Guthemberg Silvestre

Carla Sauvanaud
email: csauvana@laas.fr

Mohamed Kaâniche
email: mohamed.kaaniche@laas.fr

Karama Kanoun
email: karama.kanoun@laas.fr

An anomaly detection approach for scale-out storage systems

I. INTRODUCTION

Scale-out storage systems (SoSS) have become increasingly important to provide dependable web applications in cloud platforms. In a cloud-friendly data center environment, back-end data stores of multiple users, or tenants, share resources in order to provide horizontal scalability. Per-tenant data store is partitioned and replicated throughout different storage service instances for improving its scalability and availability. At the virtual machine (VM) level, a service instance can also share virtualized resources among different data store replicas to improve resource allocation.These SoSS allow cloud providers to enhance the platform utilization and to reduce costs. Popular SoSS include key-value stores, document stores, distributed relational databases, and block storage. So far, they have been successfully used for numerous web applications, such as threaded web conversations and posts, distributed monitoring systems, big data stores, photo tagging, user status updates, sessions and profiles.

Yet performance issues in distributed systems are still major causes of discontent among cloud tenants [START_REF] Hamilton | The cost of latency[END_REF], [START_REF] Schurman | The user and the business impact of server delays, additional bytes, and http chunking in web search[END_REF]. The root cause of unwanted performance variations of SoSS in cloud environments may depend on a wide range of uncorrelated issues, including systems permanent and transient faults, configuration errors, software bugs, limping hardware [START_REF] Do | Limplock: Understanding the impact of limpware on scale-out cloud systems[END_REF], to name a few. These performance issues can be grouped into two broad classes: failures and resource allocation problems. Failures in large SoSS are likely to be common. Although their failover mechanisms handle fail-stop failures successfully, many other failures may remain unnoticed by fault-tolerant mechanisms. For instance, Do et al. [START_REF] Do | Limplock: Understanding the impact of limpware on scale-out cloud systems[END_REF] found that a single limping network interface can cause a three orders of magnitude execution slowdown in distributed key-value data stores. Similarly, as many consolidated VMs must compete for shared resources in a single physical host, efficient resource allocation in cloud platforms becomes quite challenging. Actually, this may result in contentions of hosts resources such as main memory and network bandwidth leading to increasing completion times of operations [START_REF] Shue | Performance isolation and fairness for multi-tenant cloud storage[END_REF], [START_REF] Gulati | Proportional allocation of resources for distributed storage access[END_REF]. Therefore, we believe that cloud providers must be able to uncover these performance issues in order to enhance the dependability of SoSS.

Anomaly detection has been a widely used technique to automatically identify performance issues in large-scale distributed systems [START_REF] Chandola | Anomaly detection: A survey[END_REF]. Most of the anomaly detection approaches for distributed systems are based on a unsupervised learning method. To boost their performance, they rely on data mining techniques [START_REF] Liang | Bluegene/l failure analysis and prediction models[END_REF], [START_REF] Gujrati | A meta-learning failure predictor for blue gene/l systems[END_REF], [START_REF] Lan | Toward automated anomaly identification in large-scale systems[END_REF], [START_REF] Guan | Adaptive anomaly identification by exploring metric subspace in cloud computing infrastructures[END_REF]. Yet prediction efficiency remains the main drawback of this method [START_REF] Love | Comparing supervised and unsupervised category learning[END_REF], as it is statistically difficult to ascertain the quality of inferences drawn from the predictions of unsupervised learning algorithms. Moreover, to the best of our knowledge, there is not yet a precise analysis of the prediction efficiency of anomaly detection approaches in SoSS. In particular, none has analysed the predictability of anomalies in SoSS that use replication and load balancing to enhance data availability. This work introduces an anomaly detection approach for SoSS. We are particularly interested in identifying anomalous VMs with high prediction efficiency. To evaluate our anomaly detection approach, we built a testbed simulating a virtual data center using VMware. In this testbed, we deployed a cluster of VMs to run MongoDB [START_REF]Mongodb[END_REF], the most popular document store nowadays1 . We injected memory and network faults in VMs to provoke cloud anomalies, then we performed a precise analysis of the predictability of these anomalies. Our analysis was threefold. First, we assessed the impact of cloud anomalies on a MongoDB cluster. Then, we compared the efficiency of our prediction model implemented as supervised and unsupervised learning methods. Finally, we evaluated the quality of different monitoring data with respect to the probing sources, particularly whether it is worth monitoring directly VMs operating systems and SoSS utilization counters.

Experimental results suggest that the injected faults are likely to undermine the data availability of a MongoDB cluster regardless of its built-in fault-tolerant mechanisms, notably replication and load balancing of queries. They also highlight that a supervised learning-based implementation of our prediction model performs much better than an unsupervised one in predicting these anomalies, reducing by roughly 10% the false positive rate. Our findings point out that collecting SoSS-specific monitoring data directly from VMs provides a worthwhile improvement in the prediction efficiency of our anomaly detection approach.

II. ANOMALY DETECTION APPROACH

In this section, we describe our approach to enhance the capacity of SoSS to detct anomalies. In particular, we define our anomaly prediction model that classifies VMs in anomalous or normal based on the monitoring data of SoSS. Figure 1 provides a high-level overview of our approach. It introduces the five main functional blocks of our approach, namely SoSS, workload, fault injection, monitoring, and anomaly detection. In the remaining of the section, we detail each one of these blocks.

A. SoSS

SoSS is the subject of our anomaly detection study. Figure 2 highlights the main components a SoSS and the interactions with the workload. To enhance data availability for cloud services, SoSS provides scalable and fault-tolerant data storage. Such systems offer scalable storage service by allowing cloud operators to evenly split data in L partitions or shards across a cluster of VMs. To increase the storage capacity of the cluster, operators may add nodes as the system run. Partition replication is the most common mechanism to enforce data durability and availability in SoSS. Each partition has a predefined number of replicas K+1, which allows the system to tolerate up to K VM failures. Copies of the same partition form the so-called replica set. Depending on the scheme to maintain replicas, replication is divided in two broad groups: primary/secondary and multi-primary schemes. Essentially they differ on how requests that modify data are handled. Our approach takes into account both schemes. In addition, we assume that SoSS rely on load balancing throughout replicas in order to limit the impact of cloud anomalies on a cluster.

B. Workload

Workload functional block loads a SoSS cluster with data and runs the workload for evaluation purposes. Through this block, we can define workload settings, such as document size distribution, the rate of query per second, and the distribution popularity of documents. As depicted in Figure 2, the workload allows us to generate four types of queries, namely create, read, update, and delete (CRUD) queries.

C. Fault Injection

Fault injection allows us to emulate malfunctioning and errors events in cloud environments to cause anomalous behaviours in the SoSS. We investigate the impact of faults that are hard to identify and degrade significantly the performance of our SoSS. Our goal is to provide a collection of faulty behaviours that are likely to occur in cloud environments and that existing fault-tolerant mechanisms may fail to deal with. In this work, we focus on the following two categories of faults.

Network faults. Communication issues in distributed systems are common, and they have a significant impact on the performance of SoSS. To analyse their impact, we inject three types of network faults, namely packet loss, network latency, and limping network. Packet loss and network latency allowed us to emulate common interconnection issues of cloud platforms, such as network partition in data center networks. Limping network is intended to reproduce networks' anomalous behaviours previously observed by Do et al. [START_REF] Do | Limplock: Understanding the impact of limpware on scale-out cloud systems[END_REF]. This fault emulates a limping network interface, whose actual transfer rate does not comply with the manufacturer's specification.

Memory faults. As the price of main memory continues to drop, distributed data store operators have been encouraged to fit data to main memory in order to improve performance [START_REF] Ren | Lightweight locking for main memory database systems[END_REF]. Hence, faults in main memory have become increasingly harmful to these systems. We intend to emulate faults related to misuse of memory. At the VM level, this kind of fault allows us to make arbitrary amounts of main memory inaccessible to a SoSS. The aim is to provoke an abnormal functioning of a SoSS due to an unexpected utilisation of main memory. Consequently, there will be less main memory available and more unwanted, costly disk I/O. A typical example of this fault is a VM running out of memory due to a misconfiguration, memory leaking, overloading, or an unbalanced resource allocation.

D. Monitoring

Monitoring functional block allows us to collect data that quantifies the utilization of the cloud resources for the SoSS. To collect our datasets, we consider the monitoring system depicted in Figure 3. The monitoring system (i) collects measurements from the SoSS through probing agents; (ii) gathers and stores these measurements in a rrd file; and (iii) processes these files periodically to create our raw dataset in a relational database. To characterize the state of a VM, this system collects data from the monitoring counters of the storage service, like number of served queries, and from the cloud infrastructure, such as CPU and memory usage. Probing agents collect SoSS-specific monitoring data directly from the VMs, and generic activity monitoring data from the hypervisors or VMs.

E. Anomaly Detection

This is the main functional block of this study. We briefly introduce the main statistical learning methods that are implemented and compared in this work. Then we describe the prediction model. Finally, we present the metrics to evaluate the efficiency of our anomaly detection approach.

1) Statistical learning methods:

Statistical learning is about learning from seen data in order to predict unseen data with minimal error. Data comprises measurements represented by a feature vector x with a fixed number of dimensions d (x ∈ X ⊂ R d ) from the input space X . There are two ways of learning from data: supervised and unsupervised learning.

In supervised learning, each measurement or input is coupled with a y, a label, from the output space Y. To learn, we have N pairs (x, y) drawn independent and identically distributed (i.i.d.) from a fixed but unknown joint probability density function P r(X, Y ). This is true for both training and testing datasets. For instance, we consider the training dataset S = {x i , y i } N i=1 of N pairs (x, y). From this dataset, the supervised learning algorithm searches for a function f : X → R in a fixed function class F. State-of-the-art algorithms, such as support vector machines (SVM) [START_REF] Hastie | The elements of statistical learning[END_REF] or Adaboost [START_REF] Freund | A decision-theoretic generalization of on-line learning and an application to boosting[END_REF], aim to find f ⋆ in F with the lowest empirical risk defined as: In unsupervised learning, we have N unlabelled samples (x 1 , x 2 , . . . x N ) of a random d-vector X having probability density function P r(X). Unlike supervised learning, there is no outputs to learn from. Instead, we are interested in inferring the properties of function P r(X). This allows us to have insights into how the data is organized or clustered.

f ⋆ ∈ arg min f ∈F r emp (f ) where r emp (f ) = 1 N N i=1 I {f (x)
The unsupervised learning method allows us to make fewer assumptions about data, since they do not require labelled training datasets. To boost their efficiency, they rely on data mining techniques [START_REF] Liang | Bluegene/l failure analysis and prediction models[END_REF], [START_REF] Gujrati | A meta-learning failure predictor for blue gene/l systems[END_REF], [START_REF] Lan | Toward automated anomaly identification in large-scale systems[END_REF], [START_REF] Guan | Adaptive anomaly identification by exploring metric subspace in cloud computing infrastructures[END_REF]. Yet prediction performance remains the main drawback of these approaches [START_REF] Love | Comparing supervised and unsupervised category learning[END_REF], as it is statistically difficult to ascertain the quality of inferences drawn from the predictions of unsupervised learning method.

With supervised learning, however, there is a clear measure of success, that is quantified by the expected loss over the joint probability density function P r(X, Y ) [START_REF] Hastie | The elements of statistical learning[END_REF].

2) Anomaly prediction model: The goal of our prediction model is to detect anomalies in VMs that are likely to undermine the performance of SoSS. We intend to compare the prediction efficiency of two implementations of this model, as a supervised and as a unsupervised learning method. In our model, x represents a VM. The model classifies VMs in anomalous or normal with respect to their current state. The output of our model is defined as follows.

Ĝ

(x) = 1 if a VM is behaving anomalously. -1 if a VM is behaving normally.
This model makes predictions based on monitoring dataset, described in Subsection II-D. Monitoring dataset is preprocessed according to the statistical learning method to be performed. With supervised learning, we add labels to the training dataset. After preprocessing monitoring datasets, we set to 1 labels of input measurements of faulty VMs (Subsection II-C) in the training dataset. Hence, each line of our datasets for anomaly detection has d inputs and one output label that indicates if a VM behaves anomalously or not. This procedure is slightly different for the unsupervised learning method. Since it does not require labelled data to learn, its training dataset has d inputs with data from VMs where faults were not injected and behaved normally. To test or validate our model as an unsupervised learning method, the preprocessing procedure is the same as that for supervised learning one. We detail the use of our dataset for predictions in Subsection IV-B.

3) Prediction efficiency metric:

We evaluate the efficiency of our binary classifier with respect to two different statistical learning methods using average precision. Average precision is a single-value efficiency metric widely used in Information Retrieval [START_REF] Hastie | The elements of statistical learning[END_REF]. We chose this metric because it combines two other key efficiency metrics: precision and recall. Precision is the rate of anomalous events detected successfully over the total number of events. Recall measures the rate of anomalous events detected successfully over the total number of ground truth anomalous events. To find the average precision of a binary classifier, we compute its precision (p) averaged across all values of recall (r) between 0 and 1. It is defined as follows:

average precision = 1 0 p(r) dr (1)
Actually, the average precision corresponds to the area under the precision-recall curve. Finally, we define the prediction error of a learning model as follows:

prediction error = 1 -average precision (2)

III. EXPERIMENTAL SET-UP

This section details the experimental set-up to measure the efficiency of our anomaly detection approach.

A. Testbed Settings for Simulating a Virtual Data Center

Our experimental testbed consists of two Dell PowerEdge R620 hosts, namely target and experimentation hosts. Each host has two Xeon E5-2660 2.2 GHz processors, 64 GB of memory, and two 130 GB SATA disks. The hosts were connected by Gbit Ethernet. These resources are shared among the VMs and services running on each host. We chose VMware as the virtualization technology and ESXi 5.1.0 as hypervisor. Figure 4 depicts our virtual data center for experimentations, highlighting how we consolidated VMs across hosts. The target host runs our SoSS, monitoring agents, and our fault injection tools whereas experimentation host runs everything else, notably the workload, monitoring system, and the prototype of our anomaly detection approach. By grouping our SoSS cluster along with the fault injection tools in a single host, we aim to reduce any unwanted noise or load on the monitoring data of storage nodes.

VMs were consolidated according to their profiles. Profiles are loosely based on functional block of our approach, detailed in Section II. They essentially differ in main memory, disk, and, particularly, network capacity. To prevent a misleading network utilization, we connected storage nodes to 100Mbps virtual network. The workload functional block has two VMs with identical settings with larger amounts of main memory, which allows us to control workload settings. The VM for monitoring requires a bigger disk, just like anomaly detection's VM, that runs statistical learning algorithms over big amounts of monitoring data.

B. SoSS and Workload Settings

As SoSS, we chose MongoDB (2.4.8 release) [START_REF]Mongodb[END_REF]. We consolidated in our experimental testbed a MongoDB cluster following the SoSS definitions of Subsection II-A. In our cluster, we set the replication factor K to 1 and the number of partitions L to 2. We consolidate five VMs to deploy our cluster, query router and four document stores. Data is evenly distributed by a query router of MongoDB throughout the two partitions using the hash code of documents' keys. In MongoDB, there is a single primary replica on each replica set. Replicas regularly exchange heartbeat messages to elect the primary copies as failures occur. We used the default timeout setting of 10 seconds for primary replica election. This set-up forms a small but resilient, fault-tolerant SoSS, which allows us to measure the impact of different faulty scenarios.

We investigate the performance of MongoDB using the Yahoo! Cloud Serving Benchmark (YCSB) [START_REF] Cooper | Benchmarking cloud serving systems with ycsb[END_REF], a workload generator and benchmark tool for SoSS. In a preliminary setup, we use YCSB to load documents whose size varies from 1KB to 3KB. Once the database is loaded, we set-up YCSB to generate a mostly-read workload. The popularity of documents follows a Zipf-like distribution. We evaluated the MongoDB performance in serving an average throughput of 5000 queries per second. YCSB provides two key metrics to evaluate the performance of SoSS, average throughput rate and 99 th percentile latency. In a fault-free scenario, our MongoDB cluster serves the expected 5000 queries per second with a 99 th percentile latency smaller than 12 milliseconds.

C. Fault Injection Campaigns

We developed injection scripts to provoke cloud anomalies and to investigate the impact of a faulty cloud environment in our MongoDB cluster. These scripts allow us to inject network and memory faults detailed in Subsection II-C. We implemented network faults using Dummynet [START_REF] Carbone | Dummynet revisited[END_REF], a widely used link emulator. For memory faults, we implemented Python scripts that allocated arbitrary amounts of main memory. Table I summarizes the main characteristics of our fault injection campaigns. We can assign 10 intensity values or levels to a fault. Although the range and distribution of fault intensities had been chosen arbitrarily, they provide a fair insight about how these faults could lead to anomalies of cloud environments. Our scripts have three parameters to define a single injection: type of fault, intensity, and duration. For each VM hosting a MongoDB document store, we injected a series of faults including all types and intensities. Between two injections, script procedures ensure that the MongoDB cluster recovered completely from faults. We did not inject faults or different fault intensities concurrently.

D. Monitoring System and Datasets for Analysis

Our monitoring system, whose design is described in Subsection II-D, was built on top of Ganglia [START_REF] Massie | The ganglia distributed monitoring system: design, implementation, and experience[END_REF], a scalable distributed monitoring system for high performance computing systems. We deployed a Ganglia agent on each hypervisor and VM. Every 30 seconds, they collect data about the utilization of the cloud resources by the SoSS. These measurements, including anomaly injection information, are organized by VM and stored in a relational database every minute. To assess the effect of different probing sources on anomaly detection of VMs, measurements are organized in three source groups, namely A, B, and C, as described in Table II. The behaviour of a VM of a SoSS cluster is characterized by these 60 measurements (i.e., d = 60). Monitoring sources have a sightly similar number of measurements. They allow us to evaluate the trade-off between collecting generic VMlevel data and collecting service-level data. Sources A and B require a probing agent running inside the VM, whereas measurements from source C come from an agent probing directly the hypervisor monitoring counter. Among the probing sources, only source A is specific to the target SoSS. The list of all 60 measurements, so-called anomaly detection inputs, is available online2 . Overall, our dataset for analysis has 50,000 samples based on VMs, including 15,000 one-minute anomalous samples.

E. Anomaly Detection Implementation

The prototype of our prediction model is based on support vector machine (SVM) learning algorithm [START_REF] Hastie | The elements of statistical learning[END_REF]. Our anomaly detection module was implemented using Scikit-learn library [START_REF] Pedregosa | Scikit-learn: Machine Learning in Python[END_REF].

We designed and implemented a anomaly detection (AD) module to identify anomalies in SoSS, depicted in Figure 5. The AD has two operating phases: (i) learning and (ii) prediction. Each phase has its own dataset collected by the monitoring system. In the learning phase, monitoring data is preprocessed to generate the training dataset. Then this training dataset is fed to the AD module, whose functioning depends on the implemented statistical learning method. If the AD is based on a supervised learning method, it will identify normal and anomalous patterns, otherwise, the AD is based on an unsupervised learning method, it characterizes normal patterns only. Once the learning phase has been accomplished, AD can use its learning module in a prediction phase, as indicated in the right-hand side of Figure 5. In this phase, inputs come directly from measurements of the monitoring system, that permit classifying the behaviour of a VM into anomalous or normal.

IV. EVALUATION RESULTS

In this section, we analyse the impact of the injected faults in MongoDB cluster, we compare the prediction efficiency of these faults for two implementations of our model, and we assess the quality of information from the different sources of monitoring data for anomaly predictions.

Figures 6a and6b show the impact of our fault injection campaigns, detailed in Subsection III-C. From our experimental results, we highlight the following observations.

• Impact on the average throughput rate. Our Mon-goDB cluster served 5000 queries per second before injecting faults. Under faults, the average throughput could be reduced by more than two thirds as a single storage node has a limping network interface [START_REF] Do | Limplock: Understanding the impact of limpware on scale-out cloud systems[END_REF]. The impact of increasing network latency fault was considerably high. According to its intensity level, the average throughput ranged from 4976 to 2817 queries per second. We also observed that the impact of packet loss on the average throughput increased smoothly. It remained stable as the rate of packet loss was higher than 7.2%. Considering the average throughput rate, MongoDB seems to be very resilient against memory faults. For instance, for a memory fault intensity as high as 72% of all main memory (i.e., only 28% of main memory in available for the MongoDB instance), there was no impact on the average performance. Surprisingly, we started observing a performance degradation of its average throughput when less than a quarter of the main memory was available for all other running processes of the VM.

• Impact on the 99 th percentile latency. In a fault-free scenario, we observed a 99 th percentile latency of 12 milliseconds. When we inject a limping network fault, this metric reached a peak of 389 milliseconds, in other words, a 32-fold increase. Similar to the impact on average throughput, limping network seems to be the worst type of fault. However, as we shrank the available bandwidth to values lower than 100Kbit/s (a limping network fault with an intensity of 0.6), MongoDB identifies the faulty node and overcomes the problem by sending queries to another replica. It confirms how important network resources are for this kind of system.

Overall, all faults undermined the performance of our MongoDB cluster, despite the load balancing of queries and management of the liveness of replicas. We observed these faults had a higher impact on the 99 th percentile latency.

B. Prediction about Anomalies in Cloud Environments

The evaluation of our prediction model has three operating phases: training, cross-validation, and prediction. While training and prediction correspond exactly to the two phases of our anomaly detection framework, detailed in Subsection III-E, cross-validation actually is a preliminary phase that helps us to design our prediction model, e.g. to evaluate inputs and to select a learning method. We evaluate the efficiency of our anomaly detection framework using the dataset described in Subsection II-D. We split our dataset of 50,000 samples (described in Subsection III-D) in three parts: 60% for training dataset, 20% for cross-validation, and the remaining 20% for prediction. We remind that our model predicts a positive value, 1, whenever an anomalous VM is detected, otherwise -1.

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 500
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0

C. Cross-validation with Supervised and Unsupervised Learning Approaches

Cross-validation is an essential procedure to evaluate the quality of a learning model. In particular, this evaluation step allows us to verify if our learning model is overfitting or underfitting. Overfitting is a problem in a statistical learning method that prevents a model to generalize, in other words, the model data fitting is too specific. Underfitting has an opposite effect. If a model suffers from underfitting, it performs poorly, resulting in a high number of wrong predictions. In both cases, the model must be improved.

One way to verify overfitting and underfitting is by plotting prediction error curves of the different learning methods. These curves help us to choose the "best" learning method. In our evaluation, the error curves are based on the efficiency metric defined in Section II-E, average precision (Equation 1) and prediction error (Equation 2). Moreover, we plot the error curves as a function of the sample size to verify if our dataset has enough information for accurate predictions. Figure 7 depicts the error curves of our anomaly prediction model using an unsupervised learning and supervised learning methods. Plots have 20 points each. A point corresponds to the prediction error with a portion of the datasets. For instance, the evaluation of the first point takes into account one twentieth of training dataset and one twentieth of cross-validation dataset. As the x-axis values increase, datasets become progressively bigger. The twentieth point contains the entire training and cross-validation datasets. To compute the prediction error of a point, we train our model with its respective training dataset size and we measure error of prediction about its own training dataset and the cross-validation dataset. These plots allow us to draw the following conclusions about the models.

• None of them was underfitting. As the sample size of training and cross-validation datasets increases, the gap between the prediction error curves of both learning methods decrease significantly and then remains unchanged. This means that additional training data is unlike to help us to reduce prediction error. Our datasets seem to be large enough for training purposes of our anomaly detection model.

• Our model was overfitting when implemented as an unsupervised learning method. Our model performed predictions about anomalies with high prediction error as we used an unsupervised learning-based implementation. With this approach, prediction errors were higher than 0.2. When we implemented our model using a supervised learning method, however, it did not suffer from overfitting. Actually, we were able to reduce prediction error of our model to roughly 0.02. This result shows that the supervised-based implementation of our model generalizes properly and performs prediction with high efficiency.

To provide a wider view of the efficiency of our model based on different statistical learning methods, we compute other prediction efficiency metrics. Figure 8 shows the false positive and true negative rates of supervised and unsupervised methods. It shows that a supervised learning method is highly efficient in our context. For instance, for this set of experiments, it reduces the number of false positives, or VMs wrongly classified as anomalous, from 0.11 using an unsupervised learning method to almost 0.01 using a supervised learning method. The smaller is the false positive rate, the easier is to use predictions to trigger recovery procedures accurately. Finally, Table III summarizes other performance metrics during our cross-validation evaluation. This table shows efficiency values using the entire training and cross-validation datasets. It allows us to compare the average precision, our main metric, to other metrics of prediction efficiency. These results suggest that all efficiency metrics are consistent and confirm the superiority of the supervised learning method.

D. Measuring the Contribution of Different Monitoring Sources to Prediction Efficiency

We computed the average precision for all combinations of our three sources of data monitoring, detailed in Subsec-

V. RELATED WORK

Fault tolerance in SoSS. Distributed systems use replication and advanced request scheduling to improve data availability. Ficus [START_REF] Guy | Implementation of the ficus replicated file system[END_REF] and Bayou [START_REF] Petersen | Bayou: replicated database services for world-wide applications[END_REF] are data storage that rely on replication to ensure data availability against fail-stop failures, but they are not able to deal with transient cloud failures. Skute [START_REF] Bonvin | A self-organized, faulttolerant and scalable replication scheme for cloud storage[END_REF] provides an adaptive replication scheme that mitigate the impact of cloud failures. However, it does not provide mechanisms to ensure high data availability, such as high throughput and bounded latency. Popular SoSS, like MongoDB [START_REF]Mongodb[END_REF], CouchDB [1] and Redis [START_REF]Redis[END_REF], offer high data availability using an eventual consistency replica system and enhanced main memory data structures [START_REF] Stonebraker | Sql databases v. nosql databases[END_REF]. But, our study showed that cloud failures, like packet loss and main memory faults, can undermine considerably their data availability. Cake [START_REF] Wang | Cake: enabling high-level slos on shared storage systems[END_REF] offers a scheduling scheme to enforce highlevel data availability requirements for end users. Similarly, PARDA [START_REF] Gulati | Proportional allocation of resources for distributed storage access[END_REF] enforces proportional-share fairness in the access of data in a storage cluster. Since neither Cake nor PARDA are designed to identify faulty VMs, our work is complementary to theirs. Eriksson et al. [START_REF] Eriksson | Riskroute: a framework for mitigating network outage threats[END_REF] provide a routing framework that helps cloud operators to mitigate the impact of network failures. Alerts from our approach can contribute to enhance the assessment of network outage risk of their framework.

Anomaly detection in SoSS. Prediction models for anomaly detection are commonly implemented based on an unsupervised learning method. Liang et al. [START_REF] Liang | Bluegene/l failure analysis and prediction models[END_REF], and Gujrati et al. [START_REF] Gujrati | A meta-learning failure predictor for blue gene/l systems[END_REF] as well, provide prediction models based on event logs of supercomputers to identify platform-wide anomalies, whereas we are interested in detecting anomalous VMs based on common data center monitoring data. Chen et al. [START_REF] Chen | Failure detection in large-scale internet services by principal subspace mapping[END_REF] proposes an anomaly detection approach for large-scale systems that improves the prediction efficiency of an entropy-based information theory technique by performing a principal component analysis (PCA) of system inputs. However, this introduces computational overhead that undermines its scalability and causes a slowdown in anomaly predictions. While we focus on detecting a small set of cloud anomalies with high prediction efficiency, Lan et al. [START_REF] Lan | Toward automated anomaly identification in large-scale systems[END_REF] provides a general-purpose anomaly detection approach that strongly relies on input selection to enhance prediction efficiency. Similarly, Guan and Fu [START_REF] Guan | Adaptive anomaly identification by exploring metric subspace in cloud computing infrastructures[END_REF] performs inputs extraction based on PCA to identify the most relevant inputs for anomaly detection. Yet, our results confirm that an unsupervised-based approaches undermine the prediction efficiency of cloud anomalies.

Guan et al. [START_REF] Guan | Cda: A cloud dependability analysis framework for characterizing system dependability in cloud computing infrastructures[END_REF] implement a probabilistic prediction model based on a supervised learning method. Although their model allows us to compare the dependability of virtualized and non-virtualized cloud systems, their results suggest that their model suffers from poor prediction efficiency when it is used to predict cloud anomalies. Tan et al. [START_REF] Tan | Prepare: Predictive performance anomaly prevention for virtualized cloud systems[END_REF] propose generalpurpose prediction model to prevent cloud anomalies. Their supervised learning-based model combines 2-dependent Markov chain model with the tree-augmented Bayesian networks. But, the authors did not provide information about the prediction efficiency and the capacity of generalize of their approach.

VI. CONCLUSION

Web services rely on SoSS to provide content with high availability. However, we showed that cloud anomalies can undermine the capacity of these systems to enforce data availability requirements. In this work, we presented an anomaly detection approach to enhance the dependability of SoSS through predictions of cloud anomalies. Our results show that our approach can efficiently identify anomalous VMs. Although anomaly detection in distributed systems is commonly implemented based on an unsupervised learning method, we showed that a supervised learning-based implementation of the same prediction model reduces the false positive rate by 10%. We also showed that collecting more specific probing sources at the VM level improves the detection of anomalous VMs in SoSS. As future research, we plan to extend our approach based on our preliminary findings in order to evaluate different SoSS, under different workloads and a large number of faults.

Fig. 1 :

 1 Fig. 1: Methodology overview and the functional blocks.

Fig. 2 :

 2 Fig. 2: SoSS architecture and workload.

Fig. 3 :

 3 Fig. 3: Monitoring system and datasets.

  =yi} is computed over the training set, and I {.} is the indicator function which returns 1 if the predicate {.} is true and 0 otherwise.

Fig. 4 :

 4 Fig. 4: Experimental testbed.

Fig. 5 :

 5 Fig. 5: Anomaly detection module for SoSS.

  (a) Average throughput rate.

  99 th percentile latency.

Fig. 6 :

 6 Fig. 6: The impact of cloud anomalies caused by network and memory faults on the data availability of MongoDB.

  Error curve for unsupervised learning model.

Fig. 7 :Fig. 8 :

 78 Fig. 7: The impact of network and memory faults on the data availability of SoSS.

Fig. 9 :

 9 Fig. 9: Contribution of different monitoring sources to prediction efficiency.

TABLE II :

 II Probing sources for monitoring.

	Label	Source	Agent deployment	Measurements
	A	SoSS	VM	16
	B	TCP network layer	VM	21
	C	systat of VMs or hypervisor	VM or hypervisor	23

TABLE I :

 I Fault injection campaigns.

	Fault type	Duration (days)	Unit	0.1	0.2	0.3	0.4	Intensity levels 0.5 0.6 0.7	0.8	0.9	1.0
	Packet loss	3	%	0.8	1.6	2.4	3.2	4.0	4.8	5.6	6.4	7.2	8.0
	Network latency	3	ms.	8	16	24	32	40	48	56	64	72	80
	Limping network	3	Mbps	32	10	3.2	1.0	0.32	0.1	0.032	0.01	0.0032	0.001
	Misuse of memory	3	%	70	73	76	79	82	85	88	91	94	97

TABLE III :

 III Comparing efficiency metrics for learning methods.

			efficiency contributes to the improvement of SoSS dependabil-
			ity, we conclude that including monitoring data from probes
			installed inside the VMs is worthwhile.
			Prediction efficiency	0.80 0.85 0.90 0.95 1.00	0.8	0.9	0.95 0.93	0.98 0.98 0.98
				0.75	A	B	C	A+B A+C B+C A+B+C
							Monitoring sources
		Learning method
	Metric	Supervised	Unsupervised
	Average precision	0.98	0.75
	AUC	0.98	0.79
	Accuracy	0.99	0.83
	Positive precision	0.98	0.72
	Negative precision	0.99	0.87
	Positive recall	0.98	0.68
	Negative recall	0.99	0.69
	tion II-D. Our goal was to assess the quality of information
	provided by each monitoring source, and how they contribute
	to provide high prediction efficiency. To achieve this goal, we
	used the supervised learning version of our prediction model,
	and we considered the training and the prediction datasets.
	We selected inputs for training our model according to the

monitoring sources, namely A, B, and C. Figure

9

presents the results for different combinations of monitoring sources. If we consider only the monitoring data that comes from source C, which includes common VM-level measurements, we achieve a high prediction efficiency of 0.95. We increase the model efficiency by 0.03 when we add monitoring information from TCP-level or data storage usage sources. As any increase in

http://db-engines.com/en/ranking

http://homepages.laas.fr/gdasilva/datasets/oltp mongodb.infoA. Measuring the impact of faults on SoSS

ACKNOWLEDGMENTS

This research is partially funded by the project Secured Virtual Cloud (SVC) of the French program Investissements d'Avenir on Cloud Computing.


F Demengel

Qualitative properties of a nonlinear system involving the p-Laplacian operator and a DeGiorgi type result

Keywords:

In this article we consider the nonlinear system involving the p-

u ≥ 0, v ≥ 0 for which we prove symmetry, asymptotic behavior and non degeneracy properties. Next we prove a De Giorgi Type result in dimension 2 under some additional growth and monotonicity assumptions.

Introduction

In this article we extend some of the results obtained in [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF] in the case of the Laplacian, to the p-Laplacian case. More precisely we consider the system in R N : div(|∇u| p-2 ∇u) = (p -1)u p-1 v p div(|∇v| p-2 ∇v) = (p -1)v p-1 u p , (

where u and v are supposed to be non negative.

In the case p = 2 this problem comes from a phase separation model. As an example the Gross Pitaevskii model, [START_REF] Chang | Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates[END_REF] can be described by the non linear elliptic system 1

       -∆u + αu 3 + Λv 2 u = λ 1,Λ u in Ω -∆v + βv 3 + Λu 2 v = λ 2,Λ v in Ω u > 0, v > 0, in Ω, u = v = 0 on ∂Ω Ω u 2 = Ω v 2 = 1 (1.2)
where Ω is a smooth bounded domain in R N and α, β are positive parameters, Λ will become large. Assuming that sup(λ 1,Λ , λ 2,Λ ) ≤ C for some constant independent of Λ, formally and up to subsequences, (u Λ , v Λ ) converges to some pair (u, v) which satisfies uv = 0 and the equations

-∆u + αu 3 = λ 1,Λ u in Ω u = {x, u(x) > 0} -∆v + βv 3 = λ 2,Λ v in Ω v = {x, v(x) > 0} (1.3)
Several papers treat the convergence of (u Λ , v Λ ) away the interface γ = {x, u(x) = 0 = v(x)}, see for example [START_REF] Wei | Asymptotic behavior of solutions of planar elliptic systems with strong competition[END_REF] and [START_REF] Conti | Asymptotic estimates for the spacial segregation of competitive systems[END_REF] , [START_REF] Noris | Uniform Holder bounds for nonlinear Schrodinger systems with strong competition[END_REF] for the uniform equicontinuity of (u Λ , v Λ ). Near the interface, the profile of bounded solutions of the blow up equation for (1.2) is a system, which is completely classified in the one dimensional case, [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF], [START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF]. This system is the following

   U ′′ = U V 2 in R V ′′ = V U 2 in R, U > 0, V > 0.
In [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF], the authors expect that the same system occurs in the N dimensional case, say

   ∆U = U V 2 in R ∆V = V U 2 in R, U > 0, V > 0.
Furthermore they conjecture that for any dimension N ≤ 8, the system is in fact one dimensional. They obtain this result under some additional assumption on the growth of the solution, in dimension 2. The assumption N ≤ 8 is motivated by the case of scalar equations for which it is known that the scalar equation is not necessary one dimensional when N ≥ 9, [START_REF] Del Pino | On De Giorgi's conjecture in dimension N ≥ 9[END_REF]. The condition on the growth in the two dimensional case is satisfied in particular if the solution of the system is at most linear at infinity, as it is in the case N = 1. When N > 2, this is not sufficient, and up to now, even in the case N = 2, this asymptotic behaviour is not proved.

In [START_REF] Farina | Some Symmetry results for entire solutions of an elliptic system arising in phase transition[END_REF] A. Farina improves the result by establishing that for N = 2, as soon as u and v have at most algebraic increasing behavior and satisfy the half monotonicity condition ∂ N u > 0 or ∂ N v < 0 for some direction e N , then the solution is one dimensional. In [START_REF] Farina | Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation Accepted in[END_REF], Farina and Soave replace the half monotonicity by the condition lim x N →±∞ u(x ′ , x N )-v(x ′ , x N ) = ±∞, uniformly with respect to x ′ , conserving the assumption of algebraic growth.

Let us cite some variants on the De Giorgi result :

-In [START_REF] Wang | On the De Giorgi type conjecture for an elliptic system modeling phase separation[END_REF], K. Wang proves the one dimensional property by replacing the monotonicity condition by the fact that (u, v) is a local minimizer, and assuming in addition the at most linear growth.

-In [START_REF] Fazly | De Giorgi type results for elliptic systems[END_REF] and [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems discrete and continuous Discrete and continuous dynamical systems[END_REF] the authors consider the case of m components in place of two, and a right hand side which is the gradient of H, which satisfies a condition of orientability. Under some condition on the growth of the solutions, they prove a De Giorgi Result, which covers the case N = 2 under the assumptions of [START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF].

The present paper is motivated by the asymptotic study of the problem

       -div(|∇u| p-2 ∇u) + αu p+1 + Λv p u p-1 = λ 1,Λ u p-1 in Ω -div(|∇v| p-2 ∇v) + βv p+1 + Λu p v p-1 = λ 2,Λ v p-1 in Ω u > 0, v > 0 in Ω, u = v = 0 on ∂Ω Ω u p = Ω v p = 1 (1.4)
where Ω is a smooth bounded domain in R N and α, β are positive parameters, Λ will become large. Such a pair of solutions is a critical point for the functional

E Λ (u, v) = 1 p Ω (|∇u| p + |∇v| p ) + α p+2 Ω |u| p+2 + β p+2 Ω |v| p+2 + Λ p Ω |u| p |v| p under the constraint Ω |u| p = Ω |v| p = 1.
Assume that there exists some constant C independent on Λ with sup

Λ (λ 1,Λ , λ 2,Λ ) ≤ C,
for Λ large. As Λ goes to infinity, and up to subsequence (u Λ , v Λ ) tends formally to some pair (u, v) which satisfies uv = 0 and

-div(∇u| p-2 ∇u) + αu p+1 = λ 1,Λ u p-1 in Ω u -div(|∇v| p-2 ∇v) + βv p+1 = λ 2,Λ v p-1 in Ω v (1.5)
where

Ω u = {x, u(x) > 0}, Ω v = {x, v(x) > 0}.
It is not our purpose here to follow this way. We are interested in the one dimensional case and especially in the behavior of the limit pair of solutions (u, v) near the interface γ = {x ∈ Ω, u(x) = v(x) = 0}.

When N = 1 and Ω =]a, b[ one has the result :

Theorem 1.1. Assume that Ω =]a, b[, and that (u Λ , v Λ ) solves (1.4). There exists x Λ ∈ Ω such that u Λ (x Λ ) = m Λ = v Λ (x Λ ) goes to zero, and x Λ tends to some point x ∈]a, b[.Furthermore if ũΛ = 1 m Λ u Λ (m Λ y + x Λ ) , ṽΛ = 1 m Λ v Λ (m Λ y + x Λ ) and y ∈] a-x Λ m λ , b-x Λ m Λ [, (ũ Λ , ṽΛ ) converges locally uniformly to some pair (U, V ) which satisfies    (|U ′ | p-2 U ′ ) ′ = (p -1)V p U p-1 (|V ′ | p-2 V ′ ) ′ = (p -1)U p V p-1 on R U (0) = V (0) = 1. U, V > 0. (1.6)
Furthermore there exists some positive constant T ∞ such that

|U ′ | p + |V ′ | p -U p V p = T ∞ .
Next we are interested in the existence and in the properties of the solutions of (1.6). The existence of a non trivial solution is given in Theorem 3.1. In a second time, using the sliding method, [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF], we prove the following : Theorem 1.2. Let (U V ) be a non negative solution of

(|U ′ | p-2 U ′ ) ′ = (p -1)V p U p-1 , (|V ′ | p-2 V ′ ) ′ = (p -1)U p V p-1 on R.
(1.7)

Then up to exchanging U and V , 1) Up to translation V (y) = U (-y).

2)

U ′ > 0 everywhere, U ′ (+∞) = (T ∞ ) 1
p , and there exist some positive constants, m, M , k, K, c, C such that near -∞, me

-Kx 2 ≤ U (x) ≤ M e -kx 2 , c|x|U (x) ≤ U ′ (x) ≤ C|x|U (x).
Similar estimates hold for V exchanging -∞ and +∞.

3) Suppose that (φ, ψ) is a bounded solution of the linearized system

(|U ′ | p-2 φ ′ ) ′ = (p -1)U p-2 V p φ + pV p-1 U p-1 ψ, (|V ′ | p-2 ψ ′ ) ′ = (p -1)V p-2 U p ψ + pU p-1 V p-1 φ in R, then there exists some constant c such that (φ, ψ) = c(U ′ , V ′ ).
In a last section we present the analogous of the De Giorgi result in [START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF] in the 2 dimensional case : Theorem 1.3. Suppose that N = 2 and that (u, v) is a non negative solution of (1.1). Suppose that u ,1 > 0 and v ,1 < 0, and that there exists some constant C such that for any R large enough

1 R 2+p B 2R \B R (|u| p + |v| p ) ≤ C.
Then the solution is "one dimensional", i. e. there exists (U, V ) a solution of (1.7) and some a ∈ R 2 such that a • e 1 = 0 such that

u(x) = U (a • x), v(x) = V (a • x).
This result is not new. It is contained in [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems discrete and continuous Discrete and continuous dynamical systems[END_REF], under more general conditions both on the right hand side and on the operator which generalizes the p-Laplacian. But we present here a short proof which does not need to introduce nor stability, nor geometrical properties.

As it is done in [START_REF] Fazly | De Giorgi type results for elliptic systems[END_REF], it is probable that the results in section 4 can be extended to the case of a right hand side ∇H(u), where the function H satisfies orientability conditions as defined in [START_REF] Fazly | De Giorgi type results for elliptic systems[END_REF].

One question of interest is the following : As we mentioned above, in [START_REF] Farina | Some Symmetry results for entire solutions of an elliptic system arising in phase transition[END_REF], and [START_REF] Farina | Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation Accepted in[END_REF] the authors suppose that the solution has at most algebraic growth. Can we have the same result in the p-Laplacian case? The answer is not immediate, since the proof of Farina et al relies on some properties of the Almgren frequency function, which do not extend to the p-Laplacian case .

Another interesting direction is the following : Suppose that one replaces the Laplacian system by a Fully Nonlinear system. Of course for Pucci's operators the one dimensional system is reduced, up to constant, to the Laplacian case. But, in dimension N ≥ 2, due to the non differentiability of the Pucci's operators, the definition of stable solutions must be precised. In the same order of ideas, one can imagine to treat the case of Fully Nonlinear degenerate or singular systems, based on the model of the p-Laplacian type treated here, but not under divergence form, as the following

|∇u| α F (D 2 u) = u α+1 v α+2 |∇v| α F (D 2 v) = v α+1 u α+2 in R N
where α is some number > -1 and F is fully non linear elliptic. The reader may consult [START_REF] Birindelli | Eigenvalue, maximum Principle and Regularity for Fully non linear operators[END_REF] for properties of such operators and a convenient definition of viscosity solutions.

It would be far too long to cite all the papers written about the case of one equation. Let us cite for the p-Laplacian case the recent paper of A. Farina and Valdinoci [START_REF] Farina | 1D Symetry for solutions of semi linear and quasilinear elliptic equations Transaction Of the Am[END_REF], the very complete paper of Savin et al. [START_REF] Savin | Flat level set regularity of p Laplace phase transitions[END_REF]. For variations on the subject on De Giorgi's conjecture in the case of a single equation, the reader may consult [START_REF] Giorgi | Convergence problems for functionals and operators[END_REF], [START_REF] Farina | Symmetry for solutions of semilinear elliptic equations in R N and related conjectures[END_REF], [START_REF] Farina | Bernstein and De Giorgi type problems : new results via a geometric approach[END_REF], [START_REF] Barlow | The Liouville property and a conjecture of De Giorgi[END_REF], [START_REF] Berestycki | One dimensional symmetry of bounded entire solutions of some elliptic equations Duke Math[END_REF], [START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF], [START_REF] Ambrosio | Entire solutions of semilinear elliptic equations in R 3 and a conjecture of De Giorgi[END_REF], [START_REF] Savin | Regularity of flat level sets in phase transitions[END_REF], [START_REF] Birindelli | One-dimensional symmetry for solutions of Allen Cahn fully nonlinear equations[END_REF].

The paper is organized as follows : In Section 2, we consider the one dimensional system defining (u Λ , v Λ ) and prove Theorem 1.1. Section 3 is devoted to the proof of Theorem 1.2. Some of the technical details of this section are proved in the appendix of [START_REF] Demengel | Qualitative properties of a nonlinear system involving the p-Laplacian operator[END_REF]. In section 4 we prove Theorem 1.3.

Acknowledgment

The author wishes to thank Alberto Farina especially for his precious advises and remarks, Enrico Valdinoci and Mohamed Fazly for having pointed out several papers on the question, Isabeau Birindelli for some informal discussions about section 4, and the anonymous referee for their judicious remarks.

2 The original problem : Proof of theorem 1.1

In all that section we will frequently use subsequences in place of sequences, without mentioning it. Let us consider for α, β and Λ, λ 1,Λ , λ 2,Λ some given positive constants, (u Λ , v Λ ) which solves

   -|u ′ Λ | p-2 u ′′ Λ + αu p+1 Λ + Λv p Λ u p-1 Λ = λ 1,Λ u p-1 Λ -|v ′ Λ | p-2 v ′′ Λ + βu p+1 Λ + Λv p-1 Λ u p Λ = λ 2,Λ v p-1 Λ in ]a, b[ u Λ , v Λ > 0, u Λ (a) = u Λ (b) = v Λ (a) = v Λ (b) = 0, b a |u Λ | p = b a |v Λ | p = 1. (2.1) Such (u Λ , v Λ ) is then a (non negative) solution of the minimizing eigenvalue problem inf |{u|p=|v|p=1,(u,v)∈W 1,p (]a,b[)} E Λ (u, v),
where

E Λ (u, v) = 1 p b a |u ′ | p + 1 p b a |v ′ | p + α b a |u| p+2 p + 2 + β b a |v| p+2 p + 2 + Λ p b a |u| p |v| p . Assume that max Λ (λ 1,Λ , λ 2,λ ) ≤ C, Due to the first equation in (2.1) multiplied by u Λ , integrated over ]a, b[, one gets b a |u ′ Λ | p + α b a u p+2 Λ + Λ b a u p Λ v p Λ = λ 1,Λ . (2.2) Doing the same for v Λ , one gets that (u Λ , v Λ ) is bounded in W 1,p (]a, b[) 2 . Furthermore Lemma 2.1. There exist T Λ , C 1 and C 2 independent on Λ, such that |u ′ Λ | p p + |v ′ Λ | p p -Λ u p Λ v p Λ p -α u p+2 Λ p + 2 -β v p+2 Λ p + 2 + λ 1,Λ u p Λ p + λ 2,Λ v p Λ p = T Λ p (2.3) and 0 < C 1 ≤ T Λ ≤ C 2 < ∞ Proof
Multiply the first equation in (2.1) by u ′ Λ , the second one by v ′ Λ , and add the two equations, we obtain that

-( |u ′ Λ | p p ) ′ -( |v ′ Λ | p p ) ′ + α (u p+2 Λ ) ′ p + 2 + β (v p+2 Λ ) ′ p + 2 + Λ (v p Λ u p Λ ) ′ p = λ 1,Λ (u p Λ ) ′ p + λ 2,Λ (v p Λ ) ′ p .
Hence there exists some constant

T Λ such that (2.3) is satisfied. Integrating on [a, b] the equation defining T Λ , one gets b a |u ′ Λ | p p + b a |v ′ Λ | p p -Λ b a u p Λ v p Λ p -α b a u p+2 Λ p + 2 -β b a v p+2 Λ p + 2 +λ 1,Λ +λ 2,Λ = T Λ (b -a) p .
On the other hand, using (2.2) for u Λ and its analogous for v λ , one has

b a |u ′ Λ | p + b a |v ′ Λ | p + α b a u p+2 Λ + β b a v p+2 Λ + 2Λ b a u p Λ v p Λ = λ 1,Λ + λ 2,Λ .
Combining the two equations one gets

T Λ (b-a) = 2 b a |u ′ Λ | p +2 b a |v ′ Λ | p + 2α p + 2 b a u p+2 Λ + 2β p + 2 b a v p+2 Λ +Λ b a u p Λ v p Λ ,
and using Poincaré's inequality

b a |u ′ Λ | p ≥ C b a |u Λ | p ≥ C, one obtains that T Λ ≥ C > 0.
On the other hand since (u Λ ) and (v Λ ) are bounded in W 1,p (]a, b[) and since λ 1,Λ and λ 2,Λ are bounded, one gets that T Λ is bounded from above.

Furthermore using the equation defining T Λ , and the fact that u Λ and v Λ vanish on the end points, one gets that

u ′ Λ (a) , u ′ Λ (b), v ′ Λ (a), v ′ Λ (b)
are bounded independently on Λ. Integrating the first equation in (2.1) between a and x one gets

|u ′ Λ | p-2 u ′ Λ (x) p -1 - |u ′ Λ | p-2 u ′ Λ (a) p -1 = α x a u p+1 Λ + Λ x a v p Λ u p-1 Λ + λ 1,Λ x a u p-1 Λ using this on {x = b}, one gets Λ b a u p-1 Λ v p Λ ≤ C,

and by the positivity that |u

′ Λ | p-2 u ′ Λ (x) -|u ′ Λ | p-2 u ′ Λ (a) ≤ C.
Doing the same between x and b one obtains that |u ′ Λ | ≤ C for some constant independent on Λ. In the same manner,

|v ′ Λ | ≤ C. Theorem 2.2. Assume that (u Λ , v Λ ) solves the system (2.1) in ]a, b[. There exists x Λ such that m Λ = u(x Λ ) = v Λ (x Λ ) → 0 as Λ goes to infinity, and x Λ → x ∞ ∈]a, b[. Furthermore m 2p Λ Λ → C o > 0 and Λ 1 2p min(x Λ -a, b -x Λ ) → +∞.
Proof of Theorem 2.2 By the previous estimates (u Λ ) and (v Λ ) are relatively compact in C([a, b]). In particular up to subsequence, (u Λ ) and (v Λ ) are uniformly convergent. Let (u ∞ , v ∞ ) be the limit of such subsequence. By the identity

b a |u ∞ | p = b a |v ∞ | p ,
and by the uniform convergence , there exists x ∞ and x Λ which tends to

x ∞ , such that m Λ = u Λ (x Λ ) = v Λ (x Λ ) and x ∞ ∈ {u ∞ (x) = v ∞ (x)} = γ.
To prove that lim sup Λm 2p Λ < ∞, we argue by contradiction and define ũΛ =

1 m Λ u( y m Λ Λ 1 p + x Λ ), ṽΛ = 1 m Λ v( y m Λ Λ 1 p + x Λ ) where y ∈] -x Λ + am Λ Λ 1 p , -x Λ + bm Λ Λ 1 p [, interval which tends to R. Then (ũ Λ , ṽΛ ) satisfy the equation |ũ ′ Λ | p p + |ṽ ′ Λ | p p - ũp Λ ṽp Λ p -α ũp+2 Λ (p + 2)m p-2 Λ Λ -β ṽp+2 Λ (p + 2)m p-2 Λ + λ 1,Λ ũp Λ pm p Λ Λ + λ 2,Λ ṽp Λ pm p Λ Λ = T Λ pm 2p Λ Λ . (2.4)
Using the fact that (u ′ Λ ) is bounded independently on Λ, by the mean value's theorem

|ũ Λ (y) - 1 m Λ u(x Λ )| ≤ 1 m 2 Λ Λ 1 p |u ′ Λ | ∞
and we have analogous estimates for ṽΛ , so using u Λ (x Λ ) = m Λ = v Λ (x Λ ) one obtains that ũΛ goes to 1 uniformly, ṽΛ goes to 1, finally passing to the limit in the equation (2.3) one gets that

- 1 p = 0,
a contradiction. We have obtained that Λm 2p Λ is bounded. To end the proof, suppose by contradiction that Λm 2p Λ → 0 for a subsequence and let ũΛ (y) = 1 m Λ u Λ (m Λ y + x Λ ), then ũΛ and ṽΛ satisfy the identity

|ũ ′ Λ | p p + |ṽ ′ Λ | p p - m 2p Λ Λũ p Λ ṽp Λ p - α (p + 2) m p+2 Λ ũp+2 Λ - β (p + 2) m p+2 Λ ṽp+2 Λ + λ 1,Λ m p Λ ũp Λ p + λ 2,Λ m p Λ ṽp Λ p = T Λ p . Furthermore |ũ Λ (y) -1 m Λ u Λ (x Λ )| ≤
C|y| by the mean values theorem, which implies that ũΛ is uniformly bounded. Since |ũ ′ Λ | ∞ is bounded independently on Λ, the dominated convergence theorem implies that (up to subsequence) ũ′ Λ converges strongly in L p , in particular |ũ ′ Λ | p-2 ũ′ Λ converges in the distributional sense, and so, also does its derivative. By passing to the limit in (2.1), one obtains that (ũ Λ , ṽΛ ) tends locally uniformly to (u

∞ , v ∞ ) which satisfies |u ′ ∞ | p-2 u ′′ ∞ = 0 and |v ′ ∞ | p-2 v ′′ ∞ = 0, hence u ′ ∞ = const, v ′ ∞ = const.
Since u ∞ and v ∞ are bounded, these constant are zero, which yields a contradiction with the identity

|u ′ ∞ | p p + |v ′ ∞ | p p = T ∞ p ,
remarking that T ∞ := lim T λ = 0, by the estimates on T Λ proved before. We have obtained that m 2p Λ Λ is bounded from above by some constant > 0. We finally prove that Λ 1 2p min(x Λa, bx Λ ) → +∞. Suppose for example and by contradiction that up to a subsequence Λ

1 2p (x Λ -a) → C 1 . Define ũΛ (y) = Λ 1 2p u( y Λ 1 2p + x Λ ). Then (ũ Λ , ṽΛ ) satisfies |ũ ′ Λ | p-2 ũ′′ Λ - αũ p+1 Λ Λ p+2 2p -ṽp Λ ũp-1 Λ + λ 1,Λ ũp-1 Λ Λ 1 2 = 0, |ṽ ′ Λ | p-2 ṽ′′ Λ - βṽ p+1 Λ Λ p+2 2p -ũp Λ ṽp-1 Λ + λ 2,Λ ṽp-1 Λ Λ 1 2 = 0.
We also get from the energy estimate (2.3)

|ũ ′ Λ | p p + |ṽ ′ Λ | p p - ũp Λ ṽp Λ p - αũ p+2 Λ (p + 2)Λ p+2 2p - βṽ p+2 Λ (p + 2)Λ p+2 2p + λ 1,Λ ũp Λ 1 2 + λ 2,Λ ṽp Λ 1 2 = T Λ p .
Remark as before that when Λ goes to infinity, (ũ Λ , ṽΛ ) tends locally uniformly to some (U, V ) , which satisfies

|U ′ | p-2 U ′′ = V p U p-1 |V ′ | p-2 V ′′ = U p V p-1 and U (-C 1 ) = V (-C 1 ) = 0. Note that if (b -x Λ )Λ 1 2p → C 2 < ∞ one has U (C 2 ) =
0 and then U ′′ ≥ 0 and U ≥ 0 implies U ≡ 0. We then assume that C 2 = +∞. Using Fatou's lemma one gets

∞ -C 1 V p U p-1 ≤ lim inf (b-x Λ )Λ 1 2p (a-x Λ )Λ 1 2p ṽp Λ ũp-1 Λ = Λ b a v p Λ u p-1 Λ ≤ C
as stated in the proof of Lemma 2.1. Since U ′ is increasing and U ≥ 0, if U is not identically zero, there exists

C ′ 1 ≥ C 1 , such that U ′ (C 1 ) > 0, and U = 0 on [-C 1 , -C ′ 1 ] then U (x) ≥ U (-C ′ 1 ) + U ′ (-C ′ 1 )(x -C ′ 1 ) = U ′ (-C ′ 1 ) + (x -C ′ 1
). In the same manner there exists

C ′′ 1 ≥ C 1 , such that V (x) ≥ V (-C ′′ 1 ) + V ′ (-C ′′ 1 )(x -C ′′ 1 ).
We have obtained that near +∞, U p-1 V p ≥ Cx 2p-1 , which contradicts the fact that U p-1 V p is integrable on ] -C 1 , +∞[. Finally U = V = 0, a contradiction with the identity defining T ∞ , when passing to the limit. We have obtained that Λ

1 2p min(x Λ -a, b -x Λ ) → +∞.
This ends the proof of Theorem 1.1.

3 Qualitative properties of the p-system in the one dimensional case : Proof of Theorem 1.2

In this section we want to prove the existence of non trivial solutions to the limit system (1.7). Note that the previous existence result is obtained under the assumption (λ 1,Λ , λ 2,Λ ) ≤ C. Theorem 1.2 is a consequence of several theorems and propositions :

Theorem 3.1. There exists an entire solution for (1.7) such that U (x) = V (-x).

Proof

We argue as in [START_REF] Berestycki | On phase-separation models: asymptotics and qualitative properties[END_REF], up to technical arguments due to the non linearity of the p-Laplacian, and due to the singularity (p < 2) or the degeneracy (p > 2).

Let us consider for R large the variational problem inf

(U,V )∈Xp(R) { 1 p R -R |U ′ | p + 1 p R -R |V ′ | p + 1 p R -R |U | p |V | p }.
where

X p (R) = {(U, V ) ∈ H 1 (] -R, R[), U (x) = V (-x), U (-R) = 0, U (R) = R}.
This problem admits a unique solution (U R , V R ).

We prove that U R is non negative. Indeed one has

|U ′ R | p-2 U ′′ R = |U R | p-2 U R |V R | p ,
and a symmetric equation for V R . Multiplying by U - R and integrating by parts, using the fact that U R (-R) and U R (R) are nonnegative, one gets that U - R = 0 and then U R ≥ 0. The same is valid for V R .

By the strong maximum principle of Vasquez [START_REF] Vasquez | A strong maximum principle for some quasilinear elliptic equations[END_REF],

U R > 0 on ] -R, R[, U ′ R (-R) > 0 and since |U ′ R | p-2 U ′ R is increasing, U ′ R > 0 everywhere, finally U ′′ R ≥ 0. Analogously V ′ R < 0, then U R -V R vanishes only on zero. U R -V R > 0 on ]0, R] implies that |U ′ R | p-2 U ′′ R -|V ′ R | p-2 V ′′ R = U p-1 R V p-1 R (V R -U R ) ≤ 0 on [0, R]. First case p ≥ 2. Using V ′ R (x) = -U ′ R (-x), one has for x > 0, |V ′ R |(x) = U ′ R (-x) < U ′ R (x) and then |U ′ R | p-2 U ′′ R ≤ |U ′ R | p-2 V ′′ R . Since U ′ R > 0, this implies that U ′′ R -V ′′ R ≤ 0. From this one gets that U R ≥ V R + x for x > 0, Indeed on 0, U R (0) = V R (0) ≥ V R (0) + 0, and U R (R) = R ≥ V R (R) + R = R, in particular U R (x) > x for x > 0.
We have obtained that

|V ′ R | p-2 V ′′ R ≥ (x + ) p V p-1 R
since V ′′ R ≥ 0 everywhere, in particular on R -. Let V be the solution, ( given by Lemma (4.3) in [START_REF] Demengel | Qualitative properties of a nonlinear system involving the p-Laplacian operator[END_REF]) , of

| V ′ | p-2 V ′′ = x p | V | p-2 V , on R + , which is positive and satisfies V ′ (0) = -2. Let us extend V on R -by the linear function -2x + V (0). Since V hence defined is C 2 and is a solution of | V ′ | p-2 V ′′ = (x + ) p | V | p-2 Ṽ on both R + and R -, one gets that it is a solution on ] -R, R[. For R large enough, V R (-R) = R ≤ -2(-R) + V (0) = V (-R), while V R (R) = 0 ≤ Ṽ (R) since Ṽ is positive.
Using the comparison principle, since V and V R are respectively solution and sub-solution of the same equation, one gets that V R ≤ V on [-R, R]. Using Harnack's inequality, [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF] one gets that (U R , V R ) tends to a non trivial solution (since U R ≥ x + ) , (U, V ) which satisfies (1.7) and V (x) = U (-x). Second case p < 2:

We begin to prove that

U ′ R (x) ≥ 1 1+2 1 p-1 for x > 0. Note that for x > 0 (U ′ R ) p-1 ≤ |V ′ R | p-2 V ′ R (x) + 2(U ′ R (0)) p-1 which implies that U ′ R (x) ≤ 2 1 p-1 U ′ R (0). Hence U R (R) = R ≤ U R (0)+2 1 p-1 RU ′ R (0) On the other hand when x ∈ R - U ′ R (x) ≤ U ′ R (0) which implies by the mean value's theorem that U R (0) ≤ U R (-R) + U ′ R (0)R. We have obtained that U ′ R (0) ≥ 1 1 + 2 1 p-1
as soon as R is large enough. We derive from this that on R

+ , |V ′ R | p-2 V ′′ R ≥ 1 1+2 1 p-1 p (x + ) p V p-1 R .
We now consider the solution V of

| V ′ | p-2 V ′′ = 1 1+2 1 p-1 p x p V p-1 on R + ,
V > 0 given by Proposition 4.4 in [START_REF] Demengel | Qualitative properties of a nonlinear system involving the p-Laplacian operator[END_REF] which satisfies V ′ (0) = -2, extended by -2x + V (0) on R -.

One obtains as in the case p ≥ 2 that V R ≤ V and by Harnack's inequality, [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF], one gets that (U R , V R ) tends locally uniformly to a non trivial solution (U, V ) which satisfies (1.7), and U (x) = V (-x). Lemma 3.2. Suppose that (U, V ) satifies (1.7). Then either U ′ > 0 and V ′ < 0 or U ′ < 0 and V ′ > 0. Furthermore there exists some constant C such that

|U ′ | + |V ′ | ≤ C.

Proof

Clearly the identity

|U ′ | p + |V ′ | p -U p V p = T ∞ (3.1)
holds for some finite constant T ∞ . Since U ′′ ≥ 0, either U ′ > 0 or U ′ < 0 or there exists x 1 such that U ′ (x) > 0 for x > x 1 and conversely for x < x 1 , and analogous facts hold for V ′ . Suppose that U ′ and V ′ have the same sign somewhere, then if this sign is positif, by the increasing behavior of U ′ and V ′ , it is true also for x large .

In particular

U (+∞) = V (+∞) = +∞. The case p ≥ 2 Let ϕ = U + V . For x large enough, |ϕ ′ | p-2 = (|U ′ | + |V ′ |) p-2 ≥ c(|U ′ | p-2 + |V ′ | p-2 ) and then |ϕ ′ | p-2 ϕ ′′ ≥ c(|U ′ | p-2 + |V ′ | p-2 )(U ′′ + V ′′ ) ≥ c(|U ′ | p-2 U ′′ + |V ′ | p-2 V ′′ ) ≥ 2 p-1 (V p + U p ) ≥ (U + V ) p
as soon as x is large enough since U and V go to infinity.

Then |ϕ ′ | p-2 ϕ ′′ ≥ ϕ p , for x large enough.

In the following, c will denote some positive constant depending only on p which can vary from one line to another.

Multiplying by ϕ ′ > 0, one gets that d dx (|ϕ ′ | p ) ≥ c d dx (ϕ p+1 ). Then (ϕ ′ ) p ≥ c(ϕ p+1 -1), and since ϕ tends to infinity, for x large enough, (ϕ ′ ) p (x) ≥ cϕ p+1 (x). This implies ϕ ′ ≥ cϕ p+1 p , and then for x large enough, d dx (-ϕ -1 p ) ≥ c, which would imply that ϕ becomes negative for x large enough, a contradiction.

The case p < 2 Let ϕ = ((U ′ ) p-1 + (V ′ ) p-1 ) 1 p-1 , note that ϕ p-1 = (U ′ ) p-1 + (V ′ ) p-1 , that ϕ ≥ U ′ + V ′ and ϕ p ≤ 2 p p-1 -1 (U ′ + V ′ ) p . One has (ϕ p-1 ) ′ = d dx ((U ′ ) p-1 + (V ′ ) p-1 ) = (p -1)(U p-1 V p + V p-1 U p ) ≥ (V + U ) p ,
for x large, by the behavior at infinity of U and V .

Multiplying by

U ′ + V ′ > 0 one obtains that d dx (ϕ p-1 ) (U ′ + V ′ ) ≥ 1 p + 1 d dx ((V + U ) p+1 ).
On the other hand, by the positivity of d dx (ϕ p-1 ), and

U ′ + V ′ one has d dx (ϕ p-1 )(U ′ + V ′ ) ≤ p -1 p d dx (ϕ p )
hence integrating and using the fact that U + V goes to infinity when x goes to +∞, one gets that there exists some constant c p such that for x large enough

(U ′ + V ′ ) p ≥ c p ϕ p ≥ c p (U + V ) p+1
We end as in the case p ≥ 2 and get a contradiction. If the sign of U ′ and V ′ are both negative somewhere they are both negative for x < -x 1 . By considering the invariance of the equation by changing x in -x, and reasoning as above one gets a contradiction.

We have obtained that up to exchanging U and V , U ′ > 0 and V ′ < 0. Suppose that U ′ → +∞ somewhere, then it occurs at +∞ since U ′ is increasing, in particular U goes to +∞ at +∞, and using (3.1) so does U V .

Then (V ′ |V ′ | p-2 ) ′ = (p -1)(U V ) p-1 U → +∞, which implies that V ′ goes to +∞ at +∞, a contradiction with V ′ < 0. We have obtained that U ′ is bounded. If V ′ → -∞ somewhere, it occurs at -∞, then V goes to +∞ at -∞ , by (3.1) U V goes to +∞ at -∞ and |U ′ | p-2 U ′′ = (U V ) p-1 V → +∞ at -∞, hence U ′ becomes < 0 for x large negative, a contradiction.
We have obtained that

|U ′ | + |V ′ | ≤ C.
Proposition 3.3. Let (U, V ) be a solution such that U ′ > 0 and V ′ < 0. Then V p U p-1 → 0, U p V p-1 → 0 at ±∞. Furthermore the following hold :

U (-∞) = 0, U ′ (-∞) = 0, U ′ (+∞) = (T ∞ ) 1 p , (3.2)
V (+∞) = 0, V ′ (+∞) = 0, V ′ (-∞) = -(T ∞ ) 1 p (3.3) Proof Since d dx (|U ′ | p-2 U ′ ) ≥ 0, and d dx (|V ′ | p-2 V ′ ) ≥ 0, U ′
and V ′ have a limit at infinity. Furthermore V ′ ≤ 0 and is increasing so it converges at +∞. Its limit must be zero since if not for x large enough, V ′ ≤ -m < 0 and V would become negative for x large.

By Lemma 1.2, U ′ is bounded. Furthermore it has a positive finite limit at +infinity, hence U goes to +infinity, more precisely U behaves like an increasing linear function. By ( 3.1) U p V p has a non negative limit at +infinity, consequently [START_REF] Demengel | Qualitative properties of a nonlinear system involving the p-Laplacian operator[END_REF] one gets that V ≤ W on [x o , ∞[ and then lim x→+∞ U V = 0 as well as lim x→+∞ U p V p-1 = 0. Since V ′ → 0 at infinity, (3.1) implies that |U ′ | p (+∞) = T ∞ . In particular T ∞ > 0.

U p-1 V p = 1 U (U p V p ) → 0. Furthermore for x > x o and x o large, |V ′ | p-2 V ′′ -V p-1 ≥ 0. Let us consider W = V (x o )e -x+xo which satisfies |W ′ | p-2 W ′′ -W p-1 ≤ 0. Using lemma 4.1 in the appendix of
A symmetric result holds near -∞ exchanging U and V .

Lemma 3.4. Let U and V be as in Proposition 3.3. There exist some positive constants m, M, k, K which depend on T ∞ , such that

me -Kx 2 ≤ U (x) ≤ M e -kx 2
and some positive constants c, C such that c|x|U (x) ≤ U ′ (x) ≤ C|x|U (x), for x large negative, and analogous estimates for V near +∞. Furthermore U has two asymptotic lines {y = 0} at -∞ and {y = (T ∞ )

1 p x+ b 1 } for some b 1 ∈ R, at +∞. Similarly V has asymptotic lines 0 at +∞ and y = -(T ∞ ) 1 p x + b 2 for some b 2 at -∞.
Remark 3.5. The constant k and K can be explicitely determined in function of T ∞ .

Proof :

Let k be such that V (x) ≥ 2k|x| near -∞, define W (x) = Ce -k|x| 2 which satisfies |W ′ | p-2 W ′′ ≤ V p W p-1 for x large enough negative, where the constant has been chosen in order that W (-M ) = U (-M ), and M is conveniently large. By Lemma 4.1 in [START_REF] Demengel | Qualitative properties of a nonlinear system involving the p-Laplacian operator[END_REF], using the fact that W ′ and U ′ are bounded, one gets that U ≤ W .

The lower bound can be obtained by considering some

K such that near -∞ V (x) ≤ K|x|. Consider W (x) = Ce -Kx 2 where C is chosen so that W (-M ) = U (-M ) and M is large. |W ′ | p-2 W ′′ ≥ V p W p-1
as soon as x is large enough negative. From this one derives that W ≤ U .

We prove the assertions concerning U and U ′ . We begin to establish that U ′ ≥ U for x large negative. Indeed let us observe that for x large negative |U ′ | p-2 (x)U ′′ (x) ≥ U p-1 (x) and then multiplying by U ′ and integrating, using

U (-∞) = U ′ (-∞) = 0, one obtains (U ′ ) p (x) ≥ U p (x) for x large negative.
To prove a better estimate, observe that by the behavior of V at -∞, for

x large negative |U ′ | p-2 U ′′ (x) ≥ C|x| p U p-1 (x).
We multiply by U ′ and prove that x). Using the fact that near -∞, U ′ and |x|U tend to zero, one gets that (U ′ ) p (x) -C|x| p U p (x) ≥ 0, always for x large negative.

-x -∞ |t| p U p-1 (t)U ′ (t)dt ≥ C|x| p U p (-x), for x large positive. Indeed -x -∞ |t| p U p-1 (t)U ′ (t)dt = [|t| p U p p | -x -∞ + -x -∞ |t| p-1 U p (t)dt ≥ |x| p U p p (-
In the same manner by the behaviour of V at -∞ there exists C such that |U ′ | p-2 (x)U ′′ (x) ≤ C|x| p U p-1 (x) for x large negative. Once more we multiply by U ′ and use here

-x -∞ |t| p U p-1 (t)U ′ (t)dt = [|t| p U p p | -x -∞ + -x -∞ |t| p-1 U p (t)dt ≤ |x| p U p p (-x) + 1 |x| -x -∞ |t| p U p-1 (t)U ′ (t)dt.
From this as soon as |x| > 2, x > 0,

-x -∞ |t| p U p-1 (t)U ′ (t)dt ≤ 2|x| p U p p (-x).
We have obtained the estimate on the right U ′ (x) ≤ C|x|U (x).

To deduce from this the asymptotics of U and V , we use the previous estimates on U for V near +∞. So we have |V ′ | ≥ C|x|e -kx 2 and then by ( 3.1)

one derives that (U ′ ) p -T ∞ ≥ -C|x| p e -pkx 2 , hence U ′ (x) -T 1 p ∞ ≥ -C|x| p e -kpx 2 which implies by integrating that U (x) -T Since U is convex, (U ′ -T 1 p
∞ ) is increasing and since it tends to 0 at infinity, it is negative, hence U -T

1 p ∞ x is decreasing. Finally U -T 1 p ∞ x is decreasing and bounded from below, hence defining b 1 = lim x→+∞ (U -T 1 p ∞ x), one has (U -T 1 p ∞ x) ≥ b 1 . Of course a symmetric result holds for V . Proposition 3.6. Let (U, V ) be a solution of (1.7), with U (0) = V (0) = 1. Then V (y) = U (-y).
We assume that U ′ > 0, hence we are in the hypothesis of the previous propositions. We can assume that b 1 ≥ b 2 , since if not one can replace (U (x), V (x)) by (V (-x), U (-x)) which exchanges b 1 and b 2 . We use the sliding method of Beresticky and Nirenberg, [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF].

Let

I λ = {x, x > λ} and U λ (x) = U (2λ -x), V λ (x) = V (2λ -x) Let w 1 = U -V λ , w 2 = U λ -V .
We prove in what follows that for λ large enough and x ∈ I λ , w 1 (x) > 0 as well as w 2 (x) > 0. From the asymptotic behaviour of U and V , and since

U is convex, U (x) ≥ T 1 p ∞ x + b 1 ,
and by the asymptotic behavior of V there exists K such that V (x) ≤ -T

1 p ∞ x -+ K, this implies that w 1 (x) ≥ T 1 p ∞ (x + (2λ -x) -) + b 1 -K for x ∈ I λ . So by taking λ such that λT 1 p ∞ > K -b 1 one gets that if x ∈]λ, 2λ[, w 1 (x) ≥ T 1 p ∞ (λ + 0) + b 1 -K > 0 and for x > 2λ, w 1 (x) ≥ T 1 p ∞ (x + x -2λ) + b 1 -K ≥ 2λT 1 p ∞ + b 1 -K > 0.
We now derive from this that w 2 is also > 0 in the same I λ for large values of λ. Indeed, we have

|U ′ λ | p-2 U ′′ λ -|V ′ | p-2 V ′′ = U p (U p-1 λ -V p-1 ) + U p-1 λ (V p λ -U p ).
Multiplying this by w - 2 , integrating between λ and x and using

(U λ -V )(λ) = (U -V )(λ) = w 1 (λ) > 0 one gets - x λ (|U ′ λ | p-2 U ′ λ -|V ′ | p-2 V ′ )(w - 2 ) ′ + [ 1 p -1 (|U ′ λ | p-2 U ′ λ -|V ′ | p-2 V ′ )(w - 2 ))] x λ = x λ U p (U p-1 λ -V p-1 )w - 2 + x λ U p-1 λ (V p λ -U p )w - 2 ≤ 0 Using - x λ (|U ′ λ | p-2 U ′′ λ -|V ′ | p-2 V ′′ )(w - 2 ) ′ ≥ 0, w - 2 (
λ) = 0 and w 2 (∞) = 0, as well as the fact that U ′ and V ′ are bounded, letting x go to infinity, one gets that w - 2 = 0 and then w 2 > 0 for x ∈ I λ and λ large enough. We now define λ ⋆ = inf{λ > 0, w µ 1 (x) > 0 in I µ , for all µ > λ}. By the previous observations w µ 2 > 0 in I µ for all µ > λ ⋆ . Since U (0) = V (0) = 1, one has if λ < 0, by the increasing behaviour of U -V , (U -V )(λ) < 0, which implies that λ ⋆ ≥ 0. We want to prove that λ ⋆ = 0.

Let us observe that by continuity w λ ⋆ 1 ≥ 0 and w λ ⋆ 2 ≥ 0 on I λ ⋆ . By the strong maximum principle w λ ⋆ 1 > 0 and w λ ⋆ 2 > 0 in I λ ⋆ . By the asymptotic behaviour, there exists B 1 such that for x < B 1 < 0, V (x) + T

1 p ∞ x -b 2 < b 1 -b 2 4
Take A = sup(2λ ⋆ -B 1 , λ ⋆ ) then for x > A, and for 0 < λ < λ ⋆ ,

V (2λ -x) + T 1 p ∞ (2λ -x) -b 2 < b 1 -b 2 2 , hence for x > A and λ ∈]0, λ ⋆ [, w λ 1 (x) -T 1 p ∞ 2λ ≥ b 1 -b 2 2 . We now observe that inf [λ ⋆ ,A] w λ ⋆ 1 = m > 0, indeed w λ ⋆ 1 (λ ⋆ ) = U (λ ⋆ ) - V (λ ⋆ ) > U (0) -V (0) since U ′ > 0, V ′ < 0, U (0) = V (0) and λ ⋆ > 0. By the continuity of w λ ⋆ 1 one can choose η > 0 such that inf [λ,A] w λ ⋆ 1 ≥ 3m 4 for |λ -λ ⋆ | ≤ η. By the uniform continuity of V in a compact set, one can also impose to η < λ ⋆ that for |λ -λ ⋆ | < η, for all x ∈ [λ ⋆ -η, A], one has |V (2λ ⋆ -x) -V (2λ -x)| ≤ m
4 and then for x > λ > λ ⋆η and x < A, U (x) -V (2λx) ≥ m 2 . Finally inf [λ,A] w λ 1 > 0 for λ ⋆η < λ < λ ⋆ , and then w λ 1 > 0 on a neighborhood on the left of λ ⋆ . This contradicts the definition of λ ⋆ . We have obtained λ ⋆ = 0 and then U (x) ≥ V (-x) for x ≥ 0.

Since we have seen before that w 0 1 ≥ 0 implies w 0 2 ≥ 0, U (-x) ≥ V (x) for x > 0. We have obtained that U (x) ≥ V (-x) for x ∈ R.

Since U (0) = V (0), U (x)-V (-x) reaches its minimum at zero. This implies that (w 0 1 ) ′ (0) = U ′ (0) + V ′ (0) = 0. By the strong comparison principle one gets that U (x) = V (-x). We have obtained in the same time that b 1 = b 2 .

Part 3) in Theorem 1.2 is contained in the Proposition 3.7. Suppose that φ, ψ are bounded solutions of

(|U ′ | p-2 φ ′ ) ′ = (p -1)U p-2 V p φ + pU p-1 V p-1 ψ (|V ′ | p-2 ψ ′ ) ′ = (p -1)V p-2 U p φ + pU p-1 V p-1 φ.
Then there exists some constant c such that (φ, ψ) = c(U ′ , V ′ ).

Proof

For personal convenience we use small letters (u, v) in place of (U, V ).

We do not distinguish the case p > 2 or p < 2 for the moment. Let φ, ψ be defined as φ = u ′ φ, ψ = v ′ ψ. One has

(|u ′ | p-2 φ ′ ) ′ = (|u ′ | p-2 u ′′ ) ′ φ + p|u ′ | p-2 u ′′ φ′ + |u ′ | p-2 u ′ φ′′ = pu p-1 v p φ′ + pv p-1 u p-1 v ′ φ + (p -1)u p-2 v p u ′ φ + |u ′ | p-2 u ′ φ′′ .
On the other hand using the equation satisfied by φ one gets

p|u ′ | p-2 u ′′ φ′ + |u ′ | p-2 u ′ φ′′ = pu p-1 v p-1 v ′ ( ψ -φ).
In the same manner for v

p|v ′ | p-2 v ′′ ψ′ + |v ′ | p-2 v ′ ψ′′ = pu p-1 v p-1 u ′ ( φ -ψ).
Multiplying the first equation by u ′ φ and the second one by v ′ ψ , one gets

p|u ′ | p-2 u ′ u ′′ φ φ′ + |u ′ | p φ φ′′ + p|v ′ | p-2 v ′ v ′′ ψ ψ′ + |v ′ | p ψ ψ′′ = -p(uv) p-1 u ′ v ′ ( ψ -φ) 2 .
Let us now observe that

p|u ′ | p-2 u ′ u ′′ φ φ′ + |u ′ | p φ φ′′ = (|u ′ | p φ φ′ ) ′ -|u ′ | p ( φ′ ) 2 .
Claim : |u ′ | p φ φ′ and |v ′ | p ψ′ ψ go to zero at +∞ and -∞ This claim will end the proof since then we will have

0 = [(|u ′ | p φ φ′ ) + (|v ′ | p φ ψ′ )] ∞ -∞ = R |u ′ | p ( φ′ ) 2 + |v ′ | p ( ψ′ ) 2 -p R (uv) p-1 u ′ v ′ ( ψ -φ) 2 ≥ 0 since u ′ v ′ < 0.
This will imply φ′ = ψ′ = 0 and φ = ψ.

In the sequel we prove the claim for u and φ. The result for v and ψ can be done by obvious symmetric arguments.

Proof of the claim for u and φ

|u ′ | p φ φ′ = |u ′ | p-2 (u ′ φ)(u ′ φ′ ) = |u ′ | p-2 φu ′ φ ′ u ′ - φu ′′ (u ′ ) 2 = |u ′ | p-2 φφ ′ -φ 2 |u ′ | p-2 u ′′ u ′ = |u ′ | p-2 φφ ′ -φ 2 u p-1 v p u ′ . Indeed -xo -x u p-3 (t)u ′ (t)|t| p-1 dt = [ u p-2 |t| p-1 p -2 ] -xo -x + p -1 p -2 -xo -x u p-2 (t)|t| p-2 dt ≥ u p-2 (-x)|x| p-1 2 -p -C 1 - p -1 (2 -p)c|x| 2 -xo -x u p-3 (t)u ′ (t)|t| p-1 dt which implies that |u ′ | p-2 φ ′ (-x) ≤ C 1 -C 2 |x| p-1 u p-2 (-x) for x large enough negative and then in particular φ ′ (-x) ≤ -C 3 |x| 2-p |x| p-1 = -C 3 |x|, with C 3 > 0. This contradicts φ bounded. In the same manner if φ ≤ -m < 0 one gets |u ′ | p-2 φ ′ (-x o ) -|u ′ | p-2 φ ′ (-x) ≤ -c|x| p-1 u p-2 (-x) which implies φ ′ (-x) ≥ C 3 |x|,
| p-2 φ ′ ≥ C > 0. Then φ(-x) ≥ -x -∞ C(u ′ ) 2-p (t)dt ≥ C -x -∞ |t| 1-p u 1-p (t)u ′ (t)dt. Observe that -x -∞ |t| 1-p u 1-p (t)u ′ (t)dt ≥ c|x| 1-p u 2-p (-x). Indeed -x -∞ |t| 1-p u 1-p u ′ (t)dt = 1 2 -p -x -∞ |t| 1-p d dt (u 2-p )dt = 1 2 -p [|t| 1-p u 2-p ] -x -∞ + 1 -p 2 -p -x -∞ (-t) -p |u| 2-p (t)dt ≥ 1 2 -p |x| 1-p u 2-p (-x) - p -1 (2 -p)c|x| 2 -x -∞ |t| 1-p u 1-p u ′ (t)dt.
This ends the proof by taking |x| large enough, x < 0. We have obtained that φ(-x) ≥ C|x| 1-p u 2-p (-x) for x large enough positive and replacing in the equation satisfied by φ one gets

|u ′ | p-2 φ ′ (-x o ) -|u ′ | p-2 φ ′ (-x) ≥ C -xo -x |t| 1-p |t| p dt = C|x| 2 .
From this one derives that (|u ′ | p-2 φ ′ )(-x) ≤ -Cx 2 , a contradiction with the assumption.

The case where |u ′ | p-2 φ(-x) ≤ -C < 0 for x large enough positive can be recovered by changing φ in -φ, noting the fact that the previous computations do note use the sign of ψ. We denote ϕ = u ,N > 0 and ψ = v ,N < 0. Using the same arguments as in [START_REF] Farina | Bernstein and De Giorgi type problems : new results via a geometric approach[END_REF], lemma 2.2, one has div(B(∇u)(∇ϕ)) = (p -1)u p-2 v p ϕ + pu p-1 u p-1 ψ on R N div(B(∇v)(∇ψ)) = (p -1)v p-2 u p ψ + pv p-1 u p-1 ϕ on R N .

We now define for any direction ζ, w 1 = Letting R go to infinity one obtains ∇w 1 = ∇w 2 = 0 and also that w 1 = w 2 hence w 1 = C = w 2 . Choose ζ such that ∇u(0) • ζ = 0, then since ϕ > 0 C = 0. Let a be some unitary vector orthogonal to ζ, then u(x) = U (a • x), v(x) = V (a • x) and (U, V ) is a solution of the one dimensional system.

Remark 4 . 2 .Theorem 4 . 3 .Remark 4 . 4 .B

 424344 B(u) is a non negative symetric bilinear form. Indeed it is clear if p ≥ 2 and if p < 2, using Cauchy Schwarz inequality,|(p -2)|∇u| p-4 (∇u • p) 2 | ≤ (2p)|∇u| p-2 | p| 2 .We now have the partial De Giorgi type result : Suppose that N = 2, that p ≥ 2, that u ,N > 0 and v ,N < 0 and that there exists some constant C such that for any large R,1 R 2+p B 2R \B R (|u| p + |v| p ) ≤ C, (4.2)then the solution is one dimensional, i.e. there exists a ∈ R 2 , |a| = 1, and (U, V ) a one dimensional solution withu(x) = U (a • x), v(x) = V (a • x)The condition ( 4.2) in Theorem 4.3 is in particular satisfied when u and v are at most linear increasing at ±∞.Proof Let χ R be some function in D(B(0, 2R)) χ R = 1 on B(0, R), 0 ≤ χ R ≤ 1 and |∇χ R | ≤ C R .Testing the equation against χ p R u, and using Young's inequality one getsB 2R χ p R |∇u| p ≤ C B 2R \B R |∇χ R | p u p ≤ C R p B 2R \B R |u| p . 4R \B 2R (|u| p + |v| p ) ≤ C.

ϕ 2 2 ≤ C R 2 B 2 ) 2 R 2 R2r - 3

 2222223 ∇w 1 = ϕ∇u ,ζu ,ζ ∇ϕ ψ 2 ∇w 2 = ψ∇v ,ζv ,ζ ∇ψApply B(∇u) to the first equation and B(∇v) to the second one , and take the divergence, one gets div(ϕ 2 B(∇u)∇w 1 ) = pu p-1 v p-1 ϕψ(w 2w 1 ) (A) div(ψ 2 B(∇v)∇w 2 ) = pu p-1 v p-1 ϕψ(w 1w 2 ). (B)Let η be any function in D(B(0, 2R)), η = 1 on B(0, R), 0 ≤ η ≤ 1 and |∇η| ≤ C R . We multiply (A) by w 1 η 2 and (B) by w 2 η 2 , integrate and sum the two equations, remark that -pu p-1 v p-1 ϕψ(w 2w 1 ) 2 ≥ 0, one getsϕ 2 (B(∇u)∇w 1 , ∇(w 1 η 2 ))ψ 2 (B(∇v)∇w 2 , ∇(w 2 η 2 )) ≥ 0Using the fact that B(∇u) is a positive operator one gets thatη 2 (B(∇u)∇w 1 , ∇w 1 ) + η 2 (B(∇v)∇w 2 , ∇w 2 ) ≤ C ϕ 2 w 2 1 (B(∇u)∇η, ∇η) + ϕ 2 w 2 2 (B(∇v)∇η, ∇η) ≤ C (|∇u| p + |∇v| p )|∇η| (0,2R) (|∇u| p + |∇v| p ) ≤ Cby the previous estimates. We end the proof as in Theorem 1.2 in[START_REF] Berestycki | On entire solutions of an elliptic system modeling phase separations[END_REF] : we use for any r, Br(0) |∇u| p + |∇v| p ≤ Cr 2 , we introduce -B(0, R) 0 if not.Let y(x) = |∇u| p +|∇v| p . Note that d dr B(0,r) y(λ)dλ = d dr ( ρ cos θ, ρ sin θ)ρdρdθ = r 2π 0 y(r cos θ, r sin θ)dθ. ThenB(0,R 2 )\B(0,R) |z| -2 y(z)dz = R Cr 2 dr ≤ C + C log R.From the previous estimates, we deriveB(0,R) (B(∇u)∇w 1 , ∇w 1 )ϕ 2 + B(0,R) (B(∇v)∇w 2 , ∇w 2 )ψ 2 ≤ C (log R) 2 B(0,R 2 )\B(0,R) |z| -2 (|∇u| p + |∇v| p )(z)dz ≤ C (log R) 2 + 2 C log R .

  this still contradicts φ bounded. So we are in the hypothesis that φ tends to zero and |u ′

p ∞ x ≥ -C for x large positive.

We consider separately the cases +∞ and -∞. The case +∞. By the results in Lemmata 3.2 and 3.4, u ′ is minorated by some positive constant and v goes exponentially towards zero, hence the term φ 2 u p-1 v p u ′ on the right goes to zero.

On the other hand |u ′ | p-2 φ ′ tends to zero. Indeed, its derivative is integrable for x large by the asymptotic behavior of u and v, and the fact that φ and ψ are bounded. So it has a limit. Suppose that the limit is l = 0, then

The case -∞.

We are led to distinguish the case p ≥ 2 and p < 2

The case p ≥ 2.

Let us recall that there exist some positive constants M , k, c such that near -∞, u ≤ M e -kx 2 , and u ′ ≥ c|x|u for x large enough negative. In particular since v is linear at infinity

goes to zero at -∞. Furthermore |u ′ | p-2 φ ′ has a limit at -∞ by the equation, if this limit was = 0, this would imply that φ ′ goes to ±∞ and would contradict φ bounded. From this one obtains that |u ′ | p φ φ′ tends to zero at -∞.

The case p < 2 This case is much more involved and requires several steps.

Step 1 : There exists t p which goes to -∞ such that (|u

Suppose for a while that there does not exist t p which goes to -∞ such that (|u ′ | p-2 φ ′ )(t p ) → 0. Then there exists C > 0 such that for all t large negative either (|u

One assumes that we are in the first case and will give later the arguments in the other case. Then φ ′ > 0 near -∞, hence φ has a finite limit since φ is bounded. We begin to prove that φ tends to zero at -∞. Suppose φ does not tend to zero, then there exists m > 0 such that either φ > m or φ ≤ -m for x large negative. In the first case for some constant c > 0 which can vary from one line to another, (|u

since the last term in the first equation satisfied by φ , say pu p-1 v p-1 ψ tends to zero. Integrating between -x and -x o large negative, one gets for

We have obtained that there exists t p which goes to -∞, such that (|u ′ | p-2 φ ′ )(t p ) → 0.

Step 2: (u ′ ) p-2 φφ ′ and u p-1 φ 2 v p u ′ both tend to zero at -∞. We multiply the equation satisfied by φ, by φ and integrate between t p+1 and t p , where t p is some subsequence decreasing to -∞, given by step 1. One obtains

and since u p-1 v p-1 φψ is absolutely integrable by the estimates on u and v one gets with the positivity of (p -1)u p-2 v p φ 2 + |u ′ | p-2 (φ ′ ) 2 and summing on p that u p-2 v p φ 2 and |u ′ | p-2 (φ ′ ) 2 are integrable. Finally for all s and t going to -∞ [|u ′ | p-2 φφ ′ ] t s tends to zero, hence |u ′ | p-2 φφ ′ has a limit, and since it possesses a subsequence which tends to zero, this limit is zero.

We now prove that φ 2 u p-1 v p u ′ has a finite limit. For this it is enough to prove that its derivative is integrable at -∞.

Each of the first three terms above can be majorized near -∞ by Cu p-2 v p φ 2 and then are integrable near -∞.

we use Cauchy Schwarz's inequality as follows

We deduce that since φ 2 u p-2 v p is integrable near -∞ , so is φ 2 u p-1 v p u ′ since its derivative is integrable, it tends to zero.

Of course we would obtain symmetric properties for ψ and v near +∞. This ends the proof. [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF] The higher dimensions: Proof of Theorem 1.

3

In [START_REF] Dipierro | Geometric inequalities and symmetry results for elliptic systems discrete and continuous Discrete and continuous dynamical systems[END_REF] the author considers a more general operator than the p-Laplacian, and more general right hand side. She obtains the one dimensionality of the solution by assuming a growth property of the solution which is reduced here to

and either that uv < 0 and the solution is stable, or ∂ N u > 0 and ∂ N v < 0 for some component e N and uv > 0.

We propose here a simpler proof for the partial result in Theorem 4.3, i.e. in the case where ∂ N u > 0, ∂ N v < 0 and u > 0, v > 0.

So, the equation considered is now

where u and v are positive.

Remark 4.1. Note that if u ≥ 0 , v ≥ 0, and if u is not identically zero, u > 0. Indeed, from the properties of the p-Laplace operator, u and v are both locally bounded. Then for any compact set K in R N , there exists C K such that v ≤ C K . Then -div(|∇u| p-2 ∇u) + C p K (p -1)u p-1 ≥ 0 and the strong maximum principle of Vasquez, [START_REF] Vasquez | A strong maximum principle for some quasilinear elliptic equations[END_REF] implies the positivity of u. The same is true for v.

We define B(u) = |∇u| p-2 Id + (p -2)|∇u| p-4 (∇u ⊗ ∇u), or equivalently for any ( p, q) ∈ (R N ) 2 , (B(u)p, q) = |∇u| p-2 ( p, q) + (p -2)|∇u| p-4 (∇u • p)(∇u • q).