domain: math phys

text:
Anticipatory Caching in Small Cell Networks: A Transfer Learning Approach Ejder Baştug , Mehdi Bennis and Mérouane Debbah , Alcatel-Lucent Chair -SUPÉLEC, Gif-sur-Yvette, France Centre for Wireless Communications, University of Oulu, Finland {ejder.bastug, merouane.debbah}@supelec.fr, bennis@ee.oulu.fi

I. EXTENDED ABSTRACT

Locally caching contents at the network edge constitutes one of the five most disruptive paradigms in 5G networks [START_REF] Boccardi | Five disruptive technology directions for 5g[END_REF]. Recent results have shown that dynamic caching can significantly offload different parts of the network including the Radio access networks (RANs) and core network (CN), by smartly prefetching and storing contents closer to the endusers. In parallel to that, the era of pushing contents through the network on a best-effort basis ignoring who end-users are and what they are doing with their devices has dawned, calling for a truly context-aware and proactive networking paradigm [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF]. As a result, edge caching has taken the recent 5G literature by storm over the last few years as evidenced in [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF]- [START_REF] Golrezaei | Femtocaching: Wireless video content delivery through distributed caching helpers[END_REF]. In [START_REF] Golrezaei | Femtocaching: Wireless video content delivery through distributed caching helpers[END_REF], the concept of femtocaching was proposed addressing the problem of capacity-limited backhaul links by embedding base stations with high-storage units. In [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF], a novel edgecentric networking paradigm was proposed in which network nodes (i.e., base station (BS) and/or user terminals (UTs)) proactively cache judiciously selected contents at the network edge. Exploiting both spatial and social caching coupled with suitable device-to-device (D2D) communication was shown to efficiently offload the backhaul traffic and enhancing the overall network performance in terms of cache-hit-rates and users' satisfaction ratios. Therein, a proactive caching procedure is formulated as a supervised machine learning problem and collaborative filtering (CF) techniques are used to estimate the file popularity matrix exploiting users-files correlations. Notwithstanding this fact, the file popularity matrix remains typically large and sparse in practice, rendering CF techniques sub-optimal suffering from data sparseness and cold-start problem, which are major challenges in the machine learning community [START_REF] Pan | A survey on transfer learning[END_REF].

In this work, we build upon the work in [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF] and propose a more efficient machine learning technique, using the framework of transfer learning (TL) [START_REF] Pan | A survey on transfer learning[END_REF]. Indeed, in many real-world applications, it is expensive or even impossible to collect and label training data to build suitable prediction models. With this in mind, TL is seen as a suitable framework which allows to exploit data from other rich information sources (referred to as source domain) to further improve the prediction task in the target domain. TL has been traditionally used in data mining This research has been supported by the ERC Starting Grant 305123 MORE (Advanced Mathematical Tools for Complex Network Engineering), the SHARING project under the Finland grant 128010 and the project BESTCOM.

problems such as classification, regression and very recently in the context of CF [START_REF] Pan | A survey on transfer learning[END_REF].

1) Transfer Learning: In a nutshell, TL can be classified into inductive, transductive and unsupervised transfer learning depending on the availability of the source and target domain labels. Research issues in transfer learning boil down to: 1) what to transfer, 2) how to transfer, and 3) when to transfer.

While "what to transfer" studies which part of the knowledge can be transferred across domains or tasks, "when to transfer" deals with the issue of knowing when is best to transfer the knowledge to avoid negative transfer, notably when the source and target domains become unrelated. Finally, "how to transfer" deals with knowledge extraction which needs to be transferred. A comprehensive survey of transfer learning is found in [START_REF] Pan | A survey on transfer learning[END_REF].

2) Contribution: The basic idea of the proposed learning approach is to alleviate the data sparsity problem encountered in most CF problems, by learning and transferring the rich contextual information (i.e., source domain), to better estimate the (large-scale) file popularity matrix in the target domain. We assume that the knowledge extracted from the source domain stems from the interaction of users accessing/sharing and recommending files within their social community via D2D (Web2.0-like). Instead of learning from scratch in the target domain, the nice feature of the TL approach lies in judiciously extracting collaborative social behavior information from the source domain to aid in the learning in the target domain, which will be explained in the sequel. To the best of our knowledge this is perhaps the first contribution of transfer learning in RANs.

A. Network Model

We assume that there exists an information system S CRP in the source domain and an information system S tar in the target domain. An illustration of the scenario is given in Fig. 1.

1) Target Domain: We consider a network deployment which consists of M tar small base stations (SBSs) from the set M tar = {1, . . . , M tar } and N tar UTs from the set N tar = {1, . . . , N tar }. According to this setup, UTs seek certain files from a library F tar = {1, . . . , F tar }, where each file has length of L.

We assume that every SBS is connected to the core network via a limited backhaul link with capacity C b and every SBS has a total wireless link capacity of C w . In order to offload the backhaul (and satisfy users' requests more efficiently), every SBS needs to proactively fetch strategic contents from its CN and cache them at the edge. First, suppose that there exist D number of requests from the set D over T time slots. Suppose also a caching indicator matrix Θ ∈ {0, 1} Mtar×Ftar , where θ m,f = 1 indicates that the m-th SBS stores the f -th file and θ m,f = 0 otherwise. Then, maximization of backhaul offloading gain for a fixed Θ policy over T time slots can be formally written as:

maximize Θ 1 D d∈D 1{θ n d ,f d } subject to trace(Θ T Θ) ≤ S L , (1)
where n d and f d are the accessed SBS and file for request d, 1{.} is the indicator function and S is the total storage capacity. In the sequel, this is referred to as the target domain caching formulation. Solving this problem is highly challenging due to: i) limited SBS storage capacity; ii) large number of users and library size; iii) SBSs need to track, learn and estimate user ratings given a sparse popularity matrix.

2) Source Domain: Inspired from [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF], we exploit the contextual social network overlay composed of users' interactions within their social communities, referred to in the sequel as the source domain. The source domain considered in this work represents the behavior of users' interactions within their own social communities via D2D, modeled as a Chinese restaurant process (CRP) [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF]. That is, within every social community, users sequentially request to download their sought-after content, and when a user downloads its content, the recorded hits are recorded (i.e., history). This action affects the probability that this content will be requested by others users within the same social community, where popular contents are requested more frequently and new contents less frequently.

In the target domain, the caveat of the CF-based caching policy in [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF] is the fact that the file popularity matrix is largely unknown yielding slow convergence, and suffering from the cold-start problem. This is expected to be even more severe in settings where the number of users and files grow very large. Motivated by this fact, in this work, we propose a novel proactive caching procedure by exploiting the rich contextual information extracted from the D2D interactions via a transfer-learning procedure to more efficiently cache contents at the network edge in the target domain (i.e., higher cachehit ratios). This caching procedure is shown to outperform classical CF-based learning methods, such as in [START_REF] Baştug | Living on the Edge: The role of Proactive Caching in 5G Wireless Networks[END_REF].

B. Classical CF-based Learning

The classical CF-based learning is composed of a training and prediction part. In the training part, the goal is to estimate the popularity matrix P tar ∈ R Ntar×Ftar , where every SBS builds a model based on the already available information regarding users' ratings. As given in the previous section, let N tar and F tar denote the set of users and files associated with N tar users and F tar files. In details, P tar with entries P tar,ij is the (sparse) file popularity matrix in the target domain. R tar = {(i, j, r) : r = P tar,ij , P tar,ij = 0} refers to the set of known user ratings. In the prediction part, in order to predict the unobserved ratings in N tar , low-rank factorization techniques are used to estimate the missing entries of P tar . The goal here is to approximate the popularity matrix P tar ≈ N T tar F tar , where the factor matrices N tar ∈ R k×Ntar and F tar ∈ R k×F tar can be learned by minimizing the cost function as follows: minimize (i,j)∈Ptar (i,j)∈Ptar

n T i f j -P tar,ij 2
+ (2) λ ||N tar || 2 F + ||F tar || 2 F
where the sum is over the (i,j) user/file pairs in the training set. Additionally, n i and f j are the i-th and j-th columns of N tar and F tar respectively, and ||.|| 2 F represents Frobenius norm. In this minimization problem, the weight λ is chosen to balance between regularization and fitting training data. However, it turns out that users may rate only very few items, causing the P tar to be extremely sparse, and thus directly minimizing (2) will suffer from severe over-fitting problems.

C. TL-based Content Caching

As alluded to earlier, exploiting and transferring the vast amount of available user-file ratings from a different-yetrelated source domain can help alleviate data sparsity and solve (2) more efficiently. This is precisely the objective of this work. Formally speaking, we model the source domain S CRP , which is associated with a set of N CRP users and F CRP files denoted by N CRP and F CRP . The user-file popularity matrix in the source domain is represented by matrix P CRP ∈ R N CRP ×F CRP and likewise let R CRP = {(i, j, r) : r = P CRP,ij , P CRP,ij = 0} denote the set of observed user ratings in the source domain. In contrast with P tar , the popularity matrix in the source domain, P CRP , contains more information which helps explore hidden patterns of user social behavior for knowledge transfer. The basic principle is to smartly "borrow" judiciously-chosen user social behavior information from S CRP to better learn S tar .

The transfer learning procedure from S CRP to S tar consists of two interrelated steps. First, an item correspondence needs to be established to identify similarly-rated files in both source and target domains. Second, an optimization problem is formulated combining the source and target domains for knowledge transfer to jointly learn the popularity matrix in the target domain P tar . In this respect, we assume that both source and target domain belong to one information system s ∈ {S CRP , S tar }, which is associated with N s users and F s files denoted by N s and F s respectively. For each system s, we observe a sparse matrix P s with entries P s,ij . Let R s = {(i, j, r) : r = P s,ij , P s,ij = 0} denote the set of observed user ratings in each system. We refer to the set of shared files as F. Let N * = N CRP ∪ N tar and F * = F CRP ∪ F tar denote the union of the collections of users and files, respectively, where N * = |N * | and F * = |F * | denote the total number of unique users and files in the union of both systems.

In the proposed learning approach, we model the users N * and files F * by a user factor matrix N ∈ R k×N * and a file factor matrix F ∈ R k×F *

, where the i-th and j-th columns of these matrices are represented by n i and f j , respectively. The goal is to approximate the popularity matrix P s ≈ N T s F s , where the factor matrices N and F are learned by minimizing the following cost function:

minimize (i,j)∈Ps s α s (i,j)∈Ps n T i f j -P s,ij 2
+ (3) λ ||N|| 2 F + ||F|| 2 F
where the α s is the weight of each system. In doing so, P CRP and P tar are jointly factorized and the set of factor matrices F CRP and F tar become interdependent since the features of a shared file are required to be the same for knowledge sharing.

II. NUMERICAL RESULTS AND DISCUSSION

The datasets for numerical setup are sampled from stochastic processes and the results are obtained by averaging out 100 Monte-Carlo realizations. The evolution of the offloading gain in the target domain with respect to the storage size ratio S LFtar is shown in Fig. 2. In this figure, the following caching policies are shown for comparison:

1) Ground Truth: The popularity matrix (P tar ) is known perfectly and used for cache decision accordingly, by simply storing the most popular files for given storage size. 2) Random caching: Files are cached uniformly at random regardless of the popularity matrix. 3) CF: P tar is estimated via CF using a training set with 4% of rating density. 4) TL: P tar and P CRP are jointly factorized via TL using perfect correspondence and 12% of rating density in the training set. It can be seen that the CF method is not able to approximate the ground truth well, thus, yielding poor offloading gains similar to random caching. On the other hand, the joint estimation done by TL improves the caching performance, approaching the offloading gains of ground truth.

III. CONCLUSIONS

We showed the benefits of TL approach in a scenario where cache-enabled SBSs is giving low performance due to the poor estimation of CF. Numerical results showed that offloading gains can be improved by transferring the knowledge from source domain to target domain using TL approaches. An interesting future work would be investigating the impact of various system parameters (besides storage size) where the gains can be different depending on the numerical setting.

Figure 1 :

 1 Figure 1: An illustration of the considered scenario which consists of information systems S CRP and S tar . The knowledge obtained from social interactions via D2D communications in source domain is transferred to the target domain.

Figure 2 :

 2 Figure 2: Evolution of the offloading gain. M tar = 1, N tar = 32, F tar = 32, L = 1 MBit, C b = 1 MBit/s, C w = 128 MBit/s.


Anca-Roxana Şimon

Guillaume Gravier

Pascale Sébillot

Marie-Francine Moens

IRISA and KUL at MediaEval 2014: Search and Hyperlinking Task

This paper presents our approach and results in the hyperlinking sub-task at MediaEval 2014. A two step approach is implemented: relying on a topic segmentation technique, the first step consists in generating potential target segments; then, for each anchor, the best 20 target segments are selected according to two distinct strategies: the first one focuses on the identification of very similar targets using n-grams and named entities; the second one makes use of an intermediate structure built from topic models, which offers the possibility to control serendipity and to explain the links created.

INTRODUCTION

This paper presents the joint participation of IRISA and KUL at the MediaEval 2014 Search and Hyperlinking task, in which focus is set on the hyperlinking sub-task [START_REF] Eskevich | The Search and Hyperlinking task at MediaEval 2014[END_REF]. The goal is thus to create hyperlinks between predefined anchor segments and short video segments, called targets, which should offer complementary information not found at search time. Targets have to be automatically extracted from videos of a large collection.

The hyperlinking system we propose consists in a two step process: first, all potential target segments are extracted using a topic segmentation technique; then, the most relevant targets are selected for each anchor, with the help of content analysis and similarity measures.

In our 2013 participation [START_REF] Guinaudeau | HITS and IRISA at MediaEval 2013: Search and hyperlinking task[END_REF], we focused on precise target selection and our most efficient system consisted in a direct comparison between anchor and target segments obtained through topic segmentation using bags of n-grams to represent content, thus resulting in links between anchor and very similar targets. We thus go on exploring this direction. However, we believe that besides precise target selection, a very important aspect of hyperlinking is to offer serendipity and to explain why two video segments are linked. To address these points, we propose an intermediate structure, obtained from topic models, that allows an indirect comparison between anchors and targets. Its first advantage is that segments which do not share a consistent part of the vocabulary but are semantically related can be linked. Moreover this structure provides a basis to investigate why certain links are created. Link justification is an interesting aspect of hyperlinking which, to our knowledge, has not been addressed before.

SYSTEM OVERVIEW

The aim of our approach is to find target segments of the same topic as the anchor, or of related topics. The hyperlink generation relies on content-based comparisons exploiting spoken data obtained from automatic transcripts and manual subtitles [START_REF] Gauvain | The LIMSI broadcast news transcription system[END_REF]. Data are first lemmatized and only nouns, non modal verbs and adjectives are kept. Subsections 2.1 and 2.2 respectively detail the two parts of our system: the generation of potential target segments and the selection of the top 20 targets for each anchor.

Generating potential target segments

Each video in the test collection is partitioned into topically coherent segments with the generic topic segmentation algorithm TextSeg [START_REF] Utiyama | A statistical model for domain-independent text segmentation[END_REF], which is domain independent, needs no a priori information, is efficient on speech transcripts and on segments of varying lengths. Its main drawback concerns over-segmentation, which in our case is not problematic since the target segments must not last longer than 2 minutes.

Selection of hyperlinks targets

Each anchor segment is compared with each topically coherent segment previously obtained thanks to similarity measures. The comparison can be direct or indirect (i.e., using intermediate structures).

Four methods are proposed to select the hyperlinks targets. The baseline corresponds to the method for which we obtained our best results in 2013 (Linear+ngrams); a direct comparison between segments is done, contents being represented by bags of unigrams, bigrams and trigrams. The second method extends the previous one by using the Stanford Named Entity Recognizer (NER) [START_REF] Finkel | Incorporating non-local information into information extraction systems by Gibbs sampling[END_REF], a 3 class (person, organization and location) entity tagger (Linear+ngrams+NEs). A Jaccard similarity coefficient is used to evaluate similarities in terms of shared named entities (NEs). The n-grams and NEs similarity scores are combined; weights of 0.3 and 0.7 respectively are chosen to favor precise alignments, thus not rewarding serendipity.

For the last two methods an intermediate structure is built using Latent Dirichlet Allocation (LDA) probabilistic topic models [START_REF] Blei | Latent dirichlet allocation[END_REF] learned on the manual transcripts of the development set (1335 hours of video). Each transcript is represented as a mixture of K latent topics, where a latent topic is represented as a probability distribution over the vocabulary. Contrary to the bag of words representation, this one clusters semantically similar co-occurring terms. To construct the structure LDA is trained using Gibbs sampling, standard values for the hyperparameters α=50/K and β=0.01 [START_REF] Steyvers | Probabilistic Topic Models[END_REF] with a varying number of latent topics K: 50, 100, 150, 200, 300, 500, 700. This range for the number of topics was chosen to learn general to more specific topics. Using this structure an indirect comparison between anchor and target segments can be performed.

Our third method (TopicM ) consists in computing, for every K, the probabilities of the anchor and target segments given the topics. For each K and for each anchor-target pair, two vectors are obtained in which componenti corresponds to the probability of topici, given the words contained in the segment. Then a similarity measure is computed between these vectors, leading to a score for each anchor-target pair. To select the top targets, for each anchor a linear combination of the scores resulted for each K is done with more importance to the most specific topics.

For the last method (HierTopicM ), the previous structure with topic models is extended to form a tree-like hierarchy between the topics. This hierarchy relies on a similarity measure between the topics obtained with Ki and Kj, where Kj < Ki. Thus, each topic learned with K = 700 will be connected to the most similar topic learned with K = 500 and so on for the other K values. Having this representation a path for each anchor can be selected, starting with the bottom of the tree (K = 700) and choosing the topic with the highest probability given the words in the anchor and going on with the selection of the parents for that topic until the first level in the tree (K = 50) is reached. The targets are then selected by a linear combination of the probability values of the topics in an anchor path, given the words in the target. Thus, in the end only a part of the topics in the structure will be considered, allowing more precise control of serendipity and justification of links.

Using the structure of topics (hierarchical or not) to select the hyperlinks, serendipity can be controlled by giving more weights to values attained with more general or more specific topics. Moreover links can be justified, by looking at the top words of the topics that contributed most in the selection of those links. This structure can be used a posteriori for other methods to understand the links creation.

Considering the evaluation rules, the target segments proposed for each anchor should have a duration between 10 sec and 2 min. Therefore, a post-processing of the final targets is done to verify these constraints. If a segment is longer than 2 min, it is resegmented with a sliding window of 2 min and the best scoring window within the segment will represent the final target segment. If a segment is shorter than 10 sec it is combined with the best scoring neighbor, resulted from the topic segmentation, until the minimum length is reached.

RESULTS

Official evaluation results are reported in Table 1. Only the top 10 results were evaluated for each method, via crowdsourcing on Amazon Mechanical Turk (AMT). Other results were evaluated automatically based on the evaluation of the selected results from all participants. The best results are obtained with the topic segmentation with ngrams on the manual subtitles (Linear+ngrams). This means that turkers prefer that the content of the target segments is closely related to that of the anchor. As anticipated, on the automatic transcripts the precision decreases. Unexpectedly, adding NEs to this method and therefore favoring segments about the same people, places, organizations, diminished the precision. From a manual assessment of several target segments proposed using NEs, it seems that having targets speaking about same people (e.g., Madonna, Beckham) in different circumstances (i.e, shows on different subjects: diets, charity) is not relevant. Possibly, giving less weight to the NEs shared between anchors and targets, the precision could be improved.

Regarding the topic model-based methods, the one that uses all the topics learned (TopicM ) yields better results, comparable to those obtained with the Linear+ngrams method. From a manual assessment of the results it seems that the topics, even those learned with K = 700 are too general. The targets that were considered relevant are those for which the anchor addresses a more general topic (e.g., wildlife). The problem of generality appears also for the HierTopicM method. However having only a part of the topics considered the results are worse than with TopicM. Using such intermediate structures could be improved by learning more specific topics. Still, having the topwords of the topics that best explain the anchors and the targets can help interpret and justify the links. Additionally, with an intermediate structure, anchors are linked to targets even without sharing much vocabulary.

Table 1 :

 1 Precision values obtained for all proposed methods on the 2014 test set.

		P 5	P 10 P 20
	Linear+ngrams (LIMSI)	0.1	0.096 0.06
	Linear+ngrams (MANUAL)	0.19	0.16	0.1
	Linear+ngrams+NEs (LIMSI) 0.046 0.036 0.02
	TopicM (LIMSI)	0.093 0.093 0.048
	HierTopicM (LIMSI)	0.03	0.02	0.01


Chao Wang

Dong Ye

Some Liouville theorems for Hénon type elliptic equations

Keywords: Liouville theorem, Hénon equation, stability, finite Morse index solution MSC: 35J60, 35B45

We investigate here the nonlinear elliptic equations -∆u = |x| α e u and -∆u = |x| α |u| p-1 u with α > -2, p > 1 and N ≥ 2. In particular, we prove some Liouville type theorems for weak solutions with finite Morse index in the low dimensional Euclidean spaces or half spaces.

Introduction

In this paper, we consider two Hénon type elliptic equations as

-∆u = |x| α e u in Ω ⊂ R N , (1.1)
and

-∆u = |x| α |u| p-1 u in Ω ⊂ R N , (1.2)
where α > -2, p > 1 and N ≥ 2.

The study of stable solutions in the autonomous case, i.e. when α = 0 has been studied recently, Farina classified completely in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF] all finite Morse index classical solutions of (1.2) in R N for 1 < p < p JL , where p JL = p(N, 0) stands for the Joseph-Lundgren exponent (see [START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources[END_REF] and Theorem 1.1 below). More precisely, the equation (1.2) with α = 0 admits nontrivial classical solutions in R N which are stable outside a compact set, if and only if N ≥ 3, p = N +2 N -2 ; or N ≥ 11 and p ≥ p JL .

For the exponential case, it is shown by Farina in [START_REF] Farina | Stable solutions of -∆u = e u on R N[END_REF] that ∆u+e u = 0 has no stable classical solution in R N for 2 ≤ N ≤ 9. He proved also that any classical solution of ∆u + e u = 0 with finite Morse index in R 2 verifies e u ∈ L 1 (R 2 ), so it must be a solution classified by Chen & Li [START_REF] Chen | Classification of solutions of some nonlinear elliptic equations[END_REF], that is

u(x) = ln 32λ 2 (4 + λ 2 |x -x 0 | 2 ) 2 with λ > 0, x 0 ∈ R 2 .
Finally, when N ≥ 3, Dancer & Farina proved in [START_REF] Dancer | On the classification of solutions of -∆u = e u on R N : stability outside a compact set and applications[END_REF] that the equation (1.1) with α = 0 admits classical entire solutions which are stable outside a compact set, if and only if N ≥ 10.

A natural question is to ask if similar results can be observed for the nonautonomous case, i.e. when α = 0. The equation (1.2) has been considered by Dancer, Du & Guo in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF], they proved Theorem 1.1 Let α > -2 and u ∈ H 1 loc ∩ L ∞ loc (R N ) be a stable solution of (1.2) in R N with 1 < p < p(N, α), where

p(N, α) =    ∞, if N ≤ 10 + 4α (N -2) 2 -2(α + 2)(α + N ) + 2 (α + 2) 3 (α + 2N -2) (N -2)(N -4α -10)
, if N > 10 + 4α.

Then u ≡ 0. On the other hand, for N > 10 + 4α and p ≥ p(N, α), (1.2) admits a family of stable positive radial solutions in R N .

Dancer, Du & Guo have also studied positive solutions either in a punctured domain Ω \ {0} or in an exterior domain R N \ Ω, they obtained interesting results on asymptotic behavior when |x| tends to 0 or ∞, for solutions which are stable respectively near the origin or outside a compact set. Then they used these estimates and [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] to get some classification results.

In [START_REF] Esposito | Linear instability of entire solutions for a class of non-autonomous elliptic equations[END_REF], Esposito studied the stability of solutions to the Hénon type equations (1.1), (1.2) in R N and proved some Liouville results with α ≥ 0 and respectively bounded e u or |u|. It is also worthy to mention that in [START_REF] Du | Positive solutions of an elliptic equation with nonnegative exponent: Stability and critical power[END_REF][START_REF] Esposito | Linear instability of entire solutions for a class of non-autonomous elliptic equations[END_REF][START_REF] Dávila | On finite Morse index solutions of two equations with negative exponent[END_REF], positive entire solutions of ∆u = |x| α u -p (p > 0), with finite Morse index or stable in an exterior domain are classified.

Although we borrow many ideas from the previous works, we try to handle the problems in more general setting by three folds.

• In all previous works for (1.1) and (1.2), the authors considered solutions with locally or globally bounded e u or |u|. Here we deal with weak solutions which are not supposed a priori to be locally upper or lower bounded. For example, if α < 0 and 0 ∈ Ω, any weak solution of (1.1) or (1.2) cannot be a classical solution, due to the singularity at the origin.

• We work with general α > -2, and classify not only stable solution but also entire solution stable out of a compact set for (1.1), or with finite Morse index for (1.2) under suitable conditions on p, which were not considered in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF][START_REF] Esposito | Linear instability of entire solutions for a class of non-autonomous elliptic equations[END_REF].

• For the equation (1.2), we do not impose any sign condition for u and prove the fast decay behavior near 0 (resp. ∞) for weak solutions which are stable near the origin (resp. outside a compact set) with suitable exponent p. Finally we consider also finite Morse index solutions in the half space R N + .

In order to state our results more accurately, let us precise the meaning of weak solution and recall some basic notions. For simplicity, we assume always that Ω is a regular domain in R N and f (x,

•) : R → R is a C 1 function for almost every x ∈ Ω. Definition 1.2 We say that u is a weak solution of -∆u = f (x, u) in domain Ω ⊂ R N (bounded or not), if u ∈ H 1 loc (Ω) verifies f (x, u) ∈ L 1 loc (Ω)
and

Ω ∇u • ∇ψ -f (x, u)ψ dx = 0, ∀ ψ ∈ C 1 c (Ω). (1.3)
Here and in the following C k c (Ω) denotes the set of C k functions with compact support in Ω.

• Let u be a weak solution of -∆u = f (x, u) in Ω. We say that u is stable if

∂ u f (x, u) ∈ L 1 loc (Ω) and Q u (ψ) := Ω |∇ψ| 2 -∂ u f (x, u)ψ 2 dx ≥ 0 for all ψ ∈ C 1 c (Ω). (1.4)
Remark that when ∂ u f (x, u) ≥ 0, (1.4) holds for any ψ ∈ H 1 0 (Ω) by density argument. It is the case for the equations (1.1) and (1.2).

• The Morse index of a solution u, ind(u) is defined as the maximal dimension of all subspaces X of C 1 c (Ω) such that Q u (ψ) < 0 for any ψ ∈ X\{0}. Readily u is stable if and only if ind(u) = 0.

• A weak solution u of -∆u = f (x, u) in Ω is said to be stable outside a compact set K, if Q u (ψ) ≥ 0 for any ψ ∈ C 1 c (Ω\K).
Recall that any finite Morse index solution u is stable outside a compact set K ⊂ Ω. Indeed, for = ind(u) ≥ 0, there exists

X = span{ϕ 1 , • • • , ϕ } ⊂ C 1 c (Ω) such that Q u (ϕ) < 0 for any ϕ ∈ X\{0}, so Q u (ψ) ≥ 0 for all ψ ∈ C 1 c (Ω\K), where K = j supp(ϕ j ).
We should mention that when α = 0, it is proved in [START_REF] Dávila | Partial regularity of finite Morse index solutions to the Lane-Emden equation[END_REF] that any weak solution of (1.2) with finite Morse index and p < p JL is indeed in C 2 (Ω), therefore all results for the special case with α = 0 in (1.2) are well known thanks to [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF]. Moreover, many other interesting results for solutions stable outside a compact set can be found in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF] for the autonomous case of equation (1.2).

So our work concerns only the nonautonomous case α = 0, which is different. Furthermore, the restriction on α > -2 is necessary, seeing the following nonexistence result. Proposition 1.3 For α ≤ -2, (1.1) admits no weak solution for any domain Ω ⊂ R N containing 0.

It was proved in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF] that when α ≤ -2, (1.2) admits no positive solution over any punctured domain B(0, R)\{0}. Here and after, B(y, r) denotes the ball of center y and radius r > 0 in R N . However, as far as we are aware, it is not known if the condition α > -2 is necessary to have a sign-changing weak solution to (1.2) around the origin.

From now on, we assume that α > -2. Our main objective is to classify weak solutions of (1.1) or (1.2) in R N , which are stable outside a compact set or with finite Morse index.

Theorem 1.4 Let α > -2 and Ω = R N . For 2 ≤ N < 10 + 4α, there is no weak stable solution of (1.1).

Theorem 1.5 Let α > -2, Ω = R N and 2 < N < 10 + 4α -where α -= min(α, 0). Then (1.1) has no weak solution which is stable outside a compact set. In particular, any weak solution of (1.1) in R N has infinite Morse index if 2 < N < 10 + 4α -. Theorem 1.4 is sharp. Indeed, for N ≥ 10 + 4α and α > -2, (1.1) possesses radial stable weak solutions in R N ,

U (x) = -(2 + α) ln |x| + ln[(2 + α)(N -2)]
where |x| denotes the Euclidean norm. The stability of U is a direct consequence of the classical Hardy inequality,

R N |∇ψ| 2 dx ≥ (N -2) 2 4 R N ψ 2 |x| 2 dx, ∀ ψ ∈ H 1 (R N ). (1.5)
But we do not know if Theorem 1.5 holds under the assumptions of Theorem 1.4, i.e. when α > 0 and 10 ≤ N < 10 + 4α.

When N = 2 and α > -2, it is not difficult to see that any finite Morse index solution is indeed an energy solution, that is |x| α e u ∈ L 1 (R 2 ); Prajapat & Tarantello [START_REF] Prajapat | On a class of elliptic problems in R 2 : symmetry and uniqueness resluts[END_REF] have already classified all such solutions. Theorem 1.6 Let α > -2, Ω = R 2 and u be a weak solution of (1.1) which is stable outside a compact set, then

R 2
|x| α e u dx = 4π(α + 2).

(1.6)

Furthermore, if α ∈ 2N, we have u(x) = U * (εx) + (α + 2) ln ε where ε > 0 and

U * (x) = ln 2 + 2 ln α + 2 1 + |x| α+2 .
If α ∈ 2N, let θ be the angle of x in polar coordinates, k = α+2 2 and

U * * (x) = 2 ln 2k 1 + |x| 2k -2|x| k cos(kθ -θ 0 ) tanh ξ + ln 2 cosh 2 ξ
with ξ, θ 0 ∈ R. We have then u(x) = U * * (εx) + (α + 2) ln ε where ε > 0.

For equation (1.2), we have similar Liouville type results.

Theorem 1.7 Let α > -2, N ≥ 2. The result of Theorem 1.1 holds true for weak solution of (1.2) in R N . Moreover, let u be a weak solution of (1.2) in R N with finite Morse index. Assume that

1 < p < p(N, α -) and p = N + 2 + 2α N -2 , (1.7)
then u ≡ 0.

In [4, Theorems 1.3-1.4], Dancer, Du & Guo have considered positive solutions to (1.2) with finite Morse index, in punctured domains or in exterior domains under similar conditions on p. Using results in [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF], it was proved that either these solutions are of fast decay, or up to a suitable scaling, they converge uniformly to a positive function on S N -1 , but they did not consider the Liouville type result for entire solutions of (1.2) in R N with finite Morse index. Theorem 1.7 is sharp for α ∈ (-2, 0]. However, we don't know if it holds true for α > 0, p(N, 0) ≤ p < p(N, α) and N > 10. On the other hand, let N ≥ 3, p = N +2+2α N -2 be the critical exponent and α > -2, it is well known that (1.2) possesses radial positive entire solutions, given by

V (x) = λ N -2 2 (N + α)(N -2) 1 + λ 2+α |x| 2+α N -2 2+α
, λ > 0.

(1.8)

We see that

|x| α V (x) p-1 = O(|x| -4-α ) when |x| → ∞. As α > -2, V is clearly stable outside a compact set of R N by Hardy's inequality (1.5).
Finally we consider the half-space problem of (1.2) with the Dirichlet boundary condition.

Theorem 1.8 Let α > -2 and u be a weak solution of

-∆u = |x| α |u| p-1 u in R N + = {x ∈ R N , x N > 0}, u = 0 on ∂R N + .
Suppose that u has finite Morse index in R N + . Then u ≡ 0 under the assumption (1.7). Our proofs are based on some a priori estimations using the stability condition (1.4), in the spirit of Proposition 4 in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF] (see also Proposition 1.7 of [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF] and Proposition 5 of [START_REF] Farina | Stable solutions of -∆u = e u on R N[END_REF]), but we need to be more careful with the weak solutions since they are not supposed to be bounded a priori. Another important ingredient to handle (1.2) is the asymptotic behavior of solutions near the origin and near infinity. Theorem 1.9 Let u be a weak solution of (1.2) in Ω containing 0. Suppose that u has finite Morse index and p satisfies 1 < p < p(N, α -).

Then u ∈ C(Ω) ∩ C 2 (Ω \ {0}) and the following fast decay estimate holds:

lim |x|→0 |x| 1+ 2+α p-1 |∇u(x)| = 0. (1.9)
For the fast decay estimate as |x| goes to ∞, we need more restriction on the exponent p.

Theorem 1.10 Suppose that u is a weak stable solution of

(1.2) in R N \ K where K ⊂ R N is a compact set. Assume that α > -2 and p satisfies p(N, α) < p < p(N, α -).
where

p(N, α) = (N -2) 2 -2(α + 2)(α + N ) -2 (α + 2) 3 (α + 2N -2) (N -2)(N -4α -10)
, for all N ≥ 2.

Then we have

lim |x|→∞ |x| 2+α p-1 |u(x)| = lim |x|→∞ |x| 1+ 2+α p-1 |∇u(x)| = 0. (1.10)
The estimates (1.10) were proved in [4, Theorem 1.6] only for positive solutions. We note that the Harnack type argument used in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF] cannot work with sign-changing solutions.

We will deal with equations (1.1), (1.2) respectively in sections 2 and 3, some further remarks and open questions are exposed in section 4. In the following, C or C i denotes some generic positive constant.

Hénon equation with exponential nonlinearity

Consider -∆u = |x| α e u . First we show the necessity to working with α > -2 and a quick proof of Theorem 1.6, then we prove Theorems 1.4 and 1.5.

Nonexistence of weak solution for α ≤ -2

Arguing by contradiction, assume that α ≤ -2 and u is a weak solution of (1.1) with 0 ∈ Ω. Let B(0, R) ⊂ Ω. Define v to be the average of u over spheres centered at the origin, i.e.

v(r) := u(r, θ) = 1 ω N S N -1 u(r, θ)dθ with ω N = |S N -1 |.
By Jensen's inequality, there holds

-∆v = |x| α e u = r α e u ≥ r α e v , which means that - 1 r N -1 (r N -1 v ) ≥ r α e v .
(2.1)

Note that

-r N -1 ω N v (r) = - ∂B(0,r) ∂v ∂ν dσ = - B(0,r) ∆vdx = B(0,r)
|x| α e u dx > 0, thus v (r) < 0 for any r ∈ (0, R) and v is decreasing in (0, R). As r α e v ∈ L 1 loc (Ω) and 0 ∈ Ω, we must have α > -N . Furthermore, integrate (2.1)

in [r 1 , r] with 0 < r 1 < r < R, -r N -1 v (r) ≥ r r 1 s N -1+α e v(s) ds -r N -1 1 v (r 1 ) ≥ r r 1 s N -1+α e v(s) ds.
Tending r 1 to 0, we get

-r N -1 v (r) ≥ r 0 s N -1+α e v(s) ds ≥ r N +α e v(r) N + α . (2.2)
Hence -e -v v ≥ Cr 1+α , which yields

e -v(r) > e -v(r) -e -v(r 1 ) ≥ C r r 1 s 1+α ds ∀ 0 < r 1 < r.
As α ≤ -2, a contradiction occurs by tending r 1 to 0 since the last term goes to ∞. So Proposition 1.3 is proved.

Remark 2.1 We can remark that the above proof works for -∆u = |x| α g(u) with a positive, convex and nondecreasing nonlinearity g.

Two dimensional case

Here we prove Theorem 1.6. Assume that u is stable outside

B(0, R 0 ). Fix φ ∈ C ∞ c (R) verifying φ(t) = 1 if |t| ≤ 1 and φ(t) = 0 if |t| ≥ 2. For any R ≥ 4R 0 , let ψ R (x) = φ(R -1 |x|) -φ(R -1 0 |x|), then supp(ψ R ) ⊂ B(0, R 0 ) c , ψ R is fixed in B(0, 2R 0 ) and ψ R ≡ 1 in B(0, R) \ B(0, 2R 0 ), |∇ψ R (x)| ≤ CR -1 in B(0, R) c .
Using ψ R as the test test function in (1.4), there holds

B(0,R)\B(0,2R 0 ) |x| α e u dx ≤ R 2 |x| α e u ψ 2 R dx ≤ R 2 |∇ψ R | 2 dx ≤ B(0,2R 0 ) |∇ψ R | 2 dx + B(0,2R)\B(0,R) |∇ψ R | 2 dx ≤ C Tending R to ∞, we get |x| α e u ∈ L 1 (R 2 )
. All conclusions follow straightforwardly from Theorem 1.1 in [START_REF] Prajapat | On a class of elliptic problems in R 2 : symmetry and uniqueness resluts[END_REF].

Main technical tool

As already mentioned, our proof of Liouville type results is based on the estimate Q u (ψ) ≥ 0 with suitable test function. In fact, we have the following estimate which is an extension of result in [START_REF] Farina | Stable solutions of -∆u = e u on R N[END_REF].

Proposition 2.2 Let Ω be a domain (bounded or not) in R N , N ≥ 2. Let u be a weak and stable solution of (1.1) with α > -2. Then for any integer m ≥ 5 and any β ∈ (0, 2), there exists C > 0 depending on m, α and β such that

Ω |x| α e (2β+1)u ψ 2m dx ≤ C Ω |x| -2βα |∇ψ| 2 + |ψ||∆ψ| 2β+1 dx, (2.3
)

for all functions ψ ∈ C ∞ c (Ω) verifying ψ ∞ ≤ 1.
Proof. We use some ideas in [START_REF] Farina | Stable solutions of -∆u = e u on R N[END_REF][START_REF] Crandall | Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems[END_REF], but we need to pay more attention with weak solution. As u is not assumed to be bounded, e βu ϕ is not, a priori, a licit test function (for any β > 0), even with ϕ ∈ C ∞ c (Ω). Our idea is to consider suitable truncations of e βu and proceed as in [START_REF] Crandall | Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems[END_REF]. Let

β > 0, k ∈ N, k ≥ β -1 and ζ k (t) = e βt , if t ≤ k e βk k t, if t ≥ k. We choose also another Lipschitz function η k and a differentiable function ξ k such that η k = ζ 2 k , ξ k = η k in R. Let η k (t) = β 2 e 2βt , if t ≤ k e 2βk k 2 (t -k) + β 2 e 2βk , if t ≥ k and ξ k (t) = e 2βt 4 , if t ≤ k e 2βk 2k 2 (t -k) 2 + β 2 e 2βk (t -k) + e 2βk 4 , if t ≥ k. Since u ∈ H 1 loc (Ω), clearly ζ k (u), η k (u) ∈ H 1 loc (Ω) for any k ∈ N. Applying now (1.4) with the test function ζ k (u)ϕ, where ϕ ∈ C ∞ c (Ω), we get Ω |x| α e u ζ 2 k (u)ϕ 2 dx ≤ Ω |∇(ζ k (u)ϕ)| 2 dx = Ω |∇(ζ k (u))| 2 ϕ 2 dx + Ω ζ 2 k (u)|∇ϕ| 2 dx - Ω ζ 2 k (u) 2 ∆(ϕ 2 )dx. So |x| α e u ζ 2 k (u) ∈ L 1 loc (Ω). On the other hand, Ω |∇(ζ k (u))| 2 ϕ 2 dx = Ω ζ 2 k (u)|∇u| 2 ϕ 2 dx = Ω (∇η k (u) • ∇u)ϕ 2 dx = Ω ∇u • ∇ η k (u)ϕ 2 dx - Ω η k (u)∇u • ∇(ϕ 2 )dx = Ω |x| α e u η k (u)ϕ 2 dx - Ω η k (u)∇u • ∇(ϕ 2 )dx = Ω |x| α e u η k (u)ϕ 2 dx + Ω ξ k (u)∆(ϕ 2 )dx.
For the third line, we used the following argument: as |x| α e u > 0, (1.3) is valid for any ψ ∈ H 1 0 (Ω) by density argument. The above estimates imply then

Ω |x| α e u ζ 2 k (u)ϕ 2 dx ≤ Ω |x| α e u η k (u)ϕ 2 dx + Ω ζ 2 k (u)|∇ϕ| 2 dx + Ω ξ k (u) - ζ 2 k (u) 2 ∆(ϕ 2 )dx.
Moreover, direct calculation shows that

η k (t) ≤ β 2 + 1 4k ζ 2 k (t) in R. Therefore 1 - β 2 - 1 4k Ω |x| α e u ζ 2 k (u)ϕ 2 dx ≤ Ω ζ 2 k (u)|∇ϕ| 2 dx + Ω ξ k (u) - ζ 2 k (u) 2 ∆(ϕ 2 )dx. (2.4) Set now ϕ = ψ m with ψ ∈ C ∞ c (Ω) satisfying ψ ∞ ≤ 1 and m ∈ N * . Using ζ k (u)
≤ e βu and Hölder's inequality, there holds

Ω ζ 2 k (u)|∇ϕ| 2 dx = m 2 Ω ζ 2 k (u)ψ 2(m-1) |∇ψ| 2 dx ≤ C Ω e u ζ 2 k (u) 2β 2β+1 ψ 2(m-1) |∇ψ| 2 dx ≤ C Ω |x| α e u ζ 2 k (u)|ψ| (m-1)(2β+1) β dx 2β 2β+1 Ω |x| -2βα |∇ψ| 4β+2 dx 1 2β+1
. Take m ≥ 5 so that (m -1)(2β + 1) ≥ 2mβ for any β ∈ (0, 2). As ψ ∞ ≤ 1, we obtain

Ω ζ 2 k (u)|∇ϕ| 2 dx ≤ C Ω |x| α e u ζ 2 k (u)ψ 2m dx 2β 2β+1 Ω |x| -2βα |∇ψ| 4β+2 dx 1 2β+1
.

(2.5)

Furthermore, for any β ∈ (0, 2) there exists C > 0 depending only on β such that

e 2β 2β+1 u ≥ Ce 2β 2β+1 k 1 + (u -k) 2 ∀ u ≥ k, because e t ≥ 1 + t + t 2 2
for t ≥ 0. We deduce then for any β ∈ (0, 2), there exists C > 0 (independent of k ∈ N * ) satisfying

ξ k (u) - ζ 2 k (u) 2 ≤ C e u ζ 2 k (u) 2β 2β+1 in R.
As ∆(ϕ 2 ) = 2mψ 2m-1 ∆ψ + 2m(2m -1)ψ 2m-2 |∇ψ| 2 , proceeding as above (see also [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF]), fix m ≥ 5 and applying once again Hölder's inequality, we get

Ω ξ k (u) - ζ 2 k (u) 2 ∆(ϕ 2 )dx ≤ C Ω |x| α e u ζ 2 k (u)ψ 2m dx 2β 2β+1 Ω |x| -2βα |∇ψ| 2 + |ψ||∆ψ| 2β+1 dx 1 2β+1 . (2.6) Combining (2.4)-(2.6), 1 - β 2 - 1 4k Ω |x| α e u ζ 2 k (u)ψ 2m dx ≤ C Ω |x| α e u ζ 2 k (u)ψ 2m dx 2β 2β+1 Ω |x| -2βα |∇ψ| 2 + |ψ||∆ψ| 2β+1 dx 1 2β+1
, which means that there exists C > 0 independent of k such that

Ω |x| α e u ζ 2 k (u)ψ 2m dx ≤ C Ω |x| -2βα |∇ψ| 2 + |ψ||∆ψ| 2β+1 dx, provided 1 -β 2 -1 4k > δ > 0. Fix β ∈ (0, 2), tending k → ∞, the proof of (2.
3) is completed by the monotone convergence Theorem.

Remark 2.3

In [START_REF] Wang | Partial regularity of stable solutions to the Emden equation[END_REF], the author considered the regularity of weak stable solutions for (1.1) with α = 0. He obtained higher integrability similar to (2.3) by using the simple cut-off function u k = min(k, u). However, several arguments as the iteration process in Step 3 of the proof for [19, Lemma 2.1], are not valid for the nonautonomous equation.

Proof of Theorem 1.4

Suppose that (1.1) admits a weak and stable solution u with Ω = R N and N < 10 + 4α. Fix m ≥ 5 and choose β ∈ (0, 2) such that N -2(2β + 1) -2βα < 0.

For every R > 0, consider the function φ

R (x) = φ R -1 |x| , where φ ∈ C ∞ c (R) satisfies 0 ≤ φ ≤ 1 in R, φ(t) = 1 if |t| ≤ 1 and φ(t) = 0 if |t| ≥ 2. Applying Proposition 2.2 with ψ = φ R , B(0,R) |x| α e (2β+1)u dx ≤ R N |x| α e (2β+1)u φ 2m R dx ≤ CR N -2(2β+1)-2βα , ∀ R > 0.
Taking R → ∞, we have

R N
|x| α e (2β+1)u dx = 0, which is impossible, so we are done.

Proof of Theorem 1.5

We argue always by contradiction. Suppose that (1.1) admits a weak solution u which is stable outside a compact set K in R N . There exists R 0 > 0 such that K ⊂ B(0, R 0 ), therefore we can apply Proposition 2.2 with Ω = R N \B(0, R 0 ). We claim:

• For any β ∈ (0, 2) and any R > 2R 0 , it holds

2R 0 <|x|<R |x| α e (2β+1)u dx ≤ A + BR N -2(2β+1)-2βα (2.7)
where A, B > 0 are independent of R.

• For any β ∈ (0, 2) and any B(y, 2r) ⊂ R N \ B(0, R 0 ), it holds

B(y,r) |x| α e (2β+1)u dx ≤ Cr N -2(2β+1)-2βα , (2.8)
where C > 0 is independent of r and y.

For R > 2R 0 , let φ and φ R be as in the previous proof and define

ψ R = φ R -φ R 0 . Notice that ψ R ∈ C ∞ c (R N \K) and 0 ≤ ψ R ≤ 1 in R N and ψ R is a fixed function η 0 in B(0, 2R 0 )
. Hence Proposition 2.2 (applied with m = 5 and ψ R ) implies that for all R > 2R 0 ,

{2R 0 <|x|<R} |x| α e (2β+1)u dx ≤ R N \K |x| α e (2β+1)u ψ 2 R dx ≤ C R N \K |x| -2βα |∇ψ R | 2 + |ψ R ||∆ψ R | 2β+1 dx ≤ C R 0 ≤|x|≤2R 0 |x| -2βα |∇η 0 | 2 + η 0 |∆η 0 | 2β+1 dx + C R≤|x|≤2R |x| -2βα |∇ψ R | 2 + |ψ R ||∆ψ R | 2β+1 dx ≤ A + BR N -2(2β+1)-2βα .
As the constants A, B depend only on η 0 , R 0 , φ and β, we get the claim (2.7). Using Proposition 2.2 with the test function φ r (x -y), we obtain easily the estimate (2.8).

Consider Γ(β) = N -4β -2 -2βα -. As 3 ≤ N < 10 + 4α -, Γ(0) > 0 and Γ(2) < 0, so there exist β 1 ∈ (0, 2) and ε 0 ∈ (0, 2) such that

2β 1 + 1 ≥ 2β 1 + 1 + 2β 1 α -> θ := N 2 -ε 0 > N 2 .
Let |y| > 4R 0 and R = |y| 4 , so B(y, 2R) ⊂ R N \ B(0, R 0 ). By Hölder's inequality and (2.8),

B(y,R) (|x| α e u ) θ dx ≤ B(y,R) |x| α e (2β 1 +1)u dx θ 2β 1 +1 B(y,R) |x| 2β 1 αθ 2β 1 +1-θ dx 2β 1 +1-θ 2β 1 +1 ≤ C R N -2(2β 1 +1)-2β 1 α θ 2β 1 +1 R N + 2β 1 αθ 2β 1 +1-θ 2β 1 +1-θ 2β 1 +1 = CR N -2θ , that is, B(y,R) (|x| α e u ) θ dx ≤ CR N -2θ , ∀ |y| > 4R 0 and R = |y| 4 .
(2.9)

We claim now lim

|x|→+∞ |x| 2+α e u(x) = 0. (2.10)
To prove that, we need a well-known result of Serrin in [START_REF] Serrin | Local behavior of solutions of quasi-linear equations[END_REF] (see also Theorem 7.1.1 in [START_REF] Pucci | The maximum principle[END_REF]):

Lemma 2.4 Let θ = N 2-ε 0 , ε 0 ∈ (0, 2), q ∈ (1, ∞] and δ > 0. For any weak solution of -∆η = a(x)η in B(y, 2R) ⊂ R N , if R ε 0 a(x) L θ (B(y,2R)) ≤ δ, there holds η L ∞ (B(y,R)) ≤ CR -N q η L q (B(y,2R)) (2.11)
where C is a constant depending only on N , q, θ and δ.

Moreover, the estimate (2.11) holds also for any weak nonnegative function verifying

-∆η ≤ a(x)η in B(y, 2R) ⊂ R N assuming that R ε 0 a(x) L θ (B(y,2R)) ≤ δ. Set β 2 = N -2 2(2 + α) , λ = 2β 2 + 1 2 = N + α 2(2 + α) > 0 and w = e λu .
Then β 2 ∈ (0, 2) since 3 ≤ N < 10 + 4α and N -2(2β 2 + 1) -2β 2 α = 0. Take β = β 2 in (2.7) and tending R to ∞, we obtain

|x|≥2R 0 |x| α w 2 dx < ∞.
(2.12)

We have also -∆w -λ|x| α e u w = -λ 2 w|∇u| 2 ≤ 0 in D (R N ).

Let |y| > 8R 0 and R = |y| 8 . Using the estimate (2.9), as θ = N 2-ε 0 ,

R ε 0 λ|x| α e u L θ (B(y,2R)) ≤ λR ε 0 CR N -2θ 1 θ = C R ε 0 + N θ -2 = C .
(2.13) Applying Serrin's result with η = w, a(x) = λ|x| α e u , q = 2 and δ = C of (2.13), by (2.12),

w(y) ≤ CR -N 2 w L 2 (B(y,2R)) ≤ CR -N 2 R -α 2 |x| α 2 w L 2 (B(y,2R)) = o R -N +α 2 as |y| → ∞.
Consequently e u(y) = w(y)

1 λ = o R -N +α 2λ = o R -2-α as |y| → ∞,
hence the claim (2.10) holds true.

To finish the proof, consider v, the average of u over spheres. Fix M > 0 large satisfying α + 2 -2 (N -2)M > 0 (recall that α > -2). By (2.10), there exists R M > 0 such that

-∆v(r) = r α e u ≤ 1 M r 2 , ∀ r ≥ R M . Integrating from R M to r, we deduce then v (r) ≥ - C r N -1 - 1 (N -2)M r , ∀ r ≥ R M . As N ≥ 3, there exists R > R M such that v (r) ≥ - 2 (N -2)M r , ∀ r ≥ R .
Integrating on [R , r], we get

r 2+α e v(r) ≥ Cr α+2- 2 (N -2)M , ∀ r ≥ R , which yields sup |x|=r |x| 2+α e u(x) = r 2+α sup |x|=r e u(x) ≥ r 2+α e v(r) ≥ Cr α+2- 2 (N -2)M → ∞,
which contradicts (2.10), the proof is completed.

Hénon equation with power growth

Here we consider the equation (1.2) in R N or R N + . As for Theorem 1.4, the basic argument is always the a priori estimates resulting from the stability condition (1.4).

Proposition 3.1

Let Ω be a domain (bounded or not) in R N , N ≥ 2. Let u be a weak stable solution of (1.2) with p > 1 and α > -2. Then for any γ ∈ [1, 2p + 2 p(p -1) -1) and any integer m ≥ max p+γ p-1 , 2 , there holds

Ω |∇(|u| γ-1 2 u)| 2 + |x| α |u| γ+p |ψ| 2m dx ≤ C Ω |x| - (γ+1)α p-1 |∇ψ| 2 + |ψ||∆ψ| p+γ p-1 dx (3.1) for all test functions ψ ∈ C ∞ c (Ω) verifying ψ ∞ ≤ 1,
where the constant C depends on p, m, γ and α.

Similarly, if we suppose that the weak solution of (1.2) u belongs to H 1 loc (Ω) such that u = 0 on ∂Ω and u is stable outside a compact set K ⊂ Ω, then the estimate (3.1) holds for all test functions ψ ∈ C ∞ c (R N \K) verifying ψ ∞ ≤ 1. The proof follows the main lines of the demonstration of Proposition 1.7 in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF] or Proposition 6 in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF], with small modifications. As for Proposition 2.2, we need to consider truncations of the weak solution u, but the calculation is easier here (see also [START_REF] Dávila | Partial regularity of finite Morse index solutions to the Lane-Emden equation[END_REF]).

Set just ζ

k (t) = max(-k, min(t, k)), k ∈ N and use the test function |ζ k (u)| γ-1 2 uϕ ∈ H 1 0 (Ω) in (1.4), with ϕ ∈ C ∞ c (Ω) and ϕ ∈ C ∞ c (R N \K)
respectively. The rest of proof can be proceeded as for Proposition 1.7 in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF], and by taking k tends to ∞. We omit the details.

Assume for example that u is a weak solution of (1.2), stable outside a compact set of R N , i.e. there exists R 0 > 0 such that (3.1) holds true with Ω = R N \B(0, R 0 ). Let α > -2 and p > 1, using (3.1), we can prove

• Let γ ∈ [1, 2p + 2 p(p -1) -1) and r > 2R 0 , then 2R 0 <|x|<r |∇(u γ+1 2 )| 2 + |x| α |u| γ+p dx ≤ A + Br N - (2+α)γ+2p+α p-1 , (3.2)
where A and B are constants independent of r > 2R 0 .

• Let γ ∈ [1, 2p + 2 p(p -1) -1) and B(y, 2r) ⊂ R N \B(0, R 0 ), then

B(y,r) |∇(u γ+1
2 )| 2 + |x| α |u| γ+p dx ≤ Cr N - (2+α)γ+2p+α p-1 , (3.3)
where C > 0 is independent of y and r.

The proof is very similar to that for (2.7) and (2.8) (see Step 1 of the proof for Theorem 3.3 in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF]), we leave the details for interested readers.

3.1 Slow decay estimate and proof of Theorem 1.9

Before proving the fast decay (1.9), we prove slow decay estimates for solutions on punctured domains or at infinity for 1 < p < p(N, 0), and a special regularity result when p < p(N, α -).

Theorem 3.2 Let N ≥ 2, 1 < p < p(N, 0) and u be a weak solution to (1.2) with finite Morse index. If the domain Ω contains B(0, r) \ {0} (resp. R N \ B(0, r)), there hold u ∈ C 2,β loc (Ω \ {0}) for some β ∈ (0, 1) and

u(x) = O |x| -2+α p-1 , ∇u(x) = O |x| -1-2+α p-1 , for |x| → 0 (resp. |x| → ∞). (3.4)
Moreover, when 1 < p < p(N, α -), there exists β ∈ (0, 1) such that u ∈ C 0,β loc (Ω).

Proof. First, since each single point is of zero capacity in R N when N ≥ 2, all arguments for [6, Proposition 2.1] (see also [START_REF] Du | Positive solutions of an elliptic equation with nonnegative exponent: Stability and critical power[END_REF]) are still valid although the equation is different, that is Lemma 3.3 Let N ≥ 2 and u be a weak solution to (1.2) with finite Morse index. For every x 0 ∈ Ω, there exists r 0 > 0 such that u is stable in B(x 0 , 4r 0 ).

Similar to (3.3), by using standard cut-off function ψ ∈ C 2 0 (B(x 0 , 4r 0 )) with (3.1), we obtain |x| α |u| γ+p ∈ L 1 (B(x 0 , 3r 0 )) for any γ ∈ [1, 2p + 2 p(p -1) -1). If p < p(N, 0) and x 0 = 0, as in the proof of [4, Theorem 2.1], we can claim that

B(x 0 ,2r 0 ) |x| α |u| p-1 N 2-ε 0 dx < C for some ε 0 ∈ (0, 2) and r 0 < |x 0 | 4 ,
where C is a constant depending on α, r 0 , p and ε 0 . Applying now Lemma 2.4, since -∆u = |x| α |u| p-1 u and u ∈ L 2 loc (Ω), we get u ∈ L ∞ (B(x 0 , r 0 )). As x 0 can be any point in Ω \ {0}, it means that u ∈ L ∞ loc (Ω \ {0}). Hence |x| α |u| p-1 u ∈ L q loc (Ω \ {0}) for some q > N 2 because α > -2. We get u ∈ C 2,β loc (Ω \ {0}) by classical regularity theory.

When x 0 = 0, we can still use the above idea. But the estimate |x| α |u| p-1 ∈ L N 2-ε 0 (B(0, 2r 0 )) needs also the condition N + α(θξ-1) ξ-1 > 0 (see line 5 of page 3291 in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF]), which requires ∆(p, γ, α) < 0 so p < p(N, α). When p < p(N, α -), Lemma 2.4 yields again u ∈ L ∞ (B(0, r 0 )), therefore u is Hölder continuous at 0 by equation.

The key argument to prove (3.4) is a uniform estimate inspired by the interesting work of Phan & Souplet [START_REF] Phan | Liouville-type theorems and bounds of solutions of Hardy-Hénon equations[END_REF].

Lemma 3.4 Let 1 < p < p(N, 0), B 1 := B(0, 1) and β > 0. Assume that c ∈ C 0,β (B 1 ) verifies c C 0,β (B 1 ) ≤ C 1 and c(x) ≥ C 2 > 0 in B 1 .
Then there exists a constant C > 0 depending only on β, C 1 , C 2 , p and N , such that any stable solution u to

-∆u = c(x)|u| p-1 u in B 1 satisfies |u(x)| p-1 2 + |∇u(x)| p-1 p+1 ≤ C 1 -|x| , ∀ x ∈ B 1 .
Indeed, arguing by contradiction, we can repeat exactly the proof of Lemma 2.1 in [START_REF] Phan | Liouville-type theorems and bounds of solutions of Hardy-Hénon equations[END_REF] and arrive at a nontrivial stable solution verifying -∆v = C 0 |v| p-1 v in R N , with a constant C 0 > 0. However, this is impossible by Farina's classification result [11, Theorem 1] since p < p(N, 0).

Using the scaling argument, the proof of (3.4) is the same as for [START_REF] Phan | Liouville-type theorems and bounds of solutions of Hardy-Hénon equations[END_REF]Theorem 1.2]. Look at the situation near the origin. Applying Lemma 3.

3, u is stable in B(x 0 , R) when |x 0 | = 2R > 0 is small. Define U (y) = R 2+α p-1 u(x 0 + Ry), where y ∈ B 1 = B(0, 1). (3.5)
Therefore U is a stable solution of

-∆U = c(y)|U | p-1 U in B 1 with c(y) = x 0 R + y α . We can check that 1 ≤ c(y) ≤ 3 in B 1 and c C 1,α (B 1 ) ≤ C α , hence |U (0)| + |∇U (0)| ≤ C by Lemma 3.4, which is equivalent to say |u(x 0 )| + R|∇u(x 0 )| ≤ CR -2+α p-1 , for |x 0 | = 2R small enough.
So we are done.

Proof of Theorem 1.7 for subcritical p

Consider the subcritical cases, that is

1 < p < N + 2 + 2α N -2 and p < p(N, 0). (3.6) Let γ = 1, so N - (2 + α)γ + 2p + α p -1 = N - 2p + 2 + 2α p -1 < 0.
Consequently, since u ∈ C(R N ) by Theorem 3.2 and taking r → ∞ in (3.2), we have ∇u ∈ L 2 (R N ) and |x|

α p+1 u ∈ L p+1 (R N ). Let φ R (x) = φ R -1 |x| , where φ ∈ C ∞ c (-2, 2) is a cut-off function such that 0 ≤ φ ≤ 1 in R and φ(t) = 1 for |t| ≤ 1. As uφ R ∈ H 1 0 ∩ L ∞ is
compactly supported, by density argument, we can use it as test function in (1.2), to obtain

R N |∇u| 2 φ R dx - R N |x| α |u| p+1 φ R dx = 1 2 R N |u| 2 ∆φ R dx. (3.7)
By Hölder's inequality and the estimate (3.3), we get

R N |u| 2 ∆φ R dx ≤ R<|x|<2R |x| α p+1 |u| p+1 dx 2 p+1 R<|x|<2R |x| -2α p+1 |∆φ R | p+1 p-1 dx p-1 p+1 ≤ CR N -2p+2+2α p-1 . (3.8)
As u ∈ C 2 (R N \ {0}) by Theorem 3.2, we can apply the classical Pohozaev identity to u in Ω ε,R := B(0, R) \ B(0, ε) with R > ε > 0, then

N + α p + 1 Ω ε,R |x| α |u| p+1 dx - N -2 2 Ω ε,R |∇u| 2 dx = ∂Ω ε,R |x| α x, ν |u| p+1 dσ + ∂Ω ε,R ∂u ∂ν x, ∇u dσ - ∂Ω ε,R |∇u| 2 2
x, ν dσ.

(3.9)

Using (3.4) and p < N +2+2α N -2 , ∂B(0,R)

|x||∇u| 2 + |x| α+1 |u| p+1 dσ ≤ CR N -2- 2(2+α) p-1 → 0, if R → ∞. (3.10)
On the other hand, we have -∆u = O(|x| α ) near the origin and u ∈ C(R N ). As α > -2, by regularity theory and Sobolev embedding we can claim that ∇u ∈ L q loc (R N ) for some q > 2. Applying again Hölder's inequality,

1 ε ε 0 ∂B(0,s) |x||∇u| 2 dσ ds ≤ B(0,ε)
|∇u| 2 dx ≤ ε σ ∇u L q (B(0,ε)) , where σ > 0.

Thus there exists a sequence ε j → 0 such that lim j→∞ ∂B(0,ε j )

|x||∇u| 2 dσ = 0. ( 3.11)
Take ε = ε j in (3.9) then tend R and j to ∞. It follows from (3.10) and (3.11) that

N + α p + 1 R N |x| α u p+1 dx - N -2 2 R N |∇u| 2 dx = 0.
Combining with (3.7) and (3.8), we have

N -2 2 - N + α p + 1 R N |x| α |u| p+1 dx = 0.
As N -2 2 -N +α p+1 < 0 for subcritical p, we conclude that u ≡ 0 under the assumption (3.6).

Fast decay behavior at infinity

We show here Theorem 1. (1, ∞). If N ≤ 10 + 4α, (2 + α)γ(p) + 2p + α > N (p -1) for any p > 1. We can check also that if N > 10 + 4α, (2 + α)γ(p) + 2p + α = N (p -1) with p = p(N, α) given in Theorem 1.1. Therefore under the assumption (3.12), as p(N, α) is increasing with respect to α, there exists

γ 1 ∈ [1, 2p + 2 p(p -1) -1) such that N (p -1) -(2 + α)γ 1 -2p -α = 0. Set now β 0 = γ 1 + p 2 > 1 and ω(x) = |u(x)| β 0 ≥ 0, Using (3.
2) with γ = γ 1 and letting r → ∞, we have

|x|>2R 0 |x| α ω 2 dx < ∞. (3.13) and -∆ω -β 0 |x| α |u| p-1 ω = -β 0 (β 0 -1)|u| β 0 -2 |∇u| 2 ≤ 0 in D (R N ).
Moreover, as p < p(N, α -) ≤ p(N, 0), it follows from (3.3) that there exists ε 0 (p, N ) ∈ (0, 2) such that B(y,r)

|x| α |u| p-1 N 2-ε 0 dx ≤ Cr N -2N 2-ε 0 , ∀ B(y, 2r) ⊂ R N \ B(0, R 0 )
where the constant C is independent of y and r.

Let |y| > 8R 0 and R = |y| 8 , so B(y, 4R) ⊂ R N \ B(0, R 0 ). Therefore R ε 0 β 0 |x| α |u| p-1 L N 2-ε 0 (B(y,2R)) ≤ β 0 R ε 0 CR N -2N 2-ε 0 2-ε 0 N = C , Applying (3.13) and Lemma 2.4, ω(y) ≤ CR -N 2 ω L 2 (B(y,2R)) ≤ CR -N 2 R -α 2 |x| α 2 ω L 2 (B(y,2R)) = o R -N +α 2 as |y| → ∞. Hence |u(y)| = ω(y) 1 β 0 = o R -N +α 2β 0 = o R -2+α p-1 as |y| → ∞,
which shows the fast decay of u. To prove the second claim of (1.10), we observe that

-∆u(y) = |y| α |u| p-1 u = o R -2+α p-1 -2
as |y| → ∞.

The scaling argument and standard elliptic theory imply then Here we will use Kelvin's transformation as in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF]. Let

|∇u(y)| = o R -2+α p-1 -1 as |y| → ∞.
v(x) = |x| 2-N u x |x| 2 , for |x| > 0 small, then v verifies -∆v = |x| β |v| p-1 v with β = (N -2)(p -1) -(4 + α). We can verify that β > -2 since p > N +α N -2
. Moreover, we have the following properties (see Proposition 3.1 and the proof of Theorem 3.4 in [START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF]):

• The solution v is stable over B(0, r)\{0} for r > 0 small since u is stable outside B(0, r -1 ).

• The assumption (3.14) implies N +2+2β N -2 < p < p(N, β -). Indeed, p(N, α) < p < p(N, α) yields that β < p(N, β).

Using estimate (3.4) for v, we can prove that v is a weak stable solution of -∆v = |x| β |v| p-1 v in B(0, r). For example, there holds

B(0,r) |∇v| 2 dx ≤ C r 0 s N -3- 2(2+β) p-1 ds < ∞, since N -2 - 2(2 + β) p -1 > 0.
Moreover v is continuous in B(0, r) by Theorem 3.2 as β < p(N, β -), so we get

u(x) = |x| 2-N v x |x| 2 = O |x| 2-N = o |x| -2+α p-1 as |x| → ∞.
As p < p(N, 0), combining with (1.9) for v, we have

∇u(x) = O |x| 1-N + o |x| 2+β p-1 +1-N = o |x| -1-2+α p-1 as |x| → ∞.
The proof is completed. In fact, combining Theorems 1.9 and 1.10, the regularity result in Theorem 3.2, the supercritical exponent situation for Theorem 1.7 is a direct consequence of the following nonexistence result. As in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF][START_REF] Wei | Liouville theorems for finite Morse index solutions of biharmonic problem[END_REF], we use the Emden-Fowler change of variable u(r, σ) = r -2+α

p-1 w(t, σ), t = ln r.

Therefore w satisfies

w tt + A 1 w t + ∆ S N -1 w + A 2 w + |w| p-1 w = 0 in R × S N -1 ,
where

A 1 = N -2 - 2 + α p -1 = 0, A 2 = - 2 + α p -1 N -2 - 2 + α p -1
and ∆ S N -1 denotes the Laplace-Beltrami operator on the unit sphere S N -1 ⊂ R N . Set

w 2 t dσdt = A 1 R N w 2 t dx,
which means that w t = 0 in R N . Therefore w ≡ 0 since lim t→±∞ w(t, σ) = 0, so u ≡ 0.

Remark 3.7 We can see that the above proof works also for p(N, α) < p < min N + 2 + 2α N -2 , p(N, α -) .

Proof of Theorem 1.8

The proof is a simple adaptation of ideas for Theorem 9(b) in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF]. The main point is that we can consider v, the odd extension of u. Then v is a weak solution of (1.2) in R N . The crucial argument is that we can use test function as ζ k (u)ψ with ψ ∈ C ∞ c (R N ) as u verifies the Dirichlet boundary condition on ∂R N + , we get all the corresponding estimates since all the boundary terms are zero when doing the integration by parts (see more details in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF]). For example, u is stable outside a compact set K ⊂ R N + , so we can obtain the estimate (3.1) for ψ ∈ C ∞ c (R N \ K). By symmetry, the same estimate holds with v in R N -with ψ ∈ C ∞ c (R N \ K ) where K is the mirror symmetry of K. By taking the sum, we can claim that for any γ ∈ In other words, we refind all the regularity results and the corresponding estimates for v as for weak solution in R N with finite Morse index. For example, we can prove the corresponding result of Lemma 3.4 to solution of -∆u = c(x)|u| p-1 u in B 1 ∩ R N + verifying u = 0 on B 1 ∩ ∂R N + , where the contradiction comes from the classification result on half space, Theorem 9(b) in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF] instead of [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF]Theorem 1].

Proceeding as for Theorem 1.7, we refine then v ≡ 0, so is u.

Further remarks and open questions

In general, the Hénon equation -∆u = |x| α g(u) is more delicate to handle than the corresponding autonomous equation, i.e. when α = 0. Many properties for the autonomous equation are no longer true or less understood for the nonautonomous situation.

For example, let α > 0; we don't know if a positive and continuous solution to -∆u = |x| α u p in R N could exist with a subcritical exponent N + max(2, α) N -2 < p < p α := N + 2 + 2α N -2 .

Only very recently, Phan and Souplet proved the nonexistence of bounded positive solution in R 3 for any 1 < p < p α , see [START_REF] Phan | Liouville-type theorems and bounds of solutions of Hardy-Hénon equations[END_REF] and the references therein.

For the critical exponent case p = p α with α > -2, the existence of radial solutions in R N is known via (1.8), and they are stable outside a compact set. When α = 0 and p = N +2 N -2 , Farina proved in [START_REF] Farina | On the classification of solutions of the Lane-Emden equation on unbounded domains of R N[END_REF]Theorem 2] a very interesting equivalence between the following assumptions:

Remark 3 . 5

 35 The estimate (3.4) was proved in Theorems 5.1 and 5.2 of[START_REF] Dancer | Finite Morse index solutions of an elliptic equation with supercritical exponent[END_REF] with different method.Proof of Theorem 1.9. Thanks to Theorem 3.2, we need only to consider (1.9), which is also a direct consequence of scaling argument. Let |x 0 | > 0 be small such that B(0, 2|x 0 |) ⊂ Ω. Define V (y) = u(x 0 + Ry) in B 1 , where 2R = |x 0 |. As u ∈ C(Ω), we have ∆V ∞ ≤ CR 2+α , so |∇V (0)| ≤ CR 2+α by standard elliptic theory. Hence |∇u(x 0 )| ≤ C|x 0 | 1+α , we get easily (1.9) since α > -2.

3. 4

 4 Proof of Theorem 1.7 for supercritical p Now we are in position to prove Theorem 1.7 for p verifying p > N + 2 + 2α N -2 and p < p(N, α -).

Proposition 3 . 6

 36 Let α > -2 and p > 1 and p = N +2+2α 2+α . Then there does not exist any nontrivial weak solution of (1.2) in R N satisfying u ∈ C 2 (R N \ {0}), the estimates (1.10) and lim |x|→0 |x| 2+α p-1 |u(x)| = lim |x|→0 |x| 1+ 2+α p-1 |∇u(x)| = 0. (3.15)

E

  (w)(t) = S N -1

The estimates ( 1 .A 1 R

 11 10) and (3.15) yield lim t→±∞ w(t, σ) = lim t→±∞ |w t (t, σ)| = lim t→±∞ |∇ S N -1 w(t, σ)| = 0, hence lim t→±∞ E(w)(t) = 0. On the other hand, integrating the equation of w, we get 0 = S N -1

  [1, 2p + 2 p(p -1) -1) and m ∈ N large enough, there holdsΩ |∇(|v| γ-1 2 v)| 2 + |x| α |v| γ+p |ψ| 2m dx ≤ C for any ψ ∈ C ∞ c (R N \ (K ∪ K )) verifying ψ ∞ ≤ 1.

* D.Y. is partly supported by the French ANR project referenced ANR-08-BLAN-0335-01. We would like to thank Phan and Souplet for sending us their

(i) u is a finite Morse index solution of (1.2) in R N ;

(ii) u is a solution of (1.2) in R N and u is stable outside a compact set;

(iii) u is a solution of (1.2) in R N and ∇u ∈ L 2 (R N ).

Combining with [START_REF] Ding | On a conformally invariant elliptic equation on R N[END_REF], we get infinitely many (conformally non-equivalent) sign-changing solutions to -∆u = |u| 4 n-2 u in R N with finite Morse index.

For weak solution to (1.2) with general p ≤ p α and α > -2, the implications (i) ⇒ (ii) ⇒ (iii) hold true. For (ii) ⇒ (iii), we can take just γ = 1 in the estimate (3.2). However it is difficult to give a general positive or negative answer for implication (ii) ⇒ (i) or (iii) ⇒ (ii), which seem to be completely open, even for α = 0, except Farina's result.

An interesting work linked to that is [START_REF] Gladiali | Symmetry and nonexistence of low Morse index solutions in unbounded domains[END_REF] (see also the references therein), where the authors proved some relationships between the symmetry and low Morse index solutions for -∆u = f (|x|, u). For example, using [13, Theorem 1.5], we can claim that for α > 1, any stable solution to (1.2) in R N verifying ∇u ∈ L 2 (R N ) must be radial.

When α = 0 and p = p α , the existence of non-radial or sign-changing entire solution to (1.2) with finite Morse index seems also unknown.

When p ≥ p(N, α) and α > -2, we have the following question: Does there exist classical or weak solution u to (1.2) in R N with p ≥ p(N, α) and 0 < ind(u) < ∞?

As far as we are aware, the problem is open, even for α = 0. All known solutions for α = 0 and p ≥ p(N, 0), as radial solutions or lower dimensional solutions are either stable or have infinite Morse index in R N . Of course, we can ask the similar question for (1.1) with N ≥ 10 + 4α.

The understanding of finite Morse index but unstable solutions to (1.2) in R N for p ≥ p(N, α) is far away from evident, even they have necessarily finite energy, i.e. ∇u ∈ L 2 (R N ). For example, when α = 0, if such a solution u exists and if u → 0 as |x| → ∞, it must be non-radial and signchanging, since any positive solution is radial and all radial solutions are stable as p ≥ p(N, 0). Theorem 1.6 in [START_REF] Gladiali | Symmetry and nonexistence of low Morse index solutions in unbounded domains[END_REF] yields then ind(u) ≥ N + 1 ≥ 12, which means that u must have a large Morse index.

The assumption lim |x|→∞ u(x) = 0 seems to be reasonable for any finite Morse index solution of (1.2), but we don't know if it holds true in general for p ≥ p(N, 0).


Arnaud Marsiglietti
email: arnaud.marsiglietti@u-pem.fr

On the improvement of concavity of convex measures

Keywords: Brunn-Minkowski inequality, convex measure, Gaussian measure

.

Introduction

The classical Brunn-Minkowski inequality asserts that for all non-empty Borel sets A, B ⊂ R n and for every λ ∈ [0, 1], one has The Brunn-Minkowski inequality is a beautiful and powerful inequality in Geometry and Analysis, leading to many interesting consequences. For more information on this inequality and its influences on several mathematical theories, see the survey by Gardner [START_REF] Gardner | The Brunn-Minkowski inequality[END_REF]. See also the book by Schneider [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF], as a general reference in Convex Geometry.

|(1 -λ)A + λB| 1 n ≥ (1 -λ)|A| 1 n + λ|B| 1 n , (1)
Recently, Gardner and Zvavitch [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF] proved that the Gaussian measure γ n on R n , defined by

dγ n (x) = 1 (2π) n 2 e -|x| 2 2 dx, x ∈ R n
where | • | denotes the Euclidean norm, satisfies a Brunn-Minkowski-type inequality of the form (1) for some special classes of sets. More precisely, Gardner and Zvavitch showed that for coordinate boxes (i.e. a product of intervals) A, B ⊂ R n that contain the origin, or for A, B ⊂ R n that are dilates of the same symmetric convex set, and for every λ ∈ [0, 1], one has

γ n ((1 -λ)A + λB) 1 n ≥ (1 -λ)γ n (A) 1 n + λγ n (B) 1 n . (2)
They also conjectured that inequality [START_REF] Borell | Convex set functions in d-space[END_REF] holds for all convex symmetric sets A, B ⊂ R n . As a consequence of the Prékopa-Leindler inequality [START_REF] Prékopa | Logarithmic concave measures with application to stochastic programming[END_REF], [START_REF] Leindler | On a certain converse of Hölder's inequality, II[END_REF], [START_REF] Prékopa | On logarithmic concave measures and functions[END_REF], the Gaussian measure satisfies for all Borel sets A, B ⊂ R n and for every λ ∈ [0, 1],

γ n ((1 -λ)A + λB) ≥ γ n (A) 1-λ γ n (B) λ . (3)
Using the terminology of Borell [START_REF] Borell | Convex set functions in d-space[END_REF] (see Section 2 below for further details), inequality (3) means that the Gaussian measure is a log-concave measure.

By comparing means, one sees that inequality (2) is stronger than inequality (3), hence the results of Gardner and Zvavitch improve the concavity of the Gaussian measure by showing that this measure is 1 n -concave if restricted to special classes of sets.

We will see in this paper that the results of Gardner and Zvavitch can be extended to a more general class of measures called convex measures that includes the Gaussian measure. This is the mathematical underlying idea of the Gaussian Brunn-Minkowski inequality (2), i.e. under symmetry assumptions one can improve a certain property; in this case it is the concavity of a measure. However, we will see that symmetries are not the only hypothesis that allow improvement of concavity properties of a measure.

This paper is devoted to the study of the following question:

Question 1. For which value s ∈ [-∞, +∞],
for which class M of measures on R n and for which class C of couples of Borel subsets of R n one has

µ((1 -λ)A + λB) ≥ ((1 -λ)µ(A) s + λµ(B) s ) 1 s (4)
for every µ ∈ M, for every (A, B) ∈ C such that µ(A)µ(B) > 0 and for every λ ∈ [0, 1]?

The right-hand side of inequality (4) has to be interpreted by µ(A) 1-λ µ(B) λ for s = 0, by min(µ(A), µ(B)) for s = -∞ and by max(µ(A), µ(B)) for s = +∞.

Borell [START_REF] Borell | Convex set functions in d-space[END_REF] (see also [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF]) answered Question 1 when M is the class of sconcave measures on R n , s ∈ [-∞, +∞], and when C is the class of all couples of Borel subsets of R n (see Section 2).

Question 1 has been explored for s = 1, when restricted to the Lebesgue measure, by Bonnesen [START_REF] Bonnesen | Theorie der konvexen Körper[END_REF] and is still being studied (see e.g. [START_REF] Hernández Cifre | Refinements of the Brunn-Minkowski inequality[END_REF]).

The main results of this paper are contained in the following theorem: Theorem 1.

1. Let µ be an unconditional log-concave measure on R n and let A be an unconditional convex subset of R n . Then, for every A 1 , A 2 ∈ {αA; α > 0} and for every λ ∈ [0, 1], we obtain

µ((1 -λ)A 1 + λA 2 ) 1 n ≥ (1 -λ)µ(A 1 ) 1 n + λµ(A 2 ) 1 n .
2. Let µ i , 1 ≤ i ≤ n, be measures with densities φ i : R → R + such that φ i are non-decreasing on (-∞; 0] and non-increasing on [0; +∞). Let µ be the product measure of µ 1 , . . . , µ n and let A, B ⊂ R n be the product of n Borel subsets of R such that 0 ∈ A ∩ B. Then, for every λ ∈ [0, 1], we obtain µ((1 -λ)A + λB)

1 n ≥ (1 -λ)µ(A) 1 n + λµ(B) 1 n .
In the next section, we introduce new terminology and in the third section, we prove Theorem 1. In the last section, we discuss how these results improve concavity properties of the (extended) parallel volume.

Preliminaries

We work in the Euclidean space R n , n ≥ 1, equipped with the n 2 norm | • |. The closed unit ball is denoted by B n 2 , the unit sphere by S n-1 and the canonical basis by {e 1 , • • • , e n }. We also denote by | • | the Lebesgue measure on R n . For u ∈ S n-1 , we denote by u ⊥ the hyperplane orthogonal to u. For non-empty sets A, B in R n we define their Minkowski sum

A + B = {a + b; a ∈ A, b ∈ B}.
The parallel volume of a non-empty Borel set A ⊂ R n is the function defined on R + by t → |A + tB n 2 |.

A subset A ⊂ R n is said to be symmetric if A = -A. A function f : R n → R is said to be unconditional if for every (x 1 , • • • , x n ) ∈ R n and for every (ε 1 , • • • , ε n ) ∈ {-1, 1} n , one has f (ε 1 x 1 , • • • , ε n x n ) = f (x 1 , • • • , x n ).
A subset A ⊂ R n is said to be unconditional if the indicator function of A, denoted by 1 A , is unconditional. A measure with density function is said to be symmetric (resp. unconditional) if its density function is even (resp. unconditional).

Let us recall the terminology and the results of s-concave measures introduced by Borell in [START_REF] Borell | Convex set functions in d-space[END_REF]. One says that a measure

µ on R n is s-concave, s ∈ [-∞, +∞], if the inequality µ((1 -λ)A + λB) ≥ ((1 -λ)µ(A) s + λµ(B) s ) 1 s (5)
holds for all Borel subsets A, B ⊂ R n such that µ(A)µ(B) > 0 and for every λ ∈ [0, 1]. The limit cases are interpreted by continuity, as mentioned in the introduction. The 0-concave measures are also called log-concave measures.

Notice that an s-concave measure is r-concave for every r ≤ s. Thus, every s-concave measure is -∞-concave. The -∞-concave measures are also called convex measures.

From inequality (3), the Gaussian measure is a log-concave measure and as a consequence of the Brunn-Minkowski inequality (1), the Lebesgue measure is a 1 n -concave measure. For every s ∈ [-∞, +∞], Borell gave a complete description of s-concave measures. In particular, for s ≤ 1 n , Borell showed that every measure µ that is absolutely continuous with respect to the n-dimensional Lebesgue measure, is s-concave if and only if its density function is a γ-concave function, with

γ = s 1 -sn ∈ [- 1 n , +∞],
where a function f :

R n → R + is said to be γ-concave, with γ ∈ [-∞, +∞], if the inequality f ((1 -λ)x + λy) ≥ ((1 -λ)f (x) γ + λf (y) γ ) 1 γ
holds for every x, y ∈ R n such that f (x)f (y) > 0 and for every λ ∈ [0, 1]. As for the s-concave measures, the limit cases are interpreted by continuity. Notice that a 1-concave function is concave on its support, that f is a -∞-concave function if and only if f has convex level sets, and that f is a +∞-concave function if and only if f is constant on its support.

Measures with -∞-concave density function are a natural generalization of convex measures. The results of Borell show that a measure that has a γconcave density function with γ < -1 n , does not satisfy a concavity property of the form (5) (but satisfies other forms of concavity [START_REF] Dancs | On a class of integral inequalities and their measuretheoretic consequences[END_REF]). However, we will show that if restricted to a special class of sets, such measures are 1 n -concave.

In this paper, we call sub-convex measure a measure that has a -∞concave density function. Notice that convex measures are sub-convex.

Brunn-Minkowski type inequality for sub-convex measures

In this section, we partially answer Question 1 by investigating possible improvements of the concavity of sub-convex measures. Gardner and Zvavitch [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF] noticed in the case of the Gaussian measure, that the position of the sets A and B plays an important role. Indeed, since for s-concave probability measures µ, with s ≤ 0, the density function tends to 0 at infinity and the support can be equal to R n , one can find sets A and B such that A contains the origin and A+B 2 is far from the origin. Thus for r > 0, the inequality

µ A + B 2 r ≥ µ(A) r + µ(B) r 2
will not be satisfied. Hence, the position of sets A and B is an inherent constraint of the problem. Notice also that in the definition of s-concave measures, the condition µ(A)µ(B) > 0 is already a constraint on the position of A and B with respect to the support of µ.

Notice that Question 1 has an answer for s = +∞ if M is the class of convex measures and if C is the class of couples of Borel sets with the same measure. Indeed, one then has for every λ ∈ [0, 1],

µ((1 -λ)A + λB) ≥ inf(µ(A), µ(B)), by definition. Since µ(A) = µ(B), it follows that µ((1 -λ)A + λB) ≥ µ(A) = max(µ(A), µ(A)) = max(µ(A), µ(B)).
Notice also that for every measure µ and for all Borel sets A, B such that A ⊂ B, one has for every λ ∈ [0, 1],

µ((1 -λ)A + λB) ≥ min(µ(A), µ(B)),
since in this case one has, (1 -λ)A + λB ⊃ (1 -λ)A + λA ⊃ A.

The case of symmetric measures and symmetric sets

Under symmetry assumptions, the best concavity one can obtain is 1 n by considering, for example, the Lebesgue measure, which fulfills a lot of symmetries (unconditional), and two dilates of B n 2 (which are unconditional). This was noticed by Gardner and Zvavitch [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF] also for the Gaussian measure.

A sufficient condition to ensure that a measure µ on R n is 1 n -concave in the class of dilates of a fixed Borel set A ⊂ R n is that the function t → µ(tA) is 1 n -concave. The following proposition gives a sufficient condition for this. Proposition 3.1. Let φ : R n → R + be a measurable function such that for every x ∈ R n , the function t → φ(tx) is non-increasing on R + . Let µ be a measure with density function φ and let A be a Borel subset of R n containing 0. If the function t → µ(e t A) is log-concave on R, then the function t → µ(tA) is 1 n -concave on R + . Proof. Let µ be a measure with density function φ satisfying the assumptions of Proposition 3.1 and let A be a Borel subset of R n containing 0. Let us denote F (t) = µ(tA), for t ∈ R + . Notice that F is non-decreasing and continuous on R + . By assumption, the function t → F (e t ) is log-concave on R. It follows that the right derivative of F , denoted by F + , exists everywhere and that t → tF + (t)/F (t) is non-increasing on (0, +∞).

Notice that the function F is 1 n -concave on R + if and only if the function

t → tF + (t) F (t) F (t) 1 n
t is non-increasing on R + . A direct change of variables shows that

F (t) t n = A φ(tx) dx.
By assumption, the function t → φ(tx) is non-increasing on R + . It follows that the function t → F (t)

1 n /t is non-increasing on (0, +∞). Hence, the function t → (F (t) 1 n ) + is non-increasing on (0, +∞) as the product of two non-negative non-increasing functions on (0, +∞). We conclude that F is 1 n -concave on R + .

Remarks.

1. Proposition 3.1 is established in [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF] for the Gaussian measure by differentiating twice.

2. The assumption t → φ(tx) is non-increasing on R + is satisfied if φ is an even -∞-concave function. Proposition 3.1 is related to the (B)-conjecture. This conjecture was posed by W. Banaszczyk [START_REF] Latała | On some inequalities for Gaussian measures[END_REF] and asks whether the function t → γ n (e t A) is log-concave on R, for every convex symmetric set A ⊂ R n . The (B)conjecture was proved by Cordero-Erausquin, Fradelizi and Maurey in [START_REF] Cordero-Erausquin | The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems[END_REF]. In the same paper [START_REF] Cordero-Erausquin | The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems[END_REF], the authors have also shown that for every unconditional log-concave measure µ on R n and for every unconditional convex subset A ⊂ R n , the function t → µ(e t A) is log-concave on R. Using this and the point 2. of the previous remark, we may apply Proposition 3.1 to obtain the following corollary: Corollary 3.2. Let µ be an unconditional log-concave measure on R n and let A be an unconditional convex subset of R n . Then, the measure µ is 1 n -concave in the class of dilates of A. More precisely, for every A 1 , A 2 ∈ {αA; α > 0} and for every λ ∈ [0, 1], we obtain

µ((1 -λ)A 1 + λA 2 ) 1 n ≥ (1 -λ)µ(A 1 ) 1 n + λµ(A 2 ) 1 n .
Remark. Very recently, Livne Bar-on [START_REF] Bar-On | The (B) conjecture for uniform measures in the plane[END_REF] and Saroglou [START_REF] Saroglou | Remarks on the conjectured log-Brunn-Minkowski inequality[END_REF] proved, using different methods, that in dimension 2 for the uniform measure µ K on a symmetric convex set K ⊂ R 2 (i.e. dµ K (x) = 1 K (x) dx), the function t → µ K (e t A) is log-concave on R for every symmetric convex set A ⊂ R 2 . However, for our problem, this information is not useful since the uniform measure on a convex subset of R n is a 1 n -concave measure.

A natural question is to ask if the role of the symmetry can be relaxed. When restricted to the Gaussian measure, it has been shown by Nayar and Tkocz in [START_REF] Nayar | A note on a Brunn-Minkowski inequality for the Gaussian measure[END_REF], that in dimension 2 there exist non-symmetric convex sets A, B ⊂ R 2 satisfying 0 ∈ A ⊂ B and

γ 2 A + B 2 1 2 < γ 2 (A) 1 2 + γ 2 (B) 1 2 2 . ( 6
)
One can then construct an explicit counterexample in every dimension n ≥ 2. Moreover, the counterexample in [START_REF] Nayar | A note on a Brunn-Minkowski inequality for the Gaussian measure[END_REF] shows more than inequality [START_REF] Fradelizi | On the analogue of the concavity of entropy power in the Brunn-Minkowski theory[END_REF]. It shows that

γ 2 A + B 2 s < γ 2 (A) s + γ 2 (B) s 2 , (7)
for every s ≥ 1 -2 π . However, it could be of interest to know what happens when s ∈ (0, 1 -2 π ). Notice that the same counterexample with the following log-concave unconditional measure instead of the Gaussian measure dµ(x, y) = e -|x| e -|y| dx dy, (x, y) ∈ R 2 satisfies inequality (7) for every s > 0.

Thus, in general, the symmetry assumption on the measure is not sufficient.

On the other hand, the concavity of a non-symmetric convex measure cannot be improved in general in the class of symmetric sets even in dimension 1: Proposition 3.3. Let 0 < s < 1 and r > s. There exists an s-concave measure µ on R and symmetric sets A, B ⊂ R such that

µ A + B 2 < µ(A) r + µ(B) r 2 1 r . Proof. Let us define dµ(x) = x 1/γ 1 R + (x) dx, with γ = s 1-s > 0. Let us consider the sets A = [-a, a] and B = [-b, b] with 0 < a < b. Notice that lim a→0 µ A + B 2 = µ B 2 = µ(B) 2
1 s = lim a→0 µ(A) s + µ(B) s 2 1 s . Since µ(A) = µ(B), it follows by comparing means that µ(A) s + µ(B) s 2 1 s < µ(A) r + µ(B) r 2 1 r
.

we conclude that for sufficiently small a,

µ A + B 2 < µ(A) r + µ(B) r 2 1 r
.

Thus, in general, the symmetry assumption on the sets is not sufficient.

The case of sets with a maximal section of equal measure

In this section, we consider C to be the class of couples of Borel subsets of R n having a maximal section of equal measure. A famous result of Bonnesen [START_REF] Bonnesen | Theorie der konvexen Körper[END_REF] (for convex sets) states that if A, B ⊂ R n satisfy

sup t∈R |A ∩ (u ⊥ + tu)| n-1 = sup t∈R |B ∩ (u ⊥ + tu)| n-1 ,
for a certain u ∈ S n-1 , where |•| n-1 denotes the (n-1)-dimensional Lebesgue measure, then for every λ ∈ [0, 1], one has

|(1 -λ)A + λB| ≥ (1 -λ)|A| + λ|B|.
There exists a functional version of Bonnesen's result established by Henstock and Macbeath [START_REF] Henstock | On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik[END_REF] in dimension 1 (see Proposition 3.4 below) and later on by Dancs and Uhrin [START_REF] Dancs | On a class of integral inequalities and their measuretheoretic consequences[END_REF] in higher dimension (see Proposition 3.8 below).

Proposition 3.4 (Henstock, Macbeath [START_REF] Henstock | On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik[END_REF]). Let λ ∈ [0, 1]. Let f, g, h : R → R + be non-negative measurable functions such that max(f ) = max(g) and such that for every x, y ∈ R h((1 -λ)x + λy) ≥ min(f (x), g(y)).

Then, one has

R h(x) dx ≥ (1 -λ) R f (x) dx + λ R g(x) dx.
We deduce the following result:

Proposition 3.5. Let φ : R → R + be a -∞-concave function such that max(φ) is attained at a ∈ R. Let µ be a measure with density function φ.

Let A, B be Borel subsets of R such that a ∈ A∩B. Then, for every λ ∈ [0, 1], we have µ((1 -λ)A + λB) ≥ (1 -λ)µ(A) + λµ(B).

Proof. Let λ ∈ [0, 1]. We define, for every x ∈ R, h(x) = φ(x)1 (1-λ)A+λB (x), f (x) = φ(x)1 A (x), g(x) = φ(x)1 B (x).
Notice that for every x, y ∈ R one has

h((1 -λ)x + λy) ≥ min(f (x), g(y)),
and max(f ) = max(g) = φ(a). It follows from Proposition 3.4 that

R h(x) dx ≥ (1 -λ) R f (x) dx + λ R g(x) dx.
In other words, we obtain µ((1 -λ)A + λB) ≥ (1 -λ)µ(A) + λµ(B).

Remark. Proposition 3.5 was established in [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF] for the case where µ is the Gaussian measure on R and where A, B ⊂ R are convex. In the same paper, the authors were able to remove the convexity assumption for only one set, by using long computations. Our method bypasses the use of geometric tools and relies on the functional version Proposition 3.4.

Conversely, if a measure µ on R, with density function φ with respect to the Lebesgue measure, satisfies

µ((1 -λ)A + λB) ≥ (1 -λ)µ(A) + λµ(B),
for every λ ∈ [0, 1] and for all symmetric convex sets A, B ⊂ R, then one has for every λ ∈ [0, 1] and for every a, b ∈ R + ,

(1-λ)a+λb -((1-λ)a+λb) φ(x) dx ≥ (1 -λ) a -a φ(x) dx + λ b -b φ(x) dx.
Proposition 3.9. Let µ be a measure with density function φ : R n → R + such that φ is -1 n-1 -concave. Let A, B be Borel subsets of R n . If there exists u ∈ S n-1 such that

sup t∈R µ n-1 (A ∩ (u ⊥ + tu)) = sup t∈R µ n-1 (B ∩ (u ⊥ + tu)), then, for every λ ∈ [0, 1], we have µ((1 -λ)A + λB) ≥ (1 -λ)µ(A) + λµ(B). Proof. Let λ ∈ [0, 1]. Let us take f = φ1 A , g = φ1 B and h = φ1 (1-λ)A+λB . Then, for every x, y ∈ R n , one has h((1 -λ)x + λy) ≥ ((1 -λ)f (x) γ + λg(y) γ ) 1 γ , where γ = -1 n-1 . Moreover, u ⊥ f (x + tu) dx = A∩(u ⊥ +tu) φ(x) dx = µ n-1 (A ∩ (u ⊥ + tu)).
It follows that m u (f ) = m u (g). From Proposition 3.8, we obtain that

µ((1 -λ)A + λB) ≥ (1 -λ)µ(A) + λµ(B).

Application to the parallel volume

Let us see how improvements of the concavity of sub-convex measures can improve the concavity of a generalized form of the parallel volume. The parallel volume of a Borel subset A of R n , namely the function t → |A+tB n 2 |, is a particularly useful function in Geometry, which has been highlighted by the precursor works of Steiner [START_REF] Steiner | Über parallele Flächen[END_REF]. For more modern applications, the parallel volume and its generalized forms are still studied (see e.g. [START_REF] Hug | A local Steiner-type formula for general closed sets and applications[END_REF], [START_REF] Kampf | The parallel volume at large distances[END_REF]). Moreover, this notion of parallel volume leads to the powerful theory of mixed volumes (see [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF] for further details).

As a consequence of the Brunn-Minkowski inequality (1), one can see that if A ⊂ R n is convex, then the parallel volume of A is 1 n -concave on R + . More generally, if a measure µ is s-concave, with s ∈ [-∞; +∞], in the class of sets of the form {A + tB; t ∈ R + }, where A and B are convex subsets of R n , then the generalized parallel volume t → µ(A + tB) is s-concave on R + . Indeed, for every t 1 , t 2 ∈ R + and for every λ ∈ [0, 1], one has µ(A + ((1 -λ)t 1 + λt 2 )B) = µ((1 -λ)(A + t 1 B) + λ(A + t 2 B))

≥ ((1 -λ)µ(A + t 1 B) s + λµ(A + t 2 B) s ) 1 s .
Using this and Corollary 3.6, we obtain the following corollary: Corollary 4.1. Let µ i , 1 ≤ i ≤ n, be measures with densities φ i : R → R + such that φ i are non-decreasing on (-∞; 0] and non-increasing on [0; +∞). Let µ be the product measure of µ 1 , . . . , µ n and let A, B ⊂ R n be coordinate boxes containing the origin. Then the function t → µ(A + tB) is 1 n -concave on R + .

In the case of non-convex sets, this concavity property is false in general, even for the classical parallel volume |A + tB n 2 |. However, some conditions are given on A in [START_REF] Fradelizi | On the analogue of the concavity of entropy power in the Brunn-Minkowski theory[END_REF] for which the parallel volume of A is 1 n -concave on R + . Notice that other concavity properties of generalized forms of the classical parallel volume have been established in [START_REF] Marsiglietti | Concavity properties of extensions of the parallel volume[END_REF].

  where A + B = {a + b; a ∈ A, b ∈ B} denotes the Minkowski sum of A and B and where | • | denotes the Lebesgue measure.

3 .

 3 The converse of Proposition 3.1 is false in general, by taking, for example, φ(x) = 1 [-1,2] (x), x ∈ R, and A = [-1, 1].

The author was supported in part by the Agence Nationale de la Recherche, project GeMeCoD (ANR 2011 BS01 007 01).

It follows that the function t → φ(t) + φ(-t) is non-increasing on R + . Notice that this condition is satisfied for more general functions than -∞-concave functions attaining the maximum at 0.

However, one can use the same argument to see that if one assumes A, B ⊂ R convex containing 0 (not necessarily symmetric), then it follows that the density function φ is necessarily non-decreasing on (-∞; 0] and non-increasing on [0; +∞). Notice that this is equivalent to the fact that the density function φ is -∞-concave and max(φ) is attained at 0. By tensorization, Proposition 3.5 leads to the following corollary: Corollary 3.6. Let µ i , 1 ≤ i ≤ n, be measures with densities φ i : R → R + such that φ i are non-decreasing on (-∞; 0] and non-increasing on [0; +∞). Let µ be the product measure of µ 1 , . . . , µ n and let A, B ⊂ R n be the product of n Borel subsets of R such that 0 ∈ A ∩ B. Then, for every λ ∈ [0, 1], we have µ((1 -λ)A + λB)

Proof. We follow [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF]. By assumption, A = Π n i=1 A i and B = Π n i=1 B i , where for every i ∈ {1, . . . , n}, A i and

Using Proposition 3.5 and an inequality of Minkowski (see e.g. [START_REF] Hardy | Inequalities[END_REF]), one deduces that µ((1 -λ)A + λB)

Another consequence of Proposition 3.5 is that certain particular product measures are concave measures if A is a union of parallel slabs containing the origin. Corollary 3.7. Let µ 1 be a measure with density function φ : R → R + , such that φ is non-decreasing on (-∞; 0] and non-increasing on [0; +∞). Let µ 2 be a (n -1)-dimensional measure and let µ be the product measure of µ 1 and µ 2 . Let A = A 1 × R n-1 , where A 1 is a Borel subset of R and let B be a Borel subset of R n such that 0 ∈ A ∩ B. Then, for every λ ∈ [0, 1], we have

Inequality ( 8) was established in [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF] with power 1/n for slabs, for the case where µ is the Gaussian measure.

Proof. We follow [START_REF] Gardner | Gaussian Brunn-Minkowski inequalities[END_REF]. Let us denote B S = P e 1 (B) × R n-1 , where P e 1 (B) denotes the orthogonal projection of B onto the first coordinate axis. Then, for every λ ∈ [0, 1), one has

It follows, using Proposition 3.5, that

On the other hand, Proposition 3.4 can be generalized to the dimension n. First, let us define for a non-negative measurable function f : R n → R + and for u ∈ S n-1 ,

Proposition 3.8 (Dancs, Uhrin [START_REF] Dancs | On a class of integral inequalities and their measuretheoretic consequences[END_REF]). Let -1 n-1 ≤ γ ≤ +∞, λ ∈ [0, 1] and f, g, h : R n → R + be non-negative measurable functions such that for every

If there exists u ∈ S n-1 such that m u (f ) = m u (g) < +∞, then

Let us denote for a measure µ with density function φ, for a Borel subset A ⊂ R n and for a hyperplane H ⊂ R n ,

We deduce the following result: