of uniqueness and compactness (normalized minimizing
sequences for (1) are relatively compact and the limits after extraction of a
subsequence solves the equation), then _symmetry_ holds and $u=1$ a.e. is the
unique solution, while in the _symmetry breaking_ case, there is no rigidity
(notice that if $u$ is optimal for (1), then $|u|$ is also optimal).

In order to study the symmetry breaking phenomenon, we shall introduce a
_bifurcation_ parameter $\lambda>0$ and families of functions
$(F_{\lambda})_{\lambda>0}$ depending continuously on $\lambda$ with
$F_{\lambda}(0,1)=\lambda$. Our goal is to characterize the optimal constant
$\mu(\lambda)$ in the inequality

$F_{\lambda}\left(\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2},\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)\geq\mu(\lambda)\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}\quad\forall\,u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$
(2)

so that _symmetry_ means $\mu(\lambda)=\lambda$ while _symmetry breaking_ is
characterized by $\mu(\lambda)<\lambda$. On ${\mathbb{S}}^{d}$, there is an
intrinsic length scale, which compares with $\lambda$. The three families of
_Gagliardo-Nirenberg-Sobolev inequalities_ on ${\mathbb{S}}^{d}$ studied in
this paper are

$\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{\lambda}{p-2}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\geq\frac{\mu_{0}(p,\lambda)}{p-2}\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}\,,$
(3) $\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{\lambda}{p-2}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\geq\frac{\mu_{1}(p,\theta,\lambda)}{p-2}\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2\,\theta}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}\,,$
(4) $\left(\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{\lambda}{p-2}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)^{\theta}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}\geq\left(\frac{\mu_{2}(p,\theta,\lambda)}{p-2}\right)^{\theta}\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}\,,$
(5)

for any $u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$, where $\lambda>0$ is a
parameter while $\mu_{0}(p,\lambda)$ and $\mu_{k}(\theta,\lambda)$ with $k=1$,
$2$ are the optimal constants. Although the case $p\in[1,2)$ makes sense, we
shall for simplicity restrict our purpose to $p\in(2,\infty)$ if $d=1$, $2$
and $p\in(2,2^{*}]$ if $d\geq 3$, with $2^{*}=2\,d/(d-2)$. By convention, we
take $2^{*}=+\infty$ if $d=1$, $2$. The exponent

$\theta_{\star}:=d\,\frac{p-2}{2\,p}$ (6)

is the exponent in the Euclidean Gagliardo-Nirenberg-Sobolev inequalities

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{\theta_{\star}}\,\left\|{u}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{1-\theta_{\star}}\geq\mathcal{C}_{\mathrm{GNS}}\,\left\|{u}\right\|_{\mathrm{L}^{p}({\mathbb{R}}^{d})}^{2}\quad\forall\,u\in\mathrm{H}^{1}({\mathbb{R}}^{d},dx)$
(7)

and induces the limitation $\theta\geq\theta_{\star}$ in (5). As we shall see
later, the regime for $\lambda$ small corresponds to symmetry while the
asymptotic regime as $\lambda\to+\infty$ is related with optimal functions for
(7) on the tangent Euclidean space ${\mathbb{R}}^{d}$, which induce symmetry
breaking.

Let us give a brief account of the literature. M.-F. Bidault-Véron and L.
Véron proved Inequality (3) in [5, Corollary 6.1] as a consequence of a
rigidity result. Another proof by W. Beckner in [4] relies on spectral
methods, the Funk-Hecke formula and a duality result introduced by E. Lieb in
[27]. An earlier version corresponding to the range $p\in(2,2^{\\#})$ with
$2^{\\#}=(2\,d^{2}+1)/(d-1)^{2}$ if $d\geq 2$ and $2^{\\#}=+\infty$ if $d=1$
was established by D. Bakry and M. Emery in [1, 2], using the _carré du champ_
method and the heat flow. We refer to [3] for a general overview of the _carré
du champ_ method in the context of Markov processes or linear diffusion
equations, and to [9, 17] for the extension to nonlinear diffusion equations.
Also see Appendix A for further references in the case of the sphere.

Inequalities (4) and (5) are less standard than (3) and we are not aware of
specific references. The case $\theta=1$ in (4) and (5) corresponds to the
classical interpolation inequality (3). Inequality (5) appears (with optimal
constant) for a special value of $\theta$ in [12, Inequality (2.4)] as a
consequence of improvements of (3) based on the _carré du champ_ method. For a
presentation of _entropy methods_ and _improved interpolation inequalities_ ,
see [12] and references therein.

An important motivation for (5) arises from pure states in Lieb-Thirring
estimates and interpolation inequalities for systems studied in [8]. We refer
to [22] for an overview of functional inequalities and branches of solutions
in various frameworks, including the quite similar problem of _symmetry
breaking_ in some of the (CKN) _Caffarelli-Kohn-Nirenberg inequalities_. The
sharp threshold for the symmetry range of the parameters in such inequalities
is established in [25, 19] in the critical case and in [7, 23] in the
subcritical case. A complete parabolic proof based on entropy methods is so
far missing and results rely on _rigidity_ methods: see [18, 24] for partial
results based on flows. In the case of the sphere, Inequalities (3), (4) and
(5) have similar properties but by many aspects symmetry _versus_ symmetry
breaking issues are simpler. For instance, the regularity of the solution of
the nonlinear flow on ${\mathbb{S}}^{d}$ raises no difficulty: see for
instance [28] for some comments in this direction. Branches of optimizers
associated with Inequalities (4) and (5) for $\theta<1$ have qualitative
features which are reminiscent of those corresponding to similar (CKN)
inequalities (with also some $\theta<1$) observed in [11, 10, 13]. Beyond
numerical observations and a few estimates, these branches for (CKN) are not
well understood. Our goal is to state the corresponding properties on the
sphere.

This paper is organized as follows. In Section 2, we expose results and
strategies of proof for Inequality (3). Sections 3 and 4 are devoted to
similar properties for Inequalities (4) and (5) with a special emphasis on
symmetry and symmetry breaking. In the Appendix A, for completeness, we
provide some computations based on the _carré du champ_ method applied with a
nonlinear flow in the case of the sphere and further bibliographical
references for this technique.

##  2\. Classical Gagliardo-Nirenberg-Sobolev inequalities on
${\mathbb{S}}^{d}$

In this preliminary section, we review some known results for Inequality (3)
and give some hints for the proofs which will guide us in the study of the two
other inequalities.

######  Theorem 1.

Let $d\geq 1$, $p\in(2,2^{*})$ or $p=2^{*}$ if $d\geq 3$. Inequality (3) holds
with $\mu_{0}(p,\lambda)=\lambda$ if $\lambda\leq d$ (symmetry case) and
$\mu_{0}(p,\lambda)<\lambda$ if $\lambda>d$ (symmetry breaking case).

A standard observation is that $\lambda\mapsto\mu_{0}(p,\lambda)$ is a concave
increasing function as a consequence of

$\mu_{0}(p,\lambda)=\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus\\{0\\}}\frac{(p-2)\,\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}}$

and that the symmetry breaking point
$d=\inf\big{\\{}\lambda>0\,:\,\mu_{0}(p,\lambda)<\lambda\big{\\}}$ is such
that

$d=\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus{\mathbb{R}}}\frac{(p-2)\,\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}-\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}\,,$

where $\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus{\mathbb{R}}$ denotes
the set of non-constant functions in
$\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$.

Let us sketch the proof of Theorem 1.
$\bullet$ Symmetry breaking. Let us consider the functional

$\mathcal{F}_{\lambda}[u]:=\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}-\frac{\lambda}{p-2}\left(\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}-\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)\,.$
(8)

A Taylor expansion of $\mathcal{F}_{\lambda}[1+\varepsilon\,\varphi]$ at order
two in $\varepsilon$ for a function $\varphi$ such that
$\int_{{\mathbb{S}}^{d}}{\varphi}\,d\sigma=0$ shows that

$\lim_{\varepsilon\to
0}\frac{1}{\varepsilon^{2}}\,\mathcal{F}_{\lambda}[1+\varepsilon\,\varphi]=\|{\nabla\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}-\lambda\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\,.$

Taking for $\varphi$ an eigenfunction of the Laplace-Beltrami operator on
${\mathbb{S}}^{d}$ associated with its lowest positive eigenvalue, _i.e._ ,
such that $-\Delta\varphi=d\,\varphi$, we find that

$\|{\nabla\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}-\lambda\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}=(d-\lambda)\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}<0$

for any $\lambda>d$, which shows that $\mu_{0}(p,\lambda)<\lambda$.
$\bullet$ Symmetry. Let us consider the fast diffusion equation

$\frac{\partial\rho}{\partial
t}=\Delta\rho^{m}\,,\quad\rho(t=0,\cdot)=|u|^{p}$ (9)

for some parameter $m\in\big{[}m_{-}(d,p),m_{+}(d,p)\big{]}$ where

$m_{\pm}(d,p):=\frac{1}{(d+2)\,p}\left(d\,p+2\pm\sqrt{d\,(p-1)\,\big{(}2\,d-(d-2)\,p\big{)}}\,\right).$
(10)

A computation based on a nonlinear extension of the _carré du champ method_
shows that

$\frac{d}{dt}\mathcal{F}_{\lambda}[v(t,\cdot)]\leq-\,2\,\beta^{2}\,(d-\lambda)\int_{{\mathbb{S}}^{d}}{\big{|}\nabla
v^{1/\beta}\big{|}^{2}}\,d\sigma$

where $\beta:=2/\big{(}2-p\,(1-m)\big{)}$. References are given in Appendix A
and the key computation is detailed in Appendix A.1. Hence

$\frac{d}{dt}\mathcal{F}_{\lambda}[v(t,\cdot)]\leq
0\quad\mbox{and}\quad\lim_{t\to+\infty}\mathcal{F}_{\lambda}[v(t,\cdot)]=0$

if $\lambda\leq d$. This proves that $\mathcal{F}_{\lambda}[u]\geq 0$ for an
arbitrary $u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$. See Lemma 14 and
final comments at the end of Section A.2 for further justifications.∎

$\bullet$ Rigidity. As in [5, 6, 17], a remarkable consequence of the proof is
the following result.

######  Corollary 2.

Let $d\geq 1$, $p\in(2,2^{*})$ or $p=2^{*}$ if $d\geq 3$. If $0<\lambda\leq
d$, the equation

$-\,\Delta u+\frac{\lambda}{p-2}\,u=u^{p-1}$ (11)

admits a unique nonnegative solution, which is constant.

The proof is consequence of the _carré du champ method_ based on an idea that
goes back to [26].

$\bullet$ Asymptotics. Another piece of information is the asymptotic behavior
of the branch $\lambda\mapsto\mu_{0}(p,\lambda)$. The following result is
taken from [16, Proposition 10].

######  Proposition 3.

Let $d\geq 1$ and $p\in(2,2^{*})$. The optimal constant in (3) is such that

$\lim_{\lambda\to+\infty}\frac{\mu_{0}(p,\lambda)}{\lambda^{\theta_{\star}}}=\frac{\big{|}{\mathbb{S}}^{d}\big{|}^{-\frac{p-2}{p}}}{\theta_{\star}^{\theta_{\star}}\,(1-\theta_{\star})^{1-\theta_{\star}}}\,\mathcal{C}_{\mathrm{GNS}}\,.$

If $d\geq 3$ and $p=2^{*}$, then $\mu_{0}(p,\lambda)=\min\\{\lambda,\,d\\}$ is
not achieved in $\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$ for any
$\lambda>d$.

As $\lambda\to+\infty$, the corresponding optimal functions concentrate on
points of the sphere. A blow-up analysis shows that the projection on the
tangent Euclidean space at a blow-up point can be estimated by (7), and
optimal functions for (7) are good test functions, which allows to identify
the limit. Notice that concentration is incompatible with symmetry, so that we
recover symmetry breaking in the limit as $\lambda\to+\infty$. In the case
$d\geq 3$ and $p=2^{*}$, loss of compactness occurs and a minimizing sequence
behave like concentrating Aubin-Talenti bubbles.

$\bullet$ Reparametrization. The following observation is elementary. Let us
consider the optimal constants for Inequalities (4) and (5) seen as
variational problems, namely

$\displaystyle\mu_{1}(p,\theta,\lambda)=\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus\\{0\\}}\frac{(p-2)\,\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2\,\theta}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}}\,,$
(12)
$\displaystyle\mu_{2}(p,\theta,\lambda)=\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus\\{0\\}}\left((p-2)\,\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{\frac{2}{\theta}-2}\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{-\frac{2}{\theta}}\,.$
(13)

As already noted in Section 1, we know for free that

$\mu_{1}(p,1,\lambda)=\mu_{2}(p,1,\lambda)=\mu_{0}(p,\lambda)$

for any $\lambda>0$. The parameter $\theta$ is subject to _admissibility_
conditions that will be specified in Sections 3 and 4. Even for $\theta\neq
1$, Inequalities (4) and (5) are related with Inequality (3) as follows.

######  Proposition 4.

Let $d\geq 1$, $p\in(2,2^{*})$ and $\Lambda>0$, given. If for some admissible
$\theta$ and either $k=1$ or $k=2$, $U$ is a non-trivial minimizer for
$\mu_{k}(\theta,\Lambda)$, then for an appropriate choice of $\kappa>0$, the
function $u=\kappa\,U$ solves (11) for some $\lambda>0$. The two parameters
$\kappa$ and $\lambda$ are explicitly given in terms of
$\mu_{k}(\theta,\Lambda)$, $\|{U}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}$ and
$\|{U}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}$.

###### Proof.

Let us consider the case $k=1$. The function $U$ solves the Euler-Lagrange
equation

$-\,\Delta
U+\frac{1}{p-2}\left(\Lambda-(1-\theta)\,\mu_{1}(p,\theta,\Lambda)\,X^{2\,\theta}\right)U=\frac{\mu_{1}(p,\theta,\Lambda)\,X^{2\,\theta-2}}{(p-2)\,\|{U}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{p-2}}\,U^{p-1}$

with
$X=\|{U}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}/\|{U}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}$.
The proof is complete with the choice

$\lambda=\frac{\Lambda}{p-2}-(1-\theta)\,\mu_{1}(p,\theta,\Lambda)\,X^{2\,\theta}\quad\mbox{and}\quad\kappa^{2-p}=\frac{\mu_{1}(p,\theta,\Lambda)\,X^{2\,\theta-2}}{(p-2)\,\|{U}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{p-2}}\,.$

A similar reparametrization holds for $k=2$.∎

##  3\. A first family of refined interpolation inequalities on
${\mathbb{S}}^{d}$

Let us define

$\mathcal{Q}_{p,\theta,\lambda}^{(1)}[u]:=\frac{(p-2)\,\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2\,\theta}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}}\,.$

$\bullet$ Admissible parameters. We establish the validity range of the
parameters in (4).

######  Lemma 5.

Inequality (4) holds for any $d\geq 1$, $\theta\in[0,1]$, $p\in(2,2^{*})$ or
$p=2^{*}$ if $d\geq 3$, with

$\mu_{0}(p,\lambda)\leq\mu_{1}(p,\theta,\lambda)\leq\lambda\quad\forall\,\lambda>0\,.$

Moreover, equality holds and $\mu_{1}(p,\theta,\lambda)=\lambda$ if
$\lambda\leq d$.

###### Proof.

By Hölder’s inequality,
$\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}\leq\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}$
because $d\sigma$ is a probability measure. Using (12), we learn that

$\mu_{0}(p,\lambda)=\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus\\{0\\}}\mathcal{Q}_{p,1,\lambda}^{(1)}[u]\leq\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus\\{0\\}}\mathcal{Q}_{p,\theta,\lambda}^{(1)}[u]=\mu_{1}(p,\theta,\lambda)\leq\lambda\,,$

where the upper bound is obtained by testing
$\mathcal{Q}_{p,\theta,\lambda}^{(1)}$ with $u=1$ a.e. If $\lambda\leq d$,
there is equality because $\mu_{0}(p,\lambda)=\lambda$ (symmetry case)
according to Theorem 1.∎

We read from Lemma 5 that (4) is stronger than (3) and this is why we call
Inequality (4) a _refined Gagliardo-Nirenberg-Sobolev inequality_. Inequality
(4) can also be seen as an _improved_ inequality compared to (3) or a
_stability result_ for (3): see [12] for further considerations on these
issues and an example corresponding to a special choice of $\theta$ in (4).

As in Section 2, let us consider

$\mathcal{F}_{\lambda}^{(1)}[u]:=\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{\lambda}{p-2}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}-\frac{\lambda}{p-2}\left(\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2\,\theta}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}\right)\,.$

$\bullet$ Symmetry breaking. As in the case of (3), we obtain an estimate of
the symmetry breaking range by Taylor expanding around a constant with a
perturbation by a spherical harmonic function.

######  Lemma 6.

Let $d\geq 1$, $\theta\in[0,1]$, $p\in(2,2^{*})$ or $p=2^{*}$ if $d\geq 3$.
There is _symmetry breaking_ in (4), that is,
$\mu_{1}(p,\theta,\lambda)<\lambda$, if $\lambda>d/\theta$.

###### Proof.

Let us consider $\varphi\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$ such that
$\int_{{\mathbb{S}}^{d}}{\varphi}\,d\sigma=0$,
$u_{\varepsilon}=1+\varepsilon\,\varphi$, and compute

$\displaystyle\left\|{\nabla
u_{\varepsilon}}\right\|_{2}^{2}=\varepsilon^{2}\,\|{\nabla\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+o(\varepsilon^{2})\,,$
$\displaystyle\left\|{u_{\varepsilon}}\right\|_{2}^{2}=1+\varepsilon^{2}\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+o(\varepsilon^{2})\,,$
$\displaystyle\left\|{u_{\varepsilon}}\right\|_{p}^{2}=1+\varepsilon^{2}\,(p-1)\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+o(\varepsilon^{2})\,.$

As a consequence, we obtain that

$\lim_{\varepsilon\to
0}\frac{1}{\varepsilon^{2}}\mathcal{F}_{\lambda}^{(1)}[u_{\varepsilon}]=\|{\nabla\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}-\lambda\,\theta\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}=(d-\lambda\,\theta)\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}<0$

if we take a non-trivial function $\varphi$ such that
$-\Delta\varphi=d\,\varphi$ and $\lambda>d/\theta$.∎

$\bullet$ Symmetry. Up to a change in the range of $\lambda$, the result and
its proof are similar to the case studied in Section 2.

######  Lemma 7.

Let $d\geq 1$, $\theta\in[0,1]$, $p\in(2,2^{*})$ or $p=2^{*}$ if $d\geq 3$.
There is _symmetry_ in (4), that is, $\mu_{1}(p,\theta,\lambda)=\lambda$, if
$\lambda\leq d/\theta$.

###### Proof.

If $d=1$, one can consider a periodic interval identified with
${\mathbb{S}}^{1}$ and refer to [2] for applying the _carré du champ_ method
in this elementary setting. Assume that $d\geq 2$ and let us consider again
the fast diffusion equation (9) with $m\in\big{[}m_{-}(d,p),m_{+}(d,p)\big{]}$
where $m_{\pm}(d,p)$ is defined by (10). The same computation as in the proof
of Theorem 1 shows that

$\frac{d}{dt}\mathcal{F}_{\lambda}^{(1)}[v(t,\cdot)]\leq-\,2\,\beta^{2}\,\Big{(}d+\lambda\,\big{(}(1-\theta)\,\mathsf{x}-1\big{)}\Big{)}\int_{{\mathbb{S}}^{d}}{\big{|}\nabla
v^{1/\beta}\big{|}^{2}}\,d\sigma\quad\mbox{with}\quad\mathsf{x}:=\frac{\|{v}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2\,\theta}}{\|{v}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,\theta}}\,.$

By a Hölder inequality, we learn that $\mathsf{x}\geq 1$ and are then back to
the same arguments as in the case $\theta=1$, but now under the condition
$\lambda\leq d/\theta$. ∎

$\bullet$ Asymptotics. Heuristically, large values of $\lambda$ correspond to
highly concentrated optimal functions, which behave like the functions
obtained by a _stereographic projection_ on the tangent Euclidean space at the
point of concentration. Let us explain how we define and use the stereographic
projection. On ${\mathbb{R}}^{d}\ni x$, let $r=|x|$ and $\omega=x/|x|$ denote
spherical coordinates. On the unit sphere
${\mathbb{S}}^{d}\subset{\mathbb{R}}^{d+1}$, we consider cylindrical
coordinates $(\rho\,\omega,z)\in{\mathbb{R}}^{d}\times(-1,1)$ with
$\rho^{2}+z^{2}=1$. The _stereographic projection_
$\mathsf{S}:{\mathbb{S}}^{d}\setminus\\{\mathsf{N}\\}\to{\mathbb{R}}^{d}$,
where $\mathsf{N}\in{\mathbb{S}}^{d}$ is the North Pole defined by $z=+1$, is
such that $\mathsf{S}(\rho\,\omega,z)=r\,\omega=x$ where

$z=\frac{r^{2}-1}{r^{2}+1}\quad\mbox{and}\quad\rho=\frac{2\,r}{1+r^{2}}\,.$

If $v$ is a function on ${\mathbb{R}}^{d}$, let us consider its counterpart
$u:=\mathsf{S}^{-1}\,v$ on ${\mathbb{S}}^{d}$ obtained using the inverse
stereographic projection as

$\left(u\circ\mathsf{S}^{-1}\right)(x)=\mathsf{m}(r)^{d-2}\,v(x)\quad\forall\,x\in{\mathbb{R}}^{d}\quad\mbox{with}\quad\mathsf{m}(r)=\sqrt{(1+r^{2})/2}\,.$

For any $q\geq 1$, with $\delta(q):=2\,d-q\,(d-2)$, we obtain

$\displaystyle\int_{{\mathbb{S}}^{d}}{|u|^{q}}\,d\sigma=\big{|}{\mathbb{S}}^{d}\big{|}^{-1}\int_{{\mathbb{R}}^{d}}{\frac{|v|^{q}}{\mathsf{m}(r)^{\delta(q)}}}\,dx\,,$
$\displaystyle\int_{{\mathbb{S}}^{d}}{|\nabla
u|^{2}}\,d\sigma+\frac{1}{4}\,d\,(d-2)\int_{{\mathbb{S}}^{d}}{|u|^{2}}\,d\sigma=\big{|}{\mathbb{S}}^{d}\big{|}^{-1}\int_{{\mathbb{R}}^{d}}{|\nabla
v|^{2}}\,dx\,.$

Notice that $\delta(2)=4$. We refer to [12, Theorem 2.1] for a statement
concerning the inequality obtained from (3) and a special case of (4) by the
stereographic projection.

As another observation, we shall use the optimal constant $\mathcal{K}_{p,d}$
in the Gagliardo-Nirenberg-Sobolev inequality on ${\mathbb{R}}^{d}$

$\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}+\|{u}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}\geq\mathcal{K}_{p,d}\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{R}}^{d})}^{2}\quad\forall\,u\in\mathrm{H}^{1}({\mathbb{R}}^{d},dx)\,.$
(14)

An optimization under scaling (apply the inequality to
$u_{h}(x)=h^{d/p}\,(h\,x)$ and optimize its left-hand side with respect to
$h>0$) shows that this inequality is equivalent to (7) and, with
$\theta_{\star}$ as in (6),

$\theta_{\star}^{\theta_{\star}}\,(1-\theta_{\star})^{1-\theta_{\star}}\mathcal{K}_{p,d}=\mathcal{C}_{\mathrm{GNS}}\,.$

After these preliminaries, we can state the following result, which extends
the results for (3) of [16, Proposition 9 and 10] to Inequality (4).

######  Lemma 8.

Let $d\geq 1$, $\theta\in[0,1]$, $p\in(2,2^{*})$ or $p=2^{*}$ if $d\geq 3$.
There is an explicit constant $\mu_{1}^{\infty}(p,\theta)>0$ such that

$\mu_{1}(p,\theta,\lambda)=\mu_{1}^{\infty}(p,\theta)\,\lambda^{\gamma}\,\big{(}1+o(1)\big{)}\quad\mbox{as}\quad\lambda\to+\infty$

where $\gamma=1-d\,(p-2)\,\theta/2p=1-\,\theta\,\theta_{\star}$.

An explicit expression of $\mu_{1}^{\infty}$ can be deduced from the proof.

###### Proof.

We do a _blow-up analysis_ based on a change of variables depending on
$\lambda\to+\infty$. Let $u_{\lambda}:=\mathsf{S}^{-1}\,v_{\lambda}$ where

$v_{\lambda}(x):=v\left(2\,\sqrt{\tfrac{(\lambda-\lambda_{\star})}{p-2}}\,x\right)\quad\forall\,x\in{\mathbb{R}}^{d}\,.$

We take $\lambda_{\star}:=\frac{1}{4}\,d\,(d-2)\,(p-2)$ and $v$ as an optimal
function for (14). With $r=|x|$, we compute

$\displaystyle\mathcal{Q}_{p,\theta,\lambda}^{(1)}[u_{\lambda}]$
$\displaystyle=\frac{(p-2)\left(\|{\nabla
u_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{1}{4}\,d\,(d-2)\,\|{u_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)+(\lambda-\lambda_{\star})\,\|{u_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u_{\lambda}}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2\,\theta}\,\|{u_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}}$
$\displaystyle=\frac{(p-2)\,\|{\nabla
v_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}+(\lambda-\lambda_{\star})\,\|{\mathsf{m}(r)^{-2}\,v_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}}{|{\mathbb{S}}^{d}|^{\theta\,(1-2/p)}\,\|{\mathsf{m}(r)^{-\delta(p)/p}\,v_{\lambda}}\|_{\mathrm{L}^{p}({\mathbb{R}}^{d})}^{2\,\theta}\,\|{\mathsf{m}(r)^{-2}\,v_{\lambda}}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2\,(1-\theta)}}$
$\displaystyle=\frac{(\lambda-\lambda_{\star})\left(\tfrac{4\,(\lambda-\lambda_{\star})}{p-2}\right)^{-\,d\,\frac{p-2}{2\,p}\,\theta}\,\left(4\,\|{\nabla
v}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}+\,\|{\mathsf{m}(\varepsilon\,r)^{-2}\,v}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}\right)}{|{\mathbb{S}}^{d}|^{\theta\,(1-2/p)}\,\|{\mathsf{m}(\varepsilon\,r)^{-\delta(p)/p}\,v}\|_{\mathrm{L}^{p}({\mathbb{R}}^{d})}^{2\,\theta}\,\|{\mathsf{m}(\varepsilon\,r)^{-2}\,v}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2\,(1-\theta)}}\,.$

with $\varepsilon=\sqrt{\frac{p-2}{4\,(\lambda-\lambda_{\star})}}$. By taking
the limit as $\lambda\to+\infty$ and using Lebesgue’s theorem of dominated
convergence with $\mathsf{m}(\varepsilon\,r)\to 1/\sqrt{2}$ a.e., and by using
the fact that the quotient is proportional to the Gagliardo-Nirenberg-Sobolev
inequality up to a numerical constant, we conclude that

$\mathcal{Q}_{p,\theta,\lambda}^{(1)}[u_{\lambda}]=(\lambda-\lambda_{\star})^{1-\theta\theta_{\star}}\mu_{1}^{\infty}(p,\theta)\big{(}1+o(1)\big{)}$

as $\lambda\to+\infty,$ for a constant $\mu_{1}^{\infty}(p,\theta)$ that can
be computed explicitly, up to tedious but elementary considerations. This
proves that

$\lim_{\lambda\to+\infty}\lambda^{-\gamma}\,\mu_{1}(p,\theta,\lambda)\leq\mu_{1}^{\infty}(p,\theta).$

We have now to prove the reverse inequality. As in [16, Proposition 10], we
argue by contradiction. Let $({\lambda}_{n})_{n\in{\mathbb{N}}}$ and
$({u}_{n})_{n\in{\mathbb{N}}}$ be such that
$u_{n}\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$ with
$\|{u_{n}}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}=1$ for any $n\in{\mathbb{N}}$,
$\lim_{n\to+\infty}\lambda_{n}=+\infty$,

$\mathcal{Q}_{p,\theta,\lambda_{n}}^{(1)}[u_{n}]=\mu_{1}(p,\theta,\lambda_{n})\quad\mbox{and}\quad\lim_{n\to+\infty}\lambda_{n}^{-\gamma}\,\mu_{1}(p,\theta,\lambda_{n})\leq\mu_{1}^{\infty}(p,\theta)-\eta$
(15)

for some $\eta>0$. We learn from the expression of
$\mathcal{Q}_{p,\theta,\lambda}^{(1)}$ that
$\lambda_{n}\,\|{u_{n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\theta}\leq\mu_{1}^{\infty}(p,\theta)\,\lambda_{n}^{\gamma}$
as $n\to+\infty$ so that
$\lim_{n\to+\infty}\|{u_{n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}=0$.
Concentration occurs: there exists an eventually finite sequence
$({y}_{i})_{i\in{\mathbb{N}}}$ of points in ${\mathbb{S}}^{d}$, sequences of
positive numbers $({\zeta}_{i})_{i\in{\mathbb{N}}}$ and
$(r_{i,n})_{i,n\in{\mathbb{N}}}$ and functions
$u_{i,n}\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$ with

$u_{i,n}=u_{n}\quad\mbox{on}\quad{\mathbb{S}}^{d}\cap
B(y_{i},r_{i,n})\quad\mbox{and}\quad\mbox{supp
}u_{i,n}\subset{\mathbb{S}}^{d}\cap B(y_{i},2\,r_{i,n})$

such that, as $n\to+\infty$,

$\sum_{i}\|{\nabla
u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\sim\|{\nabla
u_{n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\quad\mbox{and}\quad\sum_{i}\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\sim\|{u_{n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\,,$

with, for all $i$,

$\lim_{n\to+\infty}r_{i,n}=0\,,\quad\Sigma_{i\in{\mathbb{N}}}\,\zeta_{i}=1\
\quad\mbox{and}\quad\lim_{n\to+\infty}\int_{{\mathbb{S}}^{d}\cap
B(y_{i},r_{i,n})}|u_{i,n}|^{p}\,d\sigma=\zeta_{i}\,.$

With a _blow up_ argument applied to $(u_{i,n})_{n\in{\mathbb{N}}}$, we prove
for all $i$ that

$\lim_{n\to+\infty}\lambda_{n}^{-\gamma}\left((p-2)\,\|{\nabla
u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda_{n}\,\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{-2\,(1-\theta)}\geq\mu_{1}^{\infty}(p,\theta)\,\zeta_{i}^{2\,\theta/p}\,.$

We may notice that

$\frac{(p-2)\,\|{\nabla
u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda_{n}\,\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}}\geq\frac{(p-2)\,\|{\nabla
u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda_{n}\,\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u_{n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}}\,.$

Let us choose an integer $N$ such that

$\Sigma_{i=1}^{N}\zeta_{i}^{2\,\theta/p}\geq\left(\Sigma_{i=1}^{N}\zeta_{i}\right)^{2\,\theta/p}>1-\frac{\eta}{2}\,\big{(}\mu_{1}^{\infty}(p,\theta)\big{)}^{-1}\,.$

For $n$ large enough, by writing

$\lambda_{n}^{-\gamma}\,\sum_{i=1}^{N}\frac{(p-2)\,\|{\nabla
u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\lambda_{n}\,\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}{\|{u_{i,n}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}}\geq\mu_{1}^{\infty}(p,\theta)\sum_{i=1}^{N}\zeta_{i}^{2\,\theta/p}\,\big{(}1+o(1)\big{)}\geq\mu_{1}^{\infty}(p,\theta)-\frac{\eta}{2}\,,$

we obtain a contradiction with (15). This concludes the proof. ∎

We collect our results on inequality (4) with optimal constant
$\mu_{1}(p,\theta,\lambda)$ as follows.

######  Theorem 9.

Let $d\geq 1$, $\theta\in[0,1]$, $p\in(2,2^{*})$ or $p=2^{*}$ if $d\geq 3$.
Inequality (3) holds with $\mu_{0}(p,\lambda)=\lambda$ if $\lambda\leq
d/\theta$ (symmetry case) and $\mu_{0}(p,\lambda)<\lambda$ if
$\lambda>d/\theta$ (symmetry breaking case). The function
$\lambda\mapsto\mu_{1}(p,\theta,\lambda)$ is monotone increasing, concave and
such that for some explicit $\mu_{1}^{\infty}(p,\theta)>0$, we have
$\mu_{1}(p,\theta,\lambda)\sim\mu_{1}^{\infty}(p,\theta)\,\lambda^{\gamma}$ as
$\lambda\to+\infty$ with $\gamma=1-2\,\theta\,\theta_{\star}$.

##  4\. A second family of refined interpolation inequalities on
${\mathbb{S}}^{d}$

In this section, we consider (5). Let us define

$\mathcal{Q}_{p,\theta,\lambda}^{(2)}[u]:=\frac{\left(\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{\lambda}{p-2}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)^{\theta}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}}{\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}}\,.$

$\bullet$ Admissible parameters. We establish the validity range of the
parameters in (5). We recall that $\theta_{\star}=d\,(p-2)/(2\,p)$ according
to the definition (6). Let us define

$q:=\frac{2\,p\,\theta}{2-p\,(1-\theta)}$ (16)

so that $q=2^{\star}$ if $\theta=\theta_{\star}$ and $q=p$ if $\theta=1$.

######  Lemma 10.

Inequality (5) holds for any $d$, $\theta$ and $p$ such that

$d\geq 1\,,\quad\theta\in[\theta_{\star},1]\cap(1-2/p,1]\,,\quad
p\in(2,2^{*})\quad\mbox{or}\quad p=2^{*}\quad\mbox{if}\quad d\geq 3$ (17)

with an optimal constant $\mu_{2}(p,\theta,\lambda)$ such that

$\mu_{0}\big{(}q,\lambda\,\tfrac{q-2}{p-2}\big{)}\leq\mu_{2}(p,\theta,\lambda)\leq\lambda\quad\forall\,\lambda>0$

where $q$ is given in terms of $p$ and $\theta$ by (16). Moreover,
$\mu_{2}(p,\theta,\lambda)=\lambda$ if $\lambda\leq
d\left(1-(1-\theta)\,\frac{p}{2}\right)$.

###### Proof.

By Hölder’s inequality,
$\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}\leq\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}$
because $d\sigma$ is a probability measure. Using (12), we learn that

$\mu_{0}\left(q,\tilde{\lambda}\right)=\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus\\{0\\}}\mathcal{Q}_{q,1,\tilde{\lambda}}^{(1)}[u]\leq\inf_{u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)\setminus\\{0\\}}\left(\mathcal{Q}_{p,\theta,\lambda}^{(2)}[u]\right)^{1/\theta}\leq\lambda$

with

$\tilde{\lambda}:=\lambda\,\frac{q-2}{p-2}\,.$

The upper bound is obtained by testing $\mathcal{Q}_{p,\theta,\lambda}^{(2)}$
with $u=1$ a.e. If $\tilde{\lambda}\leq d$, there is equality because
$\mu_{0}\big{(}q,\tilde{\lambda}\big{)}=\tilde{\lambda}$ (symmetry case)
according to Theorem 1, which gives us a _symmetry_ case. Notice that
$1-(1-\theta)\,\frac{p}{2}$ is positive if $\theta>1-2/p$.∎

As above, let us consider

$\mathcal{F}_{\lambda}^{(2)}[u]:=\left(\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}+\frac{\lambda}{p-2}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)^{\theta}\,\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2\,(1-\theta)}-\left(\frac{\lambda}{p-2}\right)^{\theta}\,\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}\,.$

$\bullet$ Symmetry breaking. Again symmetry breaking is obtained by
perturbation.

######  Lemma 11.

Assume that $d$, $\theta$ and $p$ satisfy (17). There is _symmetry breaking_
in (5), that is, $\mu_{2}(p,\theta,\lambda)<\lambda$, if $\lambda>\theta\,d$.

###### Proof.

With the same computations as in the proof of Lemma 11 and a non-trivial
spherical harmonic function $\varphi$, we obtain that

$\lim_{\varepsilon\to
0}\frac{1}{\varepsilon^{2}}\mathcal{F}_{\lambda}^{(2)}[u_{\varepsilon}]=\left(\frac{\lambda}{p-2}\right)^{\theta}\left(\theta\,\|{\nabla\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}-\lambda\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}\right)=(d\,\theta-\lambda)\,\|{\varphi}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}<0\,,$

which concludes the proof.∎

$\bullet$ Symmetry. We apply the same methods as in Sections 2 and 3.

######  Lemma 12.

Assume that $d$, $\theta$ and $p$ satisfy (17). If $\theta\leq 1/2$, there is
_symmetry_ in (5), _i.e._ , $\mu_{2}(p,\theta,\lambda)=\lambda$, if
$\lambda\leq\theta\,d$.

It is important to notice that the compatibility of the condition $\theta\leq
1/2$ with (17) induces restrictions not only on $\theta$ but also on $p$ if
$d\geq 2$: $p<2$ if $d=2$ and $p\leq 2\,d/(d-1)$ if $d\geq 3$.

###### Proof.

Let us consider an optimal function
$u\in\mathrm{H}^{1}({\mathbb{S}}^{d},d\sigma)$ for (5) such that
$\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}=1$ and let $\mathsf{a}:=\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}$ and
$\mathsf{b}:=\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}$. We assume that
$u$ is not a constant so that $\mathsf{b}<1$ by Hölder’s inequality. By
optimality, we have that

$(p-2)\,\mathsf{a}+\lambda\,\mathsf{b}=\mu_{2}(p,\theta,\lambda)\,\mathsf{b}^{\frac{1}{\theta}-1}<\lambda\,\mathsf{b}^{\frac{1}{\theta}-1}\,.$

Let us consider the same flow as in Lemma 12. At $t=0$, we know that
$\frac{d}{dt}\mathcal{F}_{\lambda}^{(2)}[v(t,\cdot)]=0$ because $u$ is a
critical point, and the computation shows that this amounts to

$\frac{\theta\,(\lambda-d)}{(p-2)\,\mathsf{a}+\lambda\,\mathsf{b}}+\frac{1-\theta}{\mathsf{b}}=0\,.$

By Lemma 10 symmetry eventually holds only if $\lambda\leq\theta\,d<d$. Let us
assume that our non symmetric minimizer corresponds to the range
$\lambda\leq\theta\,d$. Hence $\lambda-d<0$ and

$\lambda-\theta\,d>\theta\,(\lambda-d)+(1-\theta)\,\lambda\,\mathsf{b}^{\frac{1}{\theta}-2}>0$

if $\theta<1/2$ because $\mathsf{b}^{\frac{1}{\theta}-2}<1$. This is obviously
a contradiction. We conclude that there is no non-symmetric optimizer if
$\theta<1/2$ and $\lambda\leq\theta\,d$.

So far, this proof is formal as we assume that a minimizer exists. By standard
compactness results, this is always true, except for some limiting values of
$\theta$ or $p$. However, one can always approximate an inequality like (5) by
another inequality in the same family, with slightly different exponents and
pass to the limit. This concludes the proof. ∎

Let us conclude by a few remarks. The issue of _symmetry_ versus _symmetry
breaking_ is far from fully answered in case of inequality (5), which is not
unexpected in view of similar difficulties in corresponding _Caffarelli-Kohn-
Nirenberg inequalities_. This is clearly an open and interesting direction of
research. A good point is that the _carré du champ_ method at least provides
some partial answers, which are not available in similar inequalities on the
Euclidean space with weights. There are many other questions which are pending
and will be the subject of future work: asymptotics as $\lambda\to+\infty$ for
(5), reparametrization and numerical computation of the branches, expansion of
the branches near the bifurcation point, energy estimates in relation with
symmetry breaking in the asymptotic regime corresponding to $p-2>0$, small,
_etc._

##  Appendix A Nonlinear carré du champ method on the sphere

For completeness, we give a summary of the _carré du champ_ method in integral
form applied to the sphere with entropy methods and a nonlinear diffusion
equation of fast diffusion or porous medium type. More details can be found in
a series of earlier papers on manifolds (see [9, 17], [20, Section 5.2] or
[23]) with similar computations done in the perspective of uniqueness for
positive solutions of elliptic equations (rigidity) that can be found in [26,
5]. In the specific case of the sphere, it is possible to reduce the problem
to a computation based on the ultraspherical operator (see [14, 15, 21, 12,
24]). The method in the $d=1$ case is slightly different (see for instance
[20, Section 5.3]) and results do not require the use of nonlinear diffusion
equations as it is known for a long time from [1, 2], so this will not be
discussed here. There are exceptional values of the exponent ($p=6$ if $d=3$,
or $p=3$ if $d=4$) which require a specific treatment that will not be
detailed here, but the result can easily be recovered by considering the limit
as $p\to 6$ if $d=3$ and $p\to 3$ if $d=4$ in the inequalities: see for
instance [21]. Here we adopt the presentation of [17] and go along the lines
of the proof of [23, Lemma 4.3]. We adopt simplified notations and write
$\rm{Id}$ instead of using the standard metric, or simply write the Laplace-
Beltrami operator as $-\Delta$. We split the method in two steps, with a first
algebraic part closely related with the computations of [5] and entropy
methods applied to a parabolic equation in order to establish the
inequalities. Because of the regularizing effects of the parabolic equation,
there is no regularity issue in considering positive solutions and solutions
will be considered as smooth without further notice. An approximation scheme
has of course to be done for initial data in
$H^{1}({\mathbb{S}}^{d},d\sigma)$: see for instance [24] for details in the
case of the ultraspherical operator.

###  A.1. A purely algebraic computation

Let $\mathsf{H}u$ denote the _Hessian_ of a $C^{2}$ function $u$ on
${\mathbb{S}}^{d}$ and define the _trace free Hessian_ by

$\mathsf{L}u:=\mathsf{H}u-\frac{1}{d-1}\,(\Delta u)\,\mathrm{Id}\,.$

We also consider the following trace free tensor

$\mathsf{M}u:=\frac{\nabla u\otimes\nabla u}{u}-\frac{1}{d-1}\,\frac{|\nabla
u|^{2}}{u}\,\mathrm{Id}\,,$

where

$\nabla u\otimes\nabla
u:=\big{(}\partial_{i}u\,\partial_{j}u\big{)}_{ij}\quad\mbox{and}\quad\|\nabla
u\otimes\nabla u\|^{2}=|\nabla
u|^{4}=\sum_{ij}\big{(}\partial_{i}u\big{)}^{2}\,\big{(}\partial_{j}u\big{)}^{2}\,.$

Using $\mathsf{L}:\mathrm{Id}=\mathsf{M}:\mathrm{Id}=0$ and

$\displaystyle\|\mathsf{L}u\|^{2}=\|\mathsf{H}u\|^{2}-\frac{1}{d}\,(\Delta
u)^{2}\,,$ $\displaystyle\|\mathsf{M}u\|^{2}=\left\|\frac{\nabla
u\otimes\nabla u}{u}\right\|^{2}-\frac{1}{d}\,\frac{|\nabla
u|^{4}}{u^{2}}=\frac{d-1}{d}\,\frac{|\nabla u|^{4}}{u^{2}}\,,$

we deduce from

$\displaystyle\int_{{\mathbb{S}}^{d}}{\Delta u\,\frac{|\nabla
u|^{2}}{u}}\,d\sigma$ $\displaystyle=\int_{{\mathbb{S}}^{d}}{\frac{|\nabla
u|^{4}}{u^{2}}}\,d\sigma-2\int_{{\mathbb{S}}^{d}}{\mathsf{H}u:\frac{\nabla
u\otimes\nabla u}{u}}\,d\sigma$
$\displaystyle=\frac{d-1}{d-2}\int_{{\mathbb{S}}^{d}}{\|\mathsf{M}u\|^{2}}\,d\sigma-2\int_{{\mathbb{S}}^{d}}{\mathsf{L}u:\frac{\nabla
u\otimes\nabla u}{u}}\,d\sigma-\frac{2}{d}\int_{{\mathbb{S}}^{d}}{\Delta
u\,\frac{|\nabla u|^{2}}{u}}\,d\sigma$

that

$\int_{{\mathbb{S}}^{d}}{\Delta u\,\frac{|\nabla
u|^{2}}{u}}\,d\sigma=\frac{d}{d+2}\left(\frac{d}{d-1}\int_{{\mathbb{S}}^{d}}{\|\mathsf{M}u\|^{2}}\,d\sigma-2\int_{{\mathbb{S}}^{d}}{\mathsf{L}u:\frac{\nabla
u\otimes\nabla u}{u}}\,d\sigma\right)\,.$

This provides us with a first identity,

$\int_{{\mathbb{S}}^{d}}{\Delta u\,\frac{|\nabla
u|^{2}}{u}}\,d\sigma=\frac{d}{d+2}\left(\frac{d}{d-1}\int_{{\mathbb{S}}^{d}}{\|\mathsf{M}u\|^{2}}\,d\sigma-2\int_{{\mathbb{S}}^{d}}{\mathsf{L}u:\mathsf{M}u}\,d\sigma\right)\,.$
(18)

The Bochner-Lichnerowicz-Weitzenböck formula on ${\mathbb{S}}^{d}$ takes the
simple form

$\tfrac{1}{2}\,\Delta\,(|\nabla u|^{2})=\|\mathsf{H}u\|^{2}+\nabla(\Delta
u)\cdot\nabla u+(d-1)\,|\nabla u|^{2}$

where the last term, _i.e._ , $\mathrm{Ric}(\nabla u,\nabla u)=(d-1)\,|\nabla
u|^{2}$, accounts for the Ricci curvature tensor contracted with $\nabla
u\otimes\nabla u$. An integration of this formula on ${\mathbb{S}}^{d}$ shows
a second identity,

$\int_{{\mathbb{S}}^{d}}{(\Delta
u)^{2}}\,d\sigma=\frac{d}{d-1}\int_{{\mathbb{S}}^{d}}{\|\mathsf{L}u\|^{2}}\,d\sigma+d\int_{{\mathbb{S}}^{d}}{|\nabla
u|^{2}}\,d\sigma\,.$ (19)

Hence for some parameters $\beta$ and $\kappa$ to be fixed later, the
functional

$\displaystyle\mathsf{k}[u]:=$
$\displaystyle\int_{{\mathbb{S}}^{d}}{\left(\Delta u+\kappa\,\frac{|\nabla
u|^{2}}{u}\right)\left(\Delta u+(\beta-1)\,\frac{|\nabla
u|^{2}}{u}\right)}\,d\sigma$ (20) $\displaystyle=$
$\displaystyle\int_{{\mathbb{S}}^{d}}{(\Delta
u)^{2}}\,d\sigma+(\kappa+\beta-1)\int_{{\mathbb{S}}^{d}}{\Delta
u\,\frac{|\nabla
u|^{2}}{u}}\,d\sigma+\kappa\,(\beta-1)\int_{{\mathbb{S}}^{d}}{\frac{|\nabla
u|^{4}}{u^{2}}}\,d\sigma$

can be rewritten using (18) and (19) as

$\mathsf{k}[u]=\int_{{\mathbb{S}}^{d}}{\mathsf{q}[u]}\,d\sigma+d\int_{{\mathbb{S}}^{d}}{|\nabla
u|^{2}}\,d\sigma$

where

$\mathsf{q}[u]:=\frac{d}{d-1}\,\Big{(}\mathsf{a}\,\|\mathsf{L}u\|^{2}-2\,\mathsf{b}\,\mathsf{L}u:\mathsf{M}u+\mathsf{c}\,\|\mathsf{M}u\|^{2}\Big{)}$

and

$\mathsf{a}=1\,,\quad\mathsf{b}=(\kappa+\beta-1)\,\frac{d-1}{d+2}\,,\quad\mathsf{c}=(\kappa+\beta-1)\,\frac{d}{d+2}+\kappa\,(\beta-1)\,.$
(21)

Let us consider the choice of the parameters

$\kappa=\beta\,(p-2)+1\quad\mbox{and}\quad
m=1+\frac{2}{p}\left(\frac{1}{\beta}-1\right)$ (22)

and recall that by definition (10), we have

$m_{\pm}(d,p):=\frac{1}{(d+2)\,p}\left(d\,p+2\pm\sqrt{d\,(p-1)\,\big{(}2\,d-(d-2)\,p\big{)}}\,\right).$

######  Lemma 13.

Let $d\geq 2$, $p\geq 1$ if $d=2$, and $p\in[1,2^{*}]$ if $d\geq 3$. If
$m_{-}(d,p)\leq m\leq m_{+}(d,p)$, then $\mathsf{q}$ is a positive quadratic
form if $\kappa$ and $\beta$ are given in terms $m$ and $p$ by (22). As a
consequence, we have

$\mathsf{k}[u]\geq d\int_{{\mathbb{S}}^{d}}{|\nabla
u|^{2}}\,d\sigma\quad\forall\,u\in C^{2}({\mathbb{S}}^{d})\,.$

###### Proof.

It follows from the reduced discriminant condition
$\mathsf{b}^{2}-\mathsf{a}\,\mathsf{c}<0$ by tedious but elementary
computations. In the cases $p=6$ if $d=3$ and $p=3$ if $d=4$, computations
have to be done directly with $m$, without using the parameter $\beta$ (see
below).∎

In dimension $d=1$, we have a similar result with $m=\beta=1$ using the
Poincaré inequality on the circle ${\mathbb{S}}^{1}$.

###  A.2. Entropy methods and nonlinear flows

Let us introduce the parabolic evolution setting corresponding to the
nonlinear diffusion equation

$\frac{\partial w}{\partial t}=w^{2-2\beta}\left(\Delta
w+\kappa\,\frac{|\nabla w|^{2}}{w}\right)\,.$ (23)

The function

$\rho(t,x)=\big{(}w(m\,t,x)\big{)}^{\beta\,p}\quad\forall\,(t,x)\in{\mathbb{R}}^{+}\times{\mathbb{S}}^{d}$

with $\kappa$ and $\beta$ as in (22) solves the standard fast-diffusion
equation (9). From this observation, it is clear that

$\frac{d}{dt}\int_{{\mathbb{S}}^{d}}{w^{\beta\,p}}\,d\sigma=\frac{d}{dt}\int_{{\mathbb{S}}^{d}}{\rho}\,d\sigma=0\,.$

######  Lemma 14.

Let $d\geq 2$, $p\geq 1$ if $d=2$, $m>0$ and $p\in[1,2^{*}]$ if $d\geq 3$.
With $\kappa$ and $\beta$ as in (22), any smooth positive solution of (23) is
such that

$\frac{d}{dt}\int_{{\mathbb{S}}^{d}}{w^{2\,\beta}}\,d\sigma=2\,\beta^{2}\,(p-2)\int_{{\mathbb{S}}^{d}}{|\nabla
w|^{2}}\,d\sigma$

and, with $\mathsf{k}$ defined by (20),

$\frac{d}{dt}\int_{{\mathbb{S}}^{d}}{\big{|}\nabla
w^{\beta}\big{|}^{2}}\,d\sigma=-\,2\,\beta^{2}\,\mathsf{k}[w]\,.$

As a consequence, if $m_{-}(d,p)\leq m\leq m_{+}(d,p)$, then we have

$\frac{d}{dt}\int_{{\mathbb{S}}^{d}}{\big{|}\nabla
w^{\beta}\big{|}^{2}}\,d\sigma\leq-\,2\,\beta^{2}\,d\int_{{\mathbb{S}}^{d}}{|\nabla
w|^{2}}\,d\sigma$

###### Proof.

These results are easily proved using a few integrations by parts and Lemma
13.∎

A standard consequence is, for instance, the fact that

$\frac{d}{dt}\mathcal{F}_{\lambda}\big{[}w^{\beta}\big{]}\leq-\,2\,\beta^{2}\,(d-\lambda)\int_{{\mathbb{S}}^{d}}{|\nabla
w|^{2}}\,d\sigma$

if $w$ solves (23) and $\mathcal{F}_{\lambda}$ is defined by (8). Hence if
$\lambda\leq d$ and (23) is supplemented with the initial datum
$w(0,\cdot)=u^{1/\beta}$, then

$\mathcal{F}_{\lambda}[u]\geq\mathcal{F}_{\lambda}\big{[}w^{\beta}(t,\cdot)\big{]}=\lim_{s\to+\infty}\mathcal{F}_{\lambda}\big{[}w^{\beta}(s,\cdot)\big{]}=0\quad\forall\,t\geq
0\,.$

The limit as $t\to+\infty$ can be identified by analyzing $\mathsf{k}[w]=0$
and proving that this means that $w$ is then a constant if $p<2^{*}$. In the
case $p=2^{*}$, identifying $w$ as a constant can be done only by considering
$w$ as a limit as $t\to+\infty$ of a solution of (23).

Acknowledgment: This work has been partially supported by the Project EFI
(ANR-17-CE40-0030) of the French National Research Agency (ANR).
© 2022 by the authors. This paper may be reproduced, in its entirety, for non-
commercial purposes.

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