Ioan Burciu
email: ioan.burciu@insa-lyon.fr

Guillaume Villemaud

Jacques Verdier

Candidate Architecture for MIMO LTE-Advanced Receivers with Multiple Channels Capabilities and Reduced Complexity and Cost

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INTRODUCTION

Internet of things: this is the greatest challenge of mobile communications. All incoming standards need to provide higher throughputs in compact terminals. One can see that all proposed solutions converge towards same techniques: OFDM (Orthogonal Frequency Division Multiplexing), MIMO (Multiple Input Multiple Output), adaptive coding and modulation, as well as scalable bandwidths. These techniques require an increasing complexity of terminals, leading to additional cost and consumption. Particularly from the RF front-ends point of view, OFDM imposes high PAPR(Peak to Average Power Ratio) constraints and good linearity [START_REF] Van Nee | OFDM for Wireless Multimedia Communications[END_REF], MIMO conducts to the multiplication of RF chains, scalable bandwidth imposes wider bandwidth characteristics of RF components [START_REF] Kaiser | Smart Antennas: State of the Art[END_REF].

Recently, we have proposed different solutions to reduce complexity of multi-band [START_REF] Burciu | A 802.11g and UMTS Simultaneous Reception Front-End Architecture using a double IQ structure[END_REF] and multi-antenna [START_REF] Gautier | Low complexity antenna diversity front-end: Use of code multiplexing[END_REF][5] frontends. In this article we look forward to merging those principles to propose a candidate architecture for simplified LTE-Advanced terminals [START_REF] Seidel | Progress on LTE Advanced" -the new 4G standard[END_REF] -current 3GPP [7] working group for future mobile phones with very high data rates and mobility. In fact, in actual propositions for LTE-Advanced PHY-layer, the use of combined multiple-antenna capabilities and multiple separated frequency channels are very seducing for the downlink transmission. Meanwhile the use of these combined techniques supposes high cost architectures for the user embedded terminal.

This paper is organized as follows. Section II summarizes the main LTE-Advanced key points, focusing on multi-band and multi-antenna needs. Section III assesses the previously proposed architectures which separately respond to multi-band and multi-antenna problems respectively. A novel combined architecture is presented in Section IV along with a comparative complexity study. This architecture is dedicated to the multiple-antenna reception of signals composed of two arbitrarily distributed frequency channels. Finally, conclusions and forthcoming works are drawn in section V.

LTE-ADVANCED SPECIFICATION

A. Requirements LTE-Advanced, an evolved version of LTE, is currently under investigation in order to fulfill the requirements defined by the International Telecommunications Union (ITU) for next generation mobile communication systems [START_REF] Seidel | Progress on LTE Advanced" -the new 4G standard[END_REF].

Requirements for LTE-Advanced are similar to those imposed by LTE, excepting peak data rates and spectral efficiency which should be increased. The goal is to provide peak-data-rates reaching 1 Gbit/s in local areas. Such high data rates require a special spectrum allocation for a 100 MHz range in order to obtain a spectral efficiency reaching a 10 bps/Hz level. This spectrum allocation of the 100 MHz system bandwidth will use the aggregation of individual component carriers.

B. Proposed radio access techniques for LTE-Advanced

The 3GPP working groups are currently starting take into consideration the technical proposals that can be implemented in order to achieve these requirements. The main novel requirements imposed by the future LTE-Advanced concern [7]:

• Asymmetric wider transmission bandwidth

• Layered OFDMA multi-access • Advanced multi-cell transmission • Discontinuous spectrum allocations • Enhanced multi-antenna transmission techniques
In general, OFDM provide simple means to increase bandwidth by simply increasing the number of subcarrier. However, due to a fragmented spectrum, the available bandwidth might not be contiguous, as shown in Fig. 1. Some of the main challenges for 100 MHz terminals are presented below [START_REF] Seidel | Progress on LTE Advanced" -the new 4G standard[END_REF]:

C. Scenario

Among the proposed radio access techniques, multi-antenna transmission and discontinuous spectrum allocation have a direct influence on the front-end part of the receiver. Each additional antenna or each additional band induces an important complexity increase of the receiver's analog part. This study assesses the multi-antenna reception of a discontinuous spectrum signal (as shown in Fig. 1).

PRIOR WORKS: LOW COMPLEXITY FRONT-END

D. General purpose

The analog complexity issue concerning advanced receiver using digital processing has been very little addressed [START_REF] Kaiser | Smart Antennas: State of the Art[END_REF]. In fact, the performance gain achieved by diversity techniques implies an increase of the digital complexity, but also an increase of the complexity and the consumption of the analog front-end [START_REF] Tsurumi | Broadband RF stage architecture for software-defined radio inhandheld terminal applications[END_REF].

In the research domain concerning multi-band reception, we can distinguish two different categories of front-ends: nonsimultaneous receivers using switching techniques and simultaneous receiving receivers. To the knowledge of the authors, the optimal scheme when realising multiband simultaneous reception uses the front-end stack-up techniqueeach chain being dedicated to the reception of only one frequency band. Nonetheless, this architecture is characterized by some inconveniences such as the bad complexityperformance trade-off, but also the price and the physical size.

Therefore our studies deal with the analog front-end architectures dedicated to simultaneous multi-band reception. Our main goal is the improvement of the performanceconsumption-complexity trade-off offered by these structures. Two studies explore the use of a single common front-end for the processing of signals received by the antennas:

• The double IQ architecture dedicated to the simultaneous reception of two separate frequency band signals.

• The code multiplexing architecture for the multiantenna reception.

The use of a single analog chain underlies the idea of multiplexing the different RF branches. In other words, we realise a multiplexing of the different RF contributions in order to use a single frequency translation step. In the digital domain, we demultiplex the different baseband translated contributions.

E. Double IQ architecture 1) Context

Double orthogonal translation techniques are generally used in image frequency rejection front-ends [START_REF] Rudell | A 1.9GHz Wide-Band IF Double Conversion CMOS Integrated Receiver for Cordless Telephone Applications[END_REF]. The rejection of the signal having its spectrum in the image frequency band of the useful signal is a key point of the heterodyne front-end architectures. In order to perform these specifications, the double orthogonal translation technique uses a signal processing method that manages to reconstruct the useful signal starting from four baseband signals. These baseband signals are the result of a double orthogonal frequency translation. Each of them is composed of two separate contributions: the baseband components of the useful signal and of the image frequency band signal.

In other words, the double orthogonal (double IQ) structure is multiplexing the useful signal and the image frequency band signal during the translation from RF to baseband and is then demultiplexing them in the baseband domain using a signal processing technique. Starting from this structure, we proposed and developed a novel front-end architecture dedicated to the simultaneous reception of two separate frequency band signals [START_REF] Burciu | A 802.11g and UMTS Simultaneous Reception Front-End Architecture using a double IQ structure[END_REF].

2) Presentation of the structure

The proposed dual-band simultaneous reception architecture is shown in Fig. 2. The input blocks -antenna, RF filter and LNA (Low Noise Amplifier) -are parallelized in order to have a good amplification and filtering of the two useful signals S 1 and S 2 . In the same time, we concluded to a possible mutualisation of these blocks while realizing the reception of two signals representing two frequency non-overlapping channels of the same standard. Once filtered and amplified, the useful signals are translated in the baseband domain by a double IQ structure. An important aspect that has to be mentioned is the choice of the first local oscillator LO 1 frequency. This frequency is chosen in such a manner that each of the useful signals has a spectrum situated in the image frequency band of the other. In the baseband domain, the four obtained signals are digitalized. The baseband component of the two useful signals is obtained by using two dedicated basic operations processing similar with that used by the image frequency rejection structures [START_REF] Rudell | A 1.9GHz Wide-Band IF Double Conversion CMOS Integrated Receiver for Cordless Telephone Applications[END_REF].

3) Characteristics

The double orthogonal translation technique allows a theoretically perfect rejection of the image band signal for a case where the quadrature mounted mixers are perfectly matched -no phase or gain mismatch. In the meantime, design and layout defaults, such as different line lengths between the two branches and non identical gain of the mixers, generate phase and respectively gain mismatches [START_REF] Traverso | Decision Directed Channel Estimation and High I/Q Imbalance Compensation in OFDM Receivers[END_REF]. A study concerning this issue as well as the design and implementation of an original dedicated digital MMSE (Minimum Mean Square Error) method has been done.The results show that, when integrated, this MMSE method confers to the double IQ architecture the same sensitivity to IQ mismatches as that of the dedicated front-ends stack-up architecture. These results were obtained in simulation using the ADS software provided by Agilent Technologies, the input signals chosen are two non-overlapping 802.11g channels. These results were validated by using an Agilent Technologies Platform that integrates a real transmission channel and an accurate model of the proposed architecture. Fig. 3 presents these results. The BER evolution depending on the SNR (Signal to Noise Ratio) of the useful signals shows that, when integrating the digital MMSE Method, the double IQ (MMSE DIQ) architecture presents the same sensibility to the IQ defaults as the stack-up architecture (Stack-Up) -for a phase and gain mismatch of 1o and 0.3 dB.

Several studies concerning the proposed double IQ structure were realized. In the following we mention the one concerning the specifications of the ADC (Analog to Digital Converters) and that concerning the overall consumption. The ADC specification study concluded that the constraints imposed to the ADC used by this structure are almost identical (dynamics and sampling frequency) to those used by a dedicated frontend stack-up structure. The overall consumption study is comparing the double IQ structure to the front-end stack-up structure. It concludes that due to the mutualisation of several blocks as well as because of the use of fewer components, the proposed double IQ structure offers Fig. 4 Code multiplexing architecture dedicated to the simultaneous reception of a multi-antenna system. Fig. 5 802.11g simulated and measured BER evolution for a two-antenna SIMO reception using the code multiplexing architecture and the classical stack-up architecture. a significant 25% of consumption gain as well as a smaller complexity.

A. Code multiplexing architecture 1) Context

An antenna diversity system must be able of simultaneously receive different versions of the transmitted signal(s).For example, an N antennas receiver must simultaneously process N signals having their spectrums situated in the same frequency band. To answer these specifications, the RF stackup architecture is an obvious choice. It is composed of N stacked-up processing chains, each of them being dedicated to the demodulation of one of the N branches. However, this choice imposes a high complexity of the analog part of the receiver. The goal of our study concerning the multipleantenna structures aims to reduce this complexity without decreasing the baseband signal's SNR quality.

To this end, an innovative architecture is introduced based on code multiplexing. This architecture uses the direct sequence spread spectrum (DSSS) technique in order to multiplex the different antennas contributions so that a single frequency translation block is used.

2) Presentation of the structure

In order to achieve the time and frequency overlapping between each antenna contribution, decorrelation can be done by the DSSS technique. This technique is the basis of CDMA multiple access technology [START_REF] Kohno | Spread spectrum access methods for wireless communications[END_REF]. The DSSS technique consists in allocating an orthogonal spreading code to each branch. Fig. 6 Candidate Architecture LTE-Advanced Receivers capable of realizing a multi-antennas processing of a discontinuous spectrum signal. The discontinuous spectrum is received, coded in the baseband domain in order to be processed by a single front-end structure.

The signals rk(t) are multiplied by the periodical codes ck(t) which are pseudorandom sequences of N binary entities having a rate N times higher than the symbol. Thus, the resulting bandwidth of the signal's spectrum is N times larger than those of the received signal rk(t). The addition of N encoded signals is then performed in order to generate the radio-frequency multiplex signal d(t). This signal is then baseband translated by a unique IQ demodulator. The decoding of each contribution is performed in the digital domain. It consists in using matched filters composed of digital filters followed by subsampling operations.

The proposed code multiplexing architecture is shown in Fig. 4. The RF input blocks (antenna, RF filter and LNA) are parallelized in order to have a good amplification and filtration of the received radiofrequency signals.

3) Characteristics

The proposed structure uses orthogonal codes to multiplex the different branches through a single IQ demodulator. The main goal was to reduce the complexity of the analog frontend.

In [START_REF] Gautier | Low complexity antenna diversity front-end: Use of code multiplexing[END_REF], we show that this system reduces the number of ADC by using only two ADC instead of the 2N used by the classic receiver. But, for the proposed architecture, the specifications of the ADC in terms of bandwidth are much more stringent. Meanwhile, a global consumption reduction of 20% could be reached for an N= 8 antennas system.

The points revealed in [START_REF] Gautier | New antenna diversity frontend using code multiplexing[END_REF] are the evaluation of the feasibility of such a structure. The implementation of analog coding and digital decoding has been validated by BER simulations and measurements (Fig. 5). Results show that, in a Gaussian case, the bit error rate does not significantly increase when the receiver integrates this type of RF coding architecture instead of the classical front-end stack-up.

Several studies have been done concerning the influence of IQ imbalance on the signal quality. Their conclusions reveal that the proposed architecture has the same sensitivity to IQ mismatches as the stack-up architecture. We note that the resulting IQ mismatches affect in the same manner the quality of each antenna baseband translated contributions. We mention that a classical dedicated front-end stack-up will have different IQ impairments for each dedicated IQ demodulator. These theoretical performance were validated in real environment simulations. In [START_REF] Gautier | IQ imbalance reduction in a SMI multi-antenna receiver by using a code multiplexing front-end[END_REF] we implement a SMI (Sample Matrix Inversion) multi-antenna digital method in order to compare the performance of the RF code multiplexing architecture with those of a classical front-end stack up for different IQ mismatches levels.

The last important point that has to be mentioned is the capability of the proposed architecture to adapt to any multiantenna schemes. In fact, we can find different application scenarios such as MIMO reception or beamforming techniques LTE-ADVANCED RECEIVER ARCHITECTURE

B. Context

As mentioned in section II, the future LTE-Advanced standard takes in consideration a discontinuous spectrum usage, as well as MIMO techniques for the downlink transmission. In order to answer to these specifications the actual state of the art of radiofrequency receivers imposes the use of dedicated front-end stack-up architectures. For example, if we consider a dual-band discontinuous spectrum scenario where two antennas are used for the reception of a MIMO transmission, the receiver will have to integer a four front-end stack-up. Each of these front-ends is dedicated to the processing of one of the contributions obtained from the combination of two antennas and two non-adjacent frequency bands. It becomes obvious that this method, involving the stack-up of dedicated front-ends, imposes high complexity, but especially high power consumption.

One of the most important issues when designing a radiofrequency receiver is the performance-power-complexity trade-off. In order to obtain low power consumption and low complexity LTE-Advanced receiver, an obvious solution is the mutualisation of elements of the dedicated front-end used by the stacked-up architecture. If we also take into consideration the simultaneous processing constrains, we can conclude that the mutualisation of the frequency translation step imposes a multiplexing of the different RF domain contributions.

Starting from this RF multiplexing idea, we present a novel single front-end architecture able to overcome the constraints imposed to the receiver by the LTE-Advanced standard.

C. Structure presentation

Starting from the dual band simultaneous reception and the RF code spreading architectures presented in Section III, we propose a unique front-end architecture dedicated to the LTE-Advanced receivers. In this paper we chose to treat the case of a two antenna reception an OFDM dual band signal, corresponding to a handheld integration scenario. Meanwhile, the structure can be easily generalized for several antennas by increasing the number of chips used by the orthogonal codes.

In the followings, we consider that the two input signals S and S' are the result of the passage of a bi-band signal through two different transmission channels. Let Band 1 -Band 2 and Band 1 '-Band 2 ' be the two pairs of contribution composing S and respectively S'. Once received by two space separated antennas, the two input signals are separately filtered and amplified by two RF filters and two LNA respectively, as shown in Fig 6.

The multiplexing of the four contributions is realized by a two step method. To begin, we use the orthogonal code spreading technique in order to multiplex the two input signal once filtered and amplified. Let cS, cS',cBand 1 -cBand 2 and cBand 1 '-cBand 2 ' be the signals resulting from the use of the coding technique.

The orthogonal codes used here have a length of two and a chip time two times smaller than the symbol time of each of the two signals. When multiplied with the orthogonal codes, each of the S 1 and S 2 signals will see each of the components of the pairs Band 1 -Band 2 and respectively Band 1 '-Band 2 ' spread in the same manner around their own central frequency, as shown in Fig. 6. This multiplexation step is concluded by the addition of the cS 1 and cS 2 signals. Therefore, by using the code spreading technique, we multiplexed two by two the contributions having the same central frequency but which have been received on different antennas. As a result of this operation we obtain a signal having a dual band spectrum. Each of its two frequency bands is composed of the addition between cBand 1 and cBand 1 ' and respectively cBand 2 and cBand 2 '.

The second block of the architecture assessed in this section is implementing the double IQ technique similar to that used Similar to the structure presented in Section III the two digital outputs are the baseband translation of the two frequency bands composing the signal at the output of the RF coding block. Therefore, each of them is the addition between the baseband translated signals cBand 1 , cBand 1 ' and respectively cBand 2 , cBand 2 '. In order to demultiplex each of these two pairs of signals, we apply two matched filters, similar to those used for the code multiplexing architecture.

Once we separately obtain the two pairs of signals corresponding to the two antennas reception of a dual band signal, two antenna SIMO processing method are used. Each of them has one of the pairs of decoded signals as input.

D. Characteristics

The single front-end receiver presented above is capable of answering to the LTE-Advanced requirements concerning the MIMO reception of a discontinuous spectrum signal.

In order to validate the theoretical study relative to this receiver we realized several simulations. The software used for these simulations was the ADS software provided by Agilent Technologies.

Knowing that the physical layer requirements concerning LTE-Advanced are not yet finalized, we choose to use a particular RF signal model for these simulations. In order to obtain an OFDM signal having a discontinuous spectrum we use a signal composed of the addition of two 802.11g nonoverlapping channels. For the transmission channel the model chosen is the AWGN (Additive White Gaussian Noise).

The results presented in Fig. 7 show the BER evolution for different E b /N 0 of the input signal. The receivers are using either the proposed unique front-end architecture or the dedicated front -end stack-up architecture. The results showed in Fig. 7 are mainly concerning the SIMO receptors using the two different architectures when receiving the dual-channel signal -Stack-up-SIMO, DoubleStructure-SIMO-Channel1 and DoubleStructure-SIMO-Channel2. We also show the BER evolution when the receivers are realizing a SISO reception-Stack-up-SISO and DoubleStructure-SISO. Both in the SIMO and SISO case, the reception performance when using the front-end stack-up architecture are slightly better compared to those obtained when using the unique front-end structure. This is due to the fact that the codes used by the Double Structure are not completely orthogonal. This matter has been already treated in Section III. We observe that the SIMO reception offers the same performance gain when using the two architectures. Meanwhile, if a real SIMO AWGN channel we have a 3 dB gain, the simulated channel manages to offer only a quasi AWGN conditions. In fact, the two noises added to the initial signal are not completely uncorrelated. This is why the SIMO gain obtained during ADS simulation doesn't reach the 3 dB level. In conclusion we consider that the performance of the proposed architecture are the same as those of the dedicated front-end stack-up.

Parallel to this performance study, a comparative power consumption study was made. For this study we considered that the single front-end architecture, as well as the dedicated front-end stack-up imposes the same constrains to the basic blocks (LNA, filters, amplifiers, mixers and oscillators) that are composing them. A bibliography research allowed us to choose the state of the art of these basic components in terms of performance-power consumption trade-off. The conclusions of this study show a 33 % power consumption gain in favor of the single front-end architecture( 211 mW instead of 315 mW). The main reason to this important gain is due to the use of less basic blocks. In order to better illustrate this idea, Fig. 8 shows the consumption of the different types of basic blocks used by the single front-end architecture and the front-end stack-up. If we also take into account the fact that the complexity of the proposed structure is significantly smallerfewer components and image frequency filter free-, it becomes obvious that the proposed structure offers an excellent performance-consumption-complexity trade-off.

CONCLUSION

In this article, a novel low complexity architecture was presented. We consider that this type of architecture is a good candidate to the integration in the receivers dedicated to the future LTE-Advanced standard. One of the major advantages of this architecture is its low complexity and therefore low power consumption.

Knowing that the power consumption of the analog part of the receiver is directly dependent on the used number of basic processing blocks, the excellent complexity-powerperformance trade-off becomes obvious. Indeed, this is confirmed by the fact that, despite the use of a demultiplex block in the digital domain, the proposed architecture offers a significant 33% of power gain compared with the solution proposed by the actual state of the art.

The proposed architecture joins two separate innovating methods: the RF orthogonal coding and the double IQ dual band simultaneous reception technique. Expected performance of its implementation has been presented for a particular study case -SIMO reception of an OFDM discontinuous spectrum signal. The simulation results show similar performance when comparing the proposed unique front-end structure and the dedicated front-end stack-up. This validates our theoretical study.

As mentioned in section two, one of the sensible points of this structure is the impact that the IQ mismatches can have on the signal quality. One possible solution to this inconvenient is the use of a digital MMSE method in order to mitigate the IQ mismatch influence on the received signal. This represents one of the forthcoming of our activities.

Another issue that still has to be addressed aims the design and realization of a demonstrator. Real environment measurements concerning the single front-end architecture will soon be realized using a radiofrequency platform with a real SIMO transmission channel.

Fig. 1

 1 Fig. 1 LTE-Advanced spectrum decomposition and one possible discontinuous spectrum allocation. • Availability of front-end for such a large bandwidth and bandwidths of variable range • Availability of Analog Digital Converter with such a high sampling rate and quantization resolution • Increased decoding complexity e.g. for channel decoding and increased soft buffer size In this paper, we deal with theses challenges and we propose a candidate architecture for the following scenario.

Fig. 2

 2 Fig. 2 Double IQ architecture dedicated to the simultaneous reception of two separate frequency band signals.

Fig. 3

 3 Fig. 3 802.11 BER evolution during a multiband simultaneous reception using the Double IQ architecture and the Stack-up architecture integrating constant orthogonal mismatches.

Fig. 7

 7 Fig. 7 BER evolution during the simultaneous SIMO reception of a signal composed of two non-overlapping 802.11g channels.

Fig. 8

 8 Fig. 8 Consumption of the different types of basic blocks used by the single front-end architecture and the dedicated front-end stack-up. by the dual band simultaneous reception structure. The processing realized by this block realizes a frequency translation of the RF orthogonal coding block output in the baseband domain.Similar to the structure presented in Section III the two digital outputs are the baseband translation of the two frequency bands composing the signal at the output of the RF coding block. Therefore, each of them is the addition between the baseband translated signals cBand 1 , cBand 1 ' and respectively cBand 2 , cBand 2 '. In order to demultiplex each of these two pairs of signals, we apply two matched filters, similar to those used for the code multiplexing architecture.Once we separately obtain the two pairs of signals corresponding to the two antennas reception of a dual band signal, two antenna SIMO processing method are used. Each of them has one of the pairs of decoded signals as input.

ACKNOWLEDGEMENT

The authors wish to sincerely thank Orange Labs which supports this work.


Mabel Cuesta
email: cuesta@lmpa.univ-littoral.fr

Liamidi Leadi
email: leadiare@imsp-uac.org

Joseph Liouville

Positive and sign-changing solutions for a quasilinear Steklov nonlinear boundary problem with critical growth

Keywords: Critical growth, indefinite weights, Steklov boundary conditions, p-laplacian operator. 35D05, 35J60, 35J65, 35J70, 35J25, 35J35

In this work we study the existence of positive solutions and nodal solutions for the following p-laplacian problem with Steklov boundary conditions on a bounded regular domain Ω ⊂ R N ,

with given numbers p, N satisfying 1 < p < N , p * := p(N -1) N -p the critical exponent for the Sobolev trace map W 1,p (Ω) → L q (∂Ω) and functions b 0 and a, V possibly indefinite. By minimization on subsets of the associated Nehari manifold, we prove the existence of positive solutions if N ≥ max{2p-1, 3} and the parameter λ close to the principal eigenvalues of the operator -∆ p + V with weighted-Steklov boundary conditions. We also prove the existence on nodal solutions for a definite and N > max{p 2 , 2p, p p-1 , 2}. Our results show striking differences between the cases p > 2, p = 2 and p < 2.

Introduction

Consider the following problem of parameter λ -∆ p u + V (x)|u| p-2 u = 0 in Ω; |∇u| p-2 ∂u ∂ν = λa(x)|u| p-2 u + b(x)|u| p * -2 u on ∂Ω;

(1.1) for 1 < p < N , a, b two given functions in C γ (∂Ω) for some γ > 0, a ≡ 0 with b ≥ 0, V ∈ L ∞ (Ω) and p * := p(N -1) N -p . The domain Ω is a bounded subset of R N of class C 2,α for some 0 < α < 1 and N ≥ 3. Our aim is to prove the existence of solutions for λ close to the principal eigenvalues of (1.5) (see below).

In the case a ≡ 0, V ≡ 1, b ≡ 1, the quasilinear problem (1.1) arises, for instance, when searching for functions u ∈ W 1,p (Ω) for which the norm of the Sobolev's trace immersion i p * ,Ω : W 1,p (Ω) → L p * (∂Ω) is achieved:

S 0 := i p * ,Ω -p = inf u∈W 1,p (Ω)\W 1,p 0 (Ω) Ω (|∇u| p + |u| p ) dx ∂Ω |u| p * dσ p/p * , (1.2)
where σ is the restriction to ∂Ω of the the (N -1)-Hausdorff measure, which coincides with the usual Lebesgue surface measure as ∂Ω is regular enough. Due to the lack of compactness of i p * ,Ω , the existence of minimizers for (1.2) does not follows by standards methods. Following the ideas of [2], [START_REF] Bonder | Estimates for the Sobolev trace constant with critical exponent and applications[END_REF]and [START_REF] Biezuner | Best constants in Sobolev trace inequalities[END_REF], Fernandez-Bonder and Rossi proved in [START_REF] Bonder | On the existence of extremals for the Sobolev trace embedding theorem with critical exponent[END_REF] that a sufficient condition for the existence of minimizers for (1.2) is that S 0 < K -1 N,p where

K -1 N,p def = inf R N + |∇u| p dx; |∇u| ∈ L p (R N + ) and R N -1 |u| p * dy = 1 . (1.3)
In the linear case, i.e. p = 2, with b ≡ 1 and V ≡ 0, namely, for the problem (Y )

   ∆u = 0 in Ω u > 0 in Ω ∂u ∂ν + N -2 2 βu = u 2 * -1
on ∂Ω, which is related to the Yamabé problem when β = cte = mean curvature of ∂Ω, Adimurthi-Yadava [START_REF] Adimurthi | Positive solution for Neumann problem with critical non linearity on boundary[END_REF] proved that problem (Y ) has solution when β ∈ C 1 (∂Ω), N ≥ 3 and there exists a point x 0 ∈ ∂Ω such that

β(x 0 ) < h(x 0 ) := 1 N -1 N -1 i=1 ν i , (1.4)
where the ν i are the principal curvatures at x 0 ∈ ∂Ω with respect to the unit outward normal. Finally, problem (1.1) in case p = 2 and V = 0 can also be related to well known λ-parameter problem of the Brézis-Nirenberg [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] with Dirichlet boundary condition

-∆u = λu + |u| 2 * -2 u in Ω, u = 0 on ∂Ω.
Among the huge amount of improvements and generalization of this pioneering work we quote the work of Cerami-Solimini-Struwe [START_REF] Cerami | Some existence results for superlinear elliptic boundary value problems involving critical exponents[END_REF] where they stated the existence of sign changing solutions of the Dirichlet problem for λ ∈ (0, λ 1 ) and N ≥ 6. We have adapted here their approach to our quasilinear problem with nonlinear boundary conditions. Quasilinear elliptic problems with an indefinite potential V have attracted a lot of attention the last decade. After the work concerning the eigenvalue problem with Dirichlet boundary condition with an indefinite weights in [START_REF] Cuesta | A weighted eigenvalue problem for the p-Laplacian plus a potential[END_REF] and the one for the eigenvalue problem with Steklov boundary conditions in [START_REF] Leadi | A weighted eigencurve for steklov problems with a potential[END_REF], some others quasilinear problems with weights have been considered with sublinear, superlinear or concave-convex nonlinear terms. In the present work we would like to explore the effect of sign-changing weights a and V on the multiplicity of solutions for a rather simple critical-exponent quasilinear problem with a parameter λ. From the variational point of view, the geometry of the related functional associated, for example, to the eigenvalue problem -∆ p u + V |u| p-2 u = 0 in Ω, |∇u| p-2 ∂u ∂ν = λa(x)|u| p-2 u on ∂Ω

(1.5) may take in consideration the disjoint subsets ∂Ω a|u| p dσ > 0 and ∂Ω a|u| p dσ < 0.

It is well known (see [START_REF] Leadi | A weighted eigencurve for steklov problems with a potential[END_REF]) that, if a changes sign, there are two principal eigenvalues λ -1 < λ 1 for the above eigenvalue problem. We will prove in this work that positive and sign-changing solutions of problem (1.1) can also be found by minimizing the energy functional on the subset of the Nehari manifolds where ∂Ω a|u| p dσ ≷ 0. By considering indefinite weights, we improve and complete several existing results for similar problems.

This paper is organised as follows. In section 1 we study under which conditions the infimum of the associated energy functional along the Nehari manifold is achieved. We prove in Proposition 2.7 that this is the case where this infimum is less than

1 p - 1 p * K -p * p * -p 0 where K 0 def = b p p * ∞,∂Ω K -1 N,p .
In order to assure this inequality we use the well known technique of mass concentration for the fundamental solutions, i.e. functions defined in R N + realizing the infimum in (1.3). In section 2 we analyse the different Lebesgue norms of these functions and in section 3 we state our main existence result in Theorem .4.2. In section 4 we study the infimum of the associated energy Φ λ along the so called nodal subsets of the Nehari manifold. Finally Theorem 6.2 state an existence result for positive weights a.

Minimization on the Nehari manifold

Let us define the following C 1 -functional on W 1,p (Ω) by

E V (u) def = Ω (|∇u| p + V (x)|u| p ) dx, A(u) def = ∂Ω a |u| p dσ, B(u) def = ∂Ω b|u| p * dσ, Φ λ (u) def = E V (u) -λA(u).
The natural norm of W 1,p (Ω) will be denoted by

• , i.e., ∀u ∈ W 1,p (Ω), u = Ω |∇u| p dx + Ω |u| p dx 1/p .
The Lebesgue norm of L q (Ω) will be denoted by • q and the Lebesgue norm of L q (∂Ω, ρ) by • q,∂Ω , for any q ∈ [q, +∞[. Solutions of problem (1.1) will be understood in the weak sense.

As in [START_REF] Cerami | Some existence results for superlinear elliptic boundary value problems involving critical exponents[END_REF] we will make use of the Nehari manifold associated to our problem. For this end, we define the energy functional

I λ (u) = 1 p Φ λ (u) - 1 p * B(u)
and the Nehari manifold associated to

I λ N = {u ∈ W 1,p (Ω)\{0}; I λ (u), u = 0} = {u ∈ W 1,p (Ω) \ {0}; Φ λ (u) = B(u)}
that we split into three sets

A + = {u ∈ N ; A(u) > 0}, A -= {u ∈ N ; A(u) < 0}, A 0 = {u ∈ N ; A(u) = 0}.
It is well known that critical points of I λ are solutions of problem (1.1) and belong to N . Notice that I λ restricted to N is equal to

I λ (u) = 1 p - 1 p * B(u) = 1 p - 1 p * Φ λ (u).
Minimizing the functional I λ along A ± provided us with positive solutions of our problem (1.1). Precisely, let us set

C ± λ = inf u∈A ± I λ (u) (2.1)
The following result is well known, we give the proof for the sake of completeness.

Lemma 2.1. If C ± λ is achieved and C ± λ > 0 then C ± λ is a critical value of I λ associated to a positive solution of (1.1).

Proof. Let

u ∈ A + such that C + λ = I λ (u) = inf v∈A + I λ (v)
. By taking |u| instead of u we can assume that the infimum is achieve at some u ≥ 0 in A + . Furthermore, if we set

J λ = Φ λ -B, we have that u ∈ N =⇒ J λ (u) = 0, u ≡ 0 and J λ (u), u = pΦ λ (u) -p * B(u) = (p -p * )B(u).
Observe that, since 0

< C + λ = I λ (u) = 1 p -1 p * B(u) then B(u) = 0 and therefore J λ (u), u = 0. By Lagrange's Multipliers theorem there exists α ∈ R such that I λ (u) = αJ λ (u). Hence 0 = J λ (u) = I λ (u), u = α(p -p * )B(u) =⇒ α = 0. Thus I λ (u) = 0.
The aim of this section is to prove that the previous infima are achieved and that they are strictly positive. The positivity of I λ depend on whenever λ < λ 1 or λ > λ -1 , where λ 1 and λ -1 are defined as follows. Let us recall the following results on the eigenvalue problem (1.5) associated to our problem (see [START_REF] Leadi | A weighted eigencurve for steklov problems with a potential[END_REF]). By a principal eigenvalue we mean an eigenvalue having a positive eigenfunction.

Proposition 2.2 ( [START_REF] Leadi | A weighted eigencurve for steklov problems with a potential[END_REF]). Let

α a def = inf{E V (u); u p = 1, A(u) = 0}.
(2.2)

Then problem (1.5) possesses a principal eigenvalue if and only if α a > 0.

Precisely, 1. if α a > 0 and a changes sign then (1.5) admits exactly two principal eigenvalues λ -1 < λ 1 , with

λ 1 := min M + E V , (2.3)
where

M + := {u ∈ W 1,p (Ω); A(u) = 1} and λ -1 = -min M -E V , (2.4)
where M -:= {u ∈ W 1,p (Ω); A(u) = -1};

2. if α a > 0 and a is of definite sign then (1.5) admits exactly one principal eigenvalue, which are either λ 1 or λ -1 ;

3. if α a = 0 then (1.5) has a unique principal eigenvalue λ * given by

λ * = inf M + = -inf M -E V .
Moreover a function u ∈ S is an eigenfunction associated to λ * if and only if A(u) = 0 and E V (u) = α a = 0.

Remark 2.3. Actually, the hypothesis of Theorem 3.3 of [START_REF] Leadi | A weighted eigencurve for steklov problems with a potential[END_REF] are that both λ D V =first eigenvalue of u → -∆ p u + V |u| p-2 u with Dirichlet boundary condition and β(V, a) = inf{E V (u); A(u) = 0, u p,∂Ω = 1} are > 0. These two hypothesis are equivalent to α a > 0.

As a straightforward consequence of the above proposition we have Corollary 2.4. Assume α a > 0. For any λ < λ 1 (resp. for any λ > λ -1 ) there exists c > 0 such that, for all u ∈ W 1,p (Ω) satisfying A(u) ≥ 0 (resp.

A(u) ≤ 0) it holds E V (u) -λA(u) ≥ c u p . (2.5)
Remark 2.5. Weak solutions of problem (1.1) and (1.5) belong to L ∞ (Ω) ∩ L ∞ (∂Ω) according to [START_REF] Cuesta | Weighted eigenvalue problems for quasilinear elliptic operators with mixed RobinDirichlet boundary conditions[END_REF]. Consequently weak solutions are of class C 1,µ (Ω) for some 0 < µ < 1 (see [START_REF] Lieberman | Boundary regularity for solutions of degenerate elliptic equations[END_REF]).

Throughout the paper we will always assume α a > 0.

Let us now study the geometry of the fibering maps and the Nehari manifold.

Lemma 2.6.

1. Assume either λ < λ 1 or λ > λ -1 . Then for any u ∈ W 1,p (Ω) such that B(u) = 0, the function t → I λ (tu) has a local maximum at

0 < t u := Φ λ (u) B(u) 1 p * -p , (2.6)
t u u ∈ N and

I λ (t u u) = 1 p - 1 p * Φ λ (u) B(u) p/p * p * p * -p . 2. If λ < λ 1 then there exists a constant c > 0 such that ∀u ∈ A + ∪ A 0 =⇒ u ≥ c and B(u) ≥ c.
(2.7)

3. If λ > λ -1 then there exists a constant c > 0 such that ∀u ∈ A -∪ A 0 =⇒ u ≥ c and B(u) ≥ c .
4. All minimizing sequences for C ± λ are bounded.

5. C ± λ > 0.

Proof. (1) For any u ∈ W 1,p (Ω) such that B(u) > 0 one easily proved that

g u (t) = t p-1 Φ λ (u) -t p * -1 B(u)
for t > 0, vanished at t u and that the function t → I λ (tu) has a global maximum at t u . Clearly, g u (t) = 0 ⇔ tu ∈ N .

(2) We know from equation (2.5) that there exists a constant c 1 > 0 such that Φ λ (u) ≥ c 1 u p . Moreover using Sobolev's embedding from the trace we have, for some constant c 2 > 0,

B(u) ≤ c 2 b ∞ u p *
and the conclusion follows using that Φ λ (u) = B(u) because u ∈ N . One can prove (3) in a similar way. (4) Assume by contradiction that a minimizing u n ∈ A + is unbounded and take v n = un un . Thus, for a subsequence, there exists v 0 ∈ W 1,p (Ω) such that v n v 0 , strongly in L p (Ω) and L p (∂Ω). Since

1 p - 1 p * Φ λ (v n ) = I λ (u n ) u n p → 0 as n → +∞, then E V (v 0 ) -λA(v 0 ) ≤ 0. (2.8) If v 0 ≡ 0 then 0 = E V (v 0 ) -λA(v 0 ) = lim n→∞ Φ λ (v n ) =⇒ v n → 0 strongly in W 1,p (Ω),
what is in a contradiction with the fact that v n = 1. Thus v 0 ≡ 0. Also we have A(v 0 ) > 0 because the possibility A(v 0 ) = 0 is ruled out from the condition α a > 0 and (2.8). If λ < λ 1 we then have a contradiction between (2.5) and (2.8) (5) If for instance C + λ = 0 and (u n ) n is a bounded minimizing sequence converging to some u 0 weakly in W 1,p (Ω), strongly in L p (Ω) and also strongly in L p (∂Ω) hence A(u 0 ) ≥ 0 and

1 p - 1 p * Φ λ (u 0 ) ≤ lim n→∞ I λ (u n ) = C + λ = 0. (2.9)
If u 0 ≡ 0 then we will get from the last inequality that u n → 0 strongly in W 1,p (Ω), in contradiction with (2.7). Thus, u 0 ≡ 0 but now (2.9) contradicts (2.5).

In the next proposition we will prove that the values C ± λ are achieved whenever they are smaller than a certain value involving K N,p if λ is close to λ 1 . This second constraint follows from the necessity to assure that the infimum is achieved at some point lying in the open set

A + . Precisely, let us consider γ a,b def = inf{E V (u); A(u) = 0, B(u) = 1}. (2.10) Proposition 2.7. One has 1. 0 < γ a,b and
C ± λ ≤ 1 p - 1 p * γ p * p * -p a,b .
(2.11)

2. There exists δ 1 > 0 (resp. δ 2 > 0) such that

C + λ < 1 p - 1 p * γ p * p * -p a,b ∀λ ∈ (λ 1 -δ 1 , λ 1 ), ( resp. C - λ < 1 p - 1 p * γ p * p * -p a,b ∀λ ∈ (λ -1 , λ -1 + δ 2 )).
Proof. (1) It follows directly from α a > 0 that γ a,b ≥ 0. Assume by contradiction that γ a,b = 0 and let (u n ) n be a minimizing sequence for γ a,b . Assume furthermore that (u n ) n is an unbounded sequence and take v n = un un . Thus there exists a subsequence, still denoted v n , and a function v 0 such that v n v 0 , strongly in L p (Ω), in L p (∂Ω) and a.e. We have in one hand

Φ λ (v 0 ) ≤ lim inf n→+∞ Φ λ (v n ) ≤ 0, (2.12)
and in other hand A(v 0 ) = 0. Besides v 0 ≡ 0 otherwise we will deduce from (2.12) that v n → 0 strongly in W 1,p (Ω), which is in contradiction with the fact that v n = 1. Thus

α a v 0 p p ≤ E V (v 0 ) = Φ λ (v 0 ) ≤ 0,
which contradicts the hypothesis α a > 0. We conclude that the sequence (u n ) n is bounded. Hence, up to a subsequence, it converges weakly to some u 0 in W 1,p (Ω), strongly in L p (Ω) and in L p (∂Ω). Hence E V (u 0 ) ≤ 0 and A(u 0 ) = 0. If u 0 ≡ 0 we have a contradiction with the hypothesis α a > 0. If u 0 ≡ 0 hence u n converges strongly to 0, in contradiction with B(u n ) = 1.

Next, to prove for instance that

C + λ ≤ 1 p -1 p * γ p * p * -p a,b , let u n be a se- quence in W 1,p (Ω) such that A(u n ) = 0, B(u n ) = 1 and E V (u n ) → γ a,b .
We can assume also that u n ≥ 0 by taking |u n | instead of u n if necessary and, using the same argument as above, the sequence (u

n ) n is bounded in W 1,p (Ω). Let ψ ∈ C 1 (Ω) be any positive function such that supp ψ ∩ ∂Ω ⊂ {x ∈ ∂Ω; a(x) > 0}. Let us take v n = u n + ψ n . Clearly v n -u n → 0. Moreover A(v n ) = ∂Ω av p n = ∂Ω a + (u n + ψ n ) p - ∂Ω a -u p n > A(u n ) = 0, and clearly B(v n ) ≥ B(u n ) = 1.
Furthermore, using the following inequality

||x + y| q -|x| q -|y| q | ≤ C|xy| |x| q-2 + |y| q-2 , (2.13)
valid for any q ≥ 1 and any x, y ∈ R N and using also that the sequence u n is bounded we have

Φ λ (v n ) = E V (u n ) + o(1) = γ a,b + o(1).
Finally, if we consider z n := t vn v n ∈ A + , with t un defined in Proposition (2.6), it comes

C + λ ≤ 1 p - 1 p * Φ λ (v n ) B(v n ) p/p * p * p * -p → 1 p - 1 p * γ p * p * -p a,b .
(2) We only prove the estimate for C + λ . By taking t ϕ 1 ϕ 1 , where ϕ 1 is the unique positive eigenfunction associated to λ 1 such that A(ϕ 1 ) = 1 and t ϕ 1 has been defined in (2.6), by definition of C + λ one has

1 p - 1 p * -1 C + λ p * -p p * λ 1 -λ (B(ϕ 1 )) p/p * Thus, if λ ∈ R is such that λ 1 -λ (B(ϕ 1 )) p/p * < γ a,b , i.e., λ > λ 1 -(B(ϕ 1 )) p/p * γ a,b then 1 p -1 p * -1 C + λ p * -p p * < γ a,b as stated in (2.15).
Consequently, let us define

λ ± * def = inf λ ∈ R; C ± λ < 1 p - 1 p * γ p * p * -p a,b (2.14)
As a consequence of (2) in Proposition 2.7 we have λ + * < λ 1 and λ - * > λ -1 and therefore

λ ± * = sup λ ∈ R; C ± λ = 1 p - 1 p * γ p * p * -p a,b . Proposition 2.8. Let λ ∈ R. 1. If λ < λ 1 and C + λ < 1 p - 1 p * K -p * p * -p 0 (2.15) then there exists u ∈ A + ∪ A 0 such that I λ (u) = C + λ . Similarly, if λ > λ -1 and C - λ < 1 p - 1 p * K -p * p * -p 0 (2.16)
then there exists u ∈ A -∪ A 0 such that I λ (u) = C - λ .

2. If furthermore λ + * < λ < λ 1 and (2.15) holds then problem (1.1) with parameter λ possesses a positive solution u satisfying A(u) > 0 and

I λ u) = C + λ . Similarly if λ -1 < λ < λ -
* and (2.16) holds then problem (1.1) with parameter λ possesses a positive solution u satisfying A(u) < 0 and

I λ u) = C - λ .
Proof. We will only give the proof concerning C + λ since the argument is similar for C - λ . Let (u n ) n be a minimizing sequence. By (4) of Lemma 2.6 the sequence (u n ) n is bounded so assume that u n ∈ A + converges weakly to some u 0 , strongly in L p (Ω) and in L p (∂Ω). Clearly A(u 0 ) ≥ 0. Claim: We have

Φ λ (u 0 ) ≤ ( 1 p - 1 p * ) -1 C + λ p * -p p * B(u 0 ) p p * .
(2.17) Indeed, in one hand, using that (u n ) n is a minimizing sequence we have

1 p - 1 p * -1 C + λ = B(u n ) + o(1). (2.18)
Besides, we also have by the Brézis-Lieb lemma ( [START_REF] Brézis | A relation between pointwise convergence of functions and convergence of functionals[END_REF])

1 p - 1 p * -1 C + λ = Φ λ (u 0 ) + ∇(u n -u 0 ) p p + o(1) (2.19)
In other hand, let us choose > 0 such that

1 p - 1 p * -1 C + λ < (K 0 + ) -p * p * -p .
Using again Brézis-Lieb lemma and the fact that p/p * < 1, we get

B(u n ) p/p * ≤ B(u 0 ) p p * + B(u n -u) p p * + o(1),
and hence it comes from (2.18) and Lemma 2.10 (see below) gives

( 1 p - 1 p * ) -1 C + λ p/p * ≤ B(u 0 ) p p * + (K 0 + ) ∇(u n -u 0 ) p p + o(1
C + λ < (K 0 + ) -p p * -p , we obtain ( 1 p - 1 p * ) -1 C + λ p/p * ≤ B(u 0 ) p p * + 1 p - 1 p * -1 C + λ p p * -Φ λ (u 0 ) 1 p - 1 p * -1 C + λ p-p * p *
and the proof of the claim follows. Notice that u 0 ≡ 0 since, otherwise, u n → 0 strongly in W 1,p (Ω) which contradicts (3) of Lemma 2.6. As a consequence of (2.17) and that Φ λ (u 0 ) > 0 we have B(u 0 ) > 0. Finally let us prove that C + λ is achieved at t u 0 u 0 ∈ N . Indeed, again by (1) of Lemma 2.6 we have

1 p - 1 p * -1 C + λ ≤ Φ λ (u 0 ) p * p * -p B(u 0 ) p p * -p while by the claim Φ λ (u 0 ) p * p * -p B(u 0 ) p p * -p ≤ 1 p - 1 p * -1 C + λ
and the equality follows.

(

) Since λ > λ + * we get C + λ < 1 p -1 p * γ p * p * -p a,b 2
and therefore C + λ is achieved at some u ∈ A + . By replacing u by |u| if necessary, we can assume that u ≥ 0. The result then comes from Lemma 2.1. By the reularity results (see Remark 2.9) andth strong maximum principle of [START_REF] Vazquez | A strong maximum principle for some quasilinear elliptic equations[END_REF], the solution u is strictly positive up to the boundary. Remark 2.9. Notice the if λ + * < λ - * then, under the hypothesis of Proposition 2.8, we will obtain two positive solutions of problem (1.1) for any parameter λ ∈ (λ + * , λ - * ) ∩ (λ -1 , λ 1 ): one in A + and the other in A -. However, that λ + * < λ - * is not clear for general weights a, b and V . We have used in the previous proposition the following Cherrier's-type inequality that has been proved by [START_REF] Biezuner | Best constants in Sobolev trace inequalities[END_REF] in the case b ≡ 1 and can be trivially generalizes for any positive bounded weight b: Lemma 2.10. [START_REF] Biezuner | Best constants in Sobolev trace inequalities[END_REF] For any > 0 there exists C > 0 such that for all u ∈ W 1,p (Ω) it holds

∂Ω b|u| p * dσ p/p * ≤ (K 0 + ) Ω |∇u| p dx + C Ω |u| p dx where K 0 := K N,p b p/p * ∞ (2.21)
and K N p is defined in (1.3).

Estimates of the L p -norms of fundamental solutions

We turn now our attention to the problem of finding the values λ for which we have S ± λ < K -1 0 , where we denote here, for simplicity,

S ± λ = 1 p - 1 p * -1 C + λ p * -p p * . (3.1)
It is well known (see [START_REF] Nazaret | Best constants in Sobolev trace inequalities on the halfspace[END_REF]), that the value K -1 N,p defined in (1.3) is achieved at functions of the form

U ,y 0 (y, t) = -N -p p U y -y 0 , t , with y 0 ∈ R N -1 arbitrary and ∈]0 + ∞[, where
U (y, t) = 1 ((t + 1) 2 + |y| 2 ) N -p 2(p-1)
.

The functions U ,y 0 are usually called fundamental solutions. The constant K -1 N,p can be computed explicitly (see [START_REF] Bonder | Estimates for the Sobolev trace constant with critical exponent and applications[END_REF][START_REF] Nazaret | Best constants in Sobolev trace inequalities on the halfspace[END_REF]) and it is equal to

K -1 N,p = N -p p -1 p-1 π p-1 2   Γ N -1 2(p-1)
Γ p(N -1)

2(p-1)   p-1 N -1
.

Let us assume for convenience that

x 0 = 0 ∈ ∂Ω and |Ω + a ∩ B s (0)| > 0 ∀0 < s < r (3.2)
for some r > 0, where Ω + a = {x ∈ ∂Ω; a(x) > 0}. Let φ be a smooth radial function with compact support in the ball B r/2 (0) satisfying φ ≡ 1 in B r/4 (0). For any > 0 let us choose the following test functions:

u (y, t) = U (y, t)φ(y, t) = N -p p(p-1) φ(y, t) ((t + ) 2 + |y| 2 ) N -p 2(p-1)
.

(3.3)

Notice that A(u ) > 0. In order to give the asymptotic development with respect to the parameter of the quotient Φ(u ) B(u ) p/p * , we will compute each of integrals involved. Much of the work have been done by [START_REF] Bonder | Estimates for the Sobolev trace constant with critical exponent and applications[END_REF] and we refer the reader to this paper for full details. To make the computations simpler, we will choose a special parametrization of the boundary ∂Ω around 0 ∈ ∂Ω.

Since we are assuming that ∂Ω is of class C 2 , there exists c > 0 and a C 2function ρ :

{y ∈ R N -1 , |y| ≤ c} → R such that Ω ∩ B r (0) = {(y, t) ∈ Q c ; t > ρ(y)} ∂Ω ∩ B r (0) = {(y, t) ∈ Q c , ; t = ρ(y)}, (3.4)
where

Q c := {(y, t), |y| ≤ c, 0 ≤ t ≤ c} and ρ(y) = 1 2 N -1 i=1 ν i y 2 i + O(|y| 3 ) (3.5) for some ν i , i = 1, • • • , N -1.
We set hereafter

h 0 = 1 N -1 N -1 i=1 ν i . (3.6)
The value h 0 is known as the mean curvature of ∂Ω at 0 with respect to the outward normal ν.

Ω |∇u | p dx = A 1 + f 1 ( ),
where

f 1 ( ) := A 2 + O( N -p p-1 ) if p < N +1 2 -h 0 2 ω N -2 ln(1/ ) + O( ) if p = N +1 2
and

A 1 = 1 2 N -p p -1 p-1 β N -1 2 , N -1 2(p -1) ω N -2 ; A 2 = - h 0 4 N -p p -1 p β N + 1 2 , N -2p + 1 2(p -1) ω N -2 .
2.

Ω V (x)|u | p dx = f 2 ( ), f 2 ( ) :=    O( p ) if p 2 < N O( p ln( 1 )) if p 2 = N O( N -p p-1 ) if p 2 > N 3. Assume that b(0) = b ∞ and b(y, ρ(y)) -b(0) = O(|y| γ+1 ) ∀ |y| ≤ c
for some γ > 0.

Then ∂Ω b|u | p * dσ = B 1 + B 2 + o( )
where

B 1 = 1 2 b ∞ β N -1 2 , N -1 2(p -1) ω N -2 , B 2 = - 1 2 b ∞ (N -1)h 0 β N -1 2 , N -1 2(p -1) ω N -2 .
4. Assume that a ∈ C γ close to 0 for some γ > 0.

Then ∂Ω a(x)|u | p dσ = f 3 ( )
where

f 3 ( ) =            C 1 p-1 + o( p-1 ) if N > p 2 -p + 1 a(0)ω N -2 p-1 ln(1/ ) + O( p-1 ) if N = p 2 -p + 1, O( N -p p-1 ) if N < p 2 -p + 1, (3.7
) and

C 1 = 1 2 a(0)β N -1 2 , N -p 2 + p -1 2(p -1) ω N -2 .
We recall that

ω N -1 = measure of the unit sphere S N -1 of R N = 2π N 2 Γ N 2 and β(x, y) := ∞ 0 t x-1 dt (1 + t) x+y = 1 0 t x-1 (1 -t) y-1 dt = Γ(x)Γ(y) Γ(x + y)
for x, y > 0.

Proof. (1) -( 2) These estimates can be found in [START_REF] Bonder | Estimates for the Sobolev trace constant with critical exponent and applications[END_REF].

( From basic integration, we deduce for any a > -1 and b > 0 that

c/ 0 t a (1 + t 2 ) b dt =            1 2 β a+1 2 , 2b-a-1 2 + O( 2b-a-1 ) if 2b -a -1 > 0, ln(1/ ) + O(1) if 2b -a -1 = 0, O( 2b-a-1 ) if 2b -a -1 < 0. (3.8) Thus for any a, b, ∈ R + we have |y|≤c |y| a ( 2 + |y| 2 ) b dy =            ω N -2 1 2 N -1+a-2b β a+N -1 2 , 2b-a-N +1 2 + O(1) if 2b -a -N + 1 > 0, ω N -2 ln(1/ ) + O(1) if 2b -a -N + 1 = 0, O(1) if 2b -a -N + 1 < 0. ( 3.9)
By expanding (( + ρ(y)) 2 and using Taylor's theorem we find -N -p

N -1 p-1 ∂Ω∩Br(0) |y| γ+1 |u | p * dσ = |y|≤c |y| γ+1 1 + |∇ρ(y)| 2 dy (( + ρ(y)) 2 + |y| 2 ) p(N -1) 2(p-1) = |y|≤c |y| γ+1 dy ( 2 + |y| 2 ) p(N -1) 2(p-1) +O |y|≤c |y| γ+3 dy ( 2 + |y| 2 ) p(N -1) 2(p-1) =    O( γ+N -p p-1 (N -1) ) if N > p(γ + 1) -γ O(ln(1/ )) if N = p(γ + 1) -γ O(1) if N < p(γ + 1) -γ. Since N -1 p-1 + N + γ -p p-1 (N -1) = γ + 1 then
p-1 I = |y| c 1 + 1 2 |∇ρ(y)| 2 + O(|y| 4 ) ( 2 + ρ(y) 2 + 2 ρ(y) + |y| 2 ) p(N -p) 2(p-1) dy = |y| c dy ( 2 + |y| 2 ) p(N -p) 2(p-1) -p(N -p) p-1 |y| c
ρ(y)dy

( 2 + |y| 2 ) p(N -p) 2(p-1) +1 + O |y| c |y| 2 ( 2 + |y| 2 ) p(N -p) 2(p-1) = I 1 -p(N -p)
p-1

I 2 + O(I 3 )
where

I 1 = ω N -2 ×              1 2 p-N -1 p-1 β N -1 2 , N -p 2 +p-1 2(p-1) + O(1) if N > p 2 -p + 1 ln(1/ ) + O(1) if N = p 2 -p + 1 C 2 if N < p 2 -p + 1 and C 2 = 1 p 2 -p+1-N |c| p 2 -p+1-N ω N -2 > 0. Clearly I 2 = O(I 1 ) and I 3 = 2 O(I 1 ). Consequently I =                    1 2 ω N -2 p-1 β N -1 2 , N -p 2 +p-1 2(p-1) + O( N -p p-1 ) if N > p 2 -p + 1, O( N -p p-1 ln(1/ )) if N = p 2 -p + 1; O( N -p p-1 ) if N < p 2 -p + 1.
Similar computations for II give

II = N -p p-1 |y|≤c |y| γ dy ( 2 + |y| 2 ) p(N -p) 2(p-1) =              O( p-1+γ ) if N > p 2 -(p -1)(1 -γ); O N -p p-1 ln(1/ ) if N = p 2 -(p -1)(1 -γ); O( N -p p-1 ) if N < p 2 -(p -1)(1 -γ).
Since by hypothesis γ > 0 then II = I + o(I) and we conclude.

In section 5 we will need the -asymptotic of several L q -norm of the fundamental solution u defined in (3.3). Proposition 3.2.

1. ∇u 1 = f 4 ( ), where

f 4 ( ) =        O N -p p(p-1) if p > 2N -1 N , O N -p p(p-1) ln( 1 ) if p = 2N -1 N , O( N -N p ) if p ≤ 2N -1 N . 2. ∇u p-1 p-1 = O( N p -1 ).
3. u 1 = f 5 ( ), where

f 5 ( ) =        O N -p p(p-1) if p > 2N N +1 , O N -p p(p-1) ln( 1 ) if p = 2N N +1 , O( N +1-N p ) if p < 2N N +1 . 4. u p-1 p-1 = O( N p -1 ).
5. u 1,∂Ω = f 4 ( ). Proof. Let us denote by α = N -p p(p-1) . ( 1) We have

|∇u | = N -p p -1 α (t + ) 2 + |y| 2 -N -1 2(p-1) in B r/4 (0). so -α Ω |∇u | dx = N -p p -1 Ω∩B r/4 (0) (t + ) 2 + |y| 2 -γ dydt + O(1)
,

where γ = N -1 2(p-1)
. Notice that the integral on the right hand side goes to a constant as goes to 0 if 2γ -N < 0, that is, if p > 2N -1 N . In the case 1 < p ≤ 2N -1 N let us compute this integral as follows. We write Ω∩B r/4 (0)

(t + ) 2 + |y| 2 -γ dydt = Qc dtdy [(t + ) 2 + |y| 2 ] γ - Qc\Ω dtdy [(t + ) 2 + |y| 2 ] γ = I 1 -I 2 + O(1)
After firstly changing the variables t and y by t and .y respectively and secondly changing y by (t + 1).z, one gets

I 1 = C N -2γ +∞ 0 (1 + t) N -1-2γ dt +∞ 0 r N -2 (1 + r 2 ) γ dt + O(1) = O N -2γ since N -2γ = N p-2N +1 p-1 < 0 in the case 1 < p < 2N -1 N . When p = 2N -1 N
we have

I 1 = C ln 1 + O(1) = -C ln ( ) + O(1)
.

For I 2 we have

I 2 = |y| c dy ρ(y) 0 dt ((t + ) 2 + |y| 2 ) γ = |y|≤c ρ(y)dy ( 2 + |y| 2 ) γ + O |y| c |y| 4 dy ( 2 + |y| 2 ) γ+1 = O |y| c |y| 2 dy ( 2 + |y| 2 ) γ + O |y| c |y| 4 dy ( 2 + |y| 2 ) γ+1 = o N -2γ . Thus Ω |∇u | dx = α    O(1) if p > 2N -1 N , C ln( 1 ) + O(1) if p = 2N -1 N , O( N -2γ ) if p < 2N -1
N , and the conclusion follows.

(2) We have

-α(p-1) Ω |∇u | p-1 dx = N -p p -1 p-1 B r/4 (0) (t + ) 2 + |y| 2 -γ(p-1) dydt + O(1).
Since in this case 2γ(p -1) -N = -1 < 0 the integral on the right handside converges to a constant as → 0 and the result follows.

(3) Set now γ 1 = N -p 2(p-1) . We have, by letting → 0 in the integral below,

-α Ω u (y, t)dtdy = Ω∩B r/4 (0) dydt [(t + ) 2 + |y| 2 ] γ 1 = O(1) if N -2γ 1 > 0. Notice that N -2γ 1 > 0 ⇐⇒ p > 2N N +1 . If p ≤ 2N N +1 we write -α Ω u (y, t)dtdy = I 1 -I 2 + O(1)
with

I 1 = Qc dydt [(t + ) 2 + |y| 2 ] γ 1 and I 2 = Qc\Ω dydt [(t + ) 2 + |y| 2 ] γ 1 .
Following the computations of (1) (with γ 1 instead of γ) we will have

I 1 = C ln(1/ ) + O(1) if p = 2N N +1 , O( N -2γ 1 ) if p < 2N N +1 ,
and besides

I 2 = O( N -2γ 1 ).
Thus

Ω |u | dx = α    O(1) if p > 2N N +1 , C ln( 1 ) + O(1) if p = 2N N +1 , O( N -2γ 1 ) if p < 2N N +1,
and the conclusion follows.

(4) We have

-α(p-1) Ω |u | p-1 dx = Ω∩B r/4 (0) (t + ) 2 + |y| 2 -γ 1 (p-1) dydt + O(1).
In this case 2γ 1 (p -1) -N = -p < 0 the the integral on the right converges to a constant as → 0 and the result follows.

(5) In this case we have

-α ∂Ω |u | dσ = |y|≤c 1 + |∇ρ(y)| 2 dy (( + ρ(y)) 2 + |y| 2 ) γ 1 + O(1) = I 1 + O(1)
and

I 1 = |y|≤c dy (( + ρ(y)) 2 + |y| 2 ) γ 1 + O |y|≤c |y| 2 dy (( + ρ(y)) 2 + |y| 2 ) γ 1 = I 2 + I 3 .
By expanding ( + ρ(y)) 2 and using Taylor's theorem we have

I 2 = |y|≤c dy ( 2 + |y| 2 ) γ 1 -2γ 1 |y|≤c ρ(y) dy ( 2 + |y| 2 ) γ 1 +1 +O |y|≤c |y| 4 dy ( 2 + |y| 2 ) γ 1 +1 = I 1 2 + I 2 2 If 2γ 1 -N + 1 < 0, i.e., if p > 2N -1 N
then I 1 2 converges to a constant as goes to 0 and I 1 2 is O(ln( 1)) if p = 2N -1 N . In the case p < 2N -1

N

we have

I 1 2 = N -1-2γ 1 c/ 0 r N -2 (1 + r 2 ) γ 1 dt = O( N -1-2γ 1 ).
Finally one can easily check that the remaining terms can be neglected when compared with I 1 2 and the result follows. ( 6) By expanding ( + ρ(y)) 2 and using Taylor's theorem as above we have

-α(p-1) u p-1 p-1,∂Ω = |y|≤c 1 + |∇ρ(y)| 2 dy (( + ρ(y)) 2 + |y| 2 ) γ 1 (p-1 + O(1) = |y|≤c dy ( 2 + |y| 2 ) N -p 2 -2(N -p) |y|≤c ρ(y) dy ( 2 + |y| 2 ) N -p 2 +O |y|≤c |y| 2 dy ( 2 + |y| 2 ) N -p 2
and, since (N -p) -N + 1 = -p + 1 < 0, the first integral on the right converges as goes to 0 and we get the result. [START_REF] Bonder | On the existence of extremals for the Sobolev trace embedding theorem with critical exponent[END_REF] As previously and, using that 2γ

1 (p * -1) = N , -α(p * -1) ∂Ω |u | p * -1 dσ = |y|≤c 1 + |∇ρ(y)| 2 dy (( + ρ(y)) 2 + |y| 2 ) γ 1 (p * -1) + O(1) = |y|≤c dy (( + ρ(y)) 2 + |y| 2 ) N/2 +O |y|≤c |y| 2 dy (( + ρ(y)) 2 + |y| 2 ) N/2 .
By expanding ( + ρ(y)) 2 and using Taylor's theorem the first integral is now

I 1 = -1 c/ 0 r N -2 (1 + r 2 ) N/2 dt = O( -1 )
and all the other integral are negligible when compared with I 1 . Finally, since α(p * -1) = N p the result follows.

Existence of positive solutions

We can give now sufficient conditions on V, a, b et λ to fulfil the condition S ± λ < K -1 0 . By taking -a instead of a, one can prove similar results in order to have the inequality S - λ < K -1 0 . We will assume here the following hypothesis (B): there exists a point x 0 ∈ ∂Ω such that (B)

                   b ∞ is achieved at x 0 , b(x 0 ) -b(x) = O(|x -x 0 | γ+1 ) for some γ > 0, a ∈ C γ close to x 0 for some γ > 0, a(x 0 ) > 0. (4.1) Proposition 4.1. Let N ≥ 2p -1.
Assume that there exists a point x 0 ∈ ∂Ω satisfying hypothesis (B) in (4.1). Then

C + λ < 1 p - 1 p K -p * p * -p 0
holds in the following cases:

1. for any λ ∈ R if p > 2 and the mean curvature h 0 at x 0 is positive;

2. for λ, a(x 0 ) and h 0 satisfying

N -2 2 h 0 + λa(x 0 ) > 0 (4.2) if p = 2; 3. for any λ > 0 if 1 < p < 2.
Proof. By definition of S ± λ in (3.1) we have

S + λ ≤ Φ λ (u ) B(u ) p/p * .
Notice that A(u ) > 0 if and r are small enough as a consequence of the estimate (4) in Proposition 3.1. We are going to prove that there exists a positive constant Λ such that

K 0 Φ λ (u ) B(u ) p/p * = 1 -Λg( ) (4.3) with g( ) def =    if p ≥ 2, N > 2p -1; ln( 1 ) if p ≥ 2, N = 2p -1; p-1 if 1 < p < 2.
Φ λ (u ) B(u ) p/p * = A 1 + A 2 + o( ) (B 1 + B 2 + o( )) p/p * = A 1 B p/p * 1 1 + A 2 A 1 - p p * B 2 B 1 + o( ).
We have

A 2 A 1 = - 1 2 N -p N -2p + 1 (N -1)h 0 , B 2 B 1 = - 1 2 (N -1)h 0 ,
and hence we define Λ as

Λ := - A 2 A 1 - p p * B 2 B 1 = (N -p)(p -1) N -2p + 1 h 0 > 0. Notice that A 1 B p/p * 1 = K -1 0 . Second case: If p > 2 and N = 2p -1 then Φ λ (u ) B(u ) p/p * = A 1 -h 0 ω N -2 2 ln(1/ ) + O( ) (B 1 + O( )) p/p * = A 1 B p/p * 1 1 - h 0 ω N -2 2A 1 ln(1/ ) + O( )
Here we define

Λ := h 0 ω N -2 2A 1 > 0.
Third case: If 1 < p < 2 then

K 0 Φ λ (u ) B(u ) p/p * = A 1 -λC 1 p-1 + o( p-1 ) (B 1 + o( p-1 )) p/p * = A 1 B p/p * 1 -λ C 1 B p/p * 1 p-1 + o( p-1 )
Notice that if 1 < p < 2 one has ln(1/ ) = o( p-1 ). Here

Λ := λ C 1 A 1 > 0
in the case λ > 0 and a(x 0 ) > 0.

Fourth case:

If p = 2 and N > 2p -1 = 3 then Φ λ (u ) B(u ) p/p * = A 1 + (A 2 -λC 1 ) + o( ) (B 1 + B 2 + o( )) p/p * = A 1 B p/p * 1 1 + A 2 -λC 1 A 1 - p p * B 2 B 1 + o( ).
Now we have (in this case)

-Λ := A 2 -λC 1 A 1 - p p * B 2 B 1 = - N -2 N -3 h 0 -λ 2 N -3 a(x 0 )
and therefore, if (4.2) holds, Λ > 0.

Fifth case:

If p = 2 and N = 2p -1 = 3 then Φ λ (u ) B(u ) p/p * = A 1 + -h 0 ω 1 2 -λa(x 0 )ω 1 ln(1 ) + O( ) (B 1 + O( )) p/p * = = A 1 B p/p * 1 1 + -h 0 2 -λa(x 0 ) A 1 ω 1 ln(1/ ) + O( ).
Here we have

Λ := h 0 2 + λa(x 0 ) A 1 ω 1 .
Thus, if (4.2) holds, Λ > 0.

As a direct consequence of Proposition 4.1 and Proposition 2.8 we can now formulate the main result of this section. with λ + * defined in (2.14). Then problem (1.1) possesses a positive solution u satisfying A(u) > 0 in the following cases:

1. if p > 2 and the mean curvature at x 0 satisfies h 0 > 0, 2. if 1 < p < 2, λ > 0, 3. if p = 2 and a(x 0 ), h 0 satisfy (4.2).

We also have the analogous result when considering -a instead of a: Theorem 4.3. Let N ≥ 2p -1 and assume that there exists a point x 0 ∈ ∂Ω such that hypothesis (B) of (4.1) are satisfied for b and -a.

Let λ ∈ R satisfying λ -1 < λ < λ - * (4.6)
with λ - * defined in (2.14) Then the problem (1.1) possesses a positive solution u satisfying A(u) < 0 in the following cases:

1. if p > 2 and the mean curvature at x 0 satisfies h 0 > 0, 2. if 1 < p < 2, λ < 0 and a(x 0 ) < 0, 3. if p = 2 and a(x 0 ), h 0 satisfy (4.2). Remark 4.4. [START_REF] Adimurthi | Positive solution for Neumann problem with critical non linearity on boundary[END_REF] for the Yamabe problem (Y ).

Notice that if

p = 2, V ≡ 0,a = β then λ 1 = 0, λ + * = -∞. Condition (4.2) for λ = 1 is condition (1.4) of
2. In order to have λ > 0 in the case 1 < p < 2, one can required that λ 1 > 0 (resp. λ -1 < 0). Thus, it will be enough to ask for instance that inf{E V (u); u p = 1} > 0, a condition weaker than α a > 0.

Minimisation along nodal subsets of the Nehari manifold

In order to find nodal solutions we introduce the following nodal subsets of the Nehari manifold:

N + = {u ∈ W 1,p (Ω); u ± ∈ A + }, N -= {u ∈ W 1,p (Ω); u ± ∈ A -} (5.1)
and let us define

D ± λ = inf u∈N ± I λ (u).
(5.2)

Clearly 2C ± λ ≤ D ± λ for any λ ∈ R since I λ (u) = I λ (u + ) + I λ (u -) ≥ C + λ + C + λ .
In what follows we are going to show, under certain conditions on λ, p, a and b, similar to those of the first section, that both D ± λ are achieved, providing us with a pair of nodal solutions of problem (1.1). Our intention is to prove now that the Palais-Smale condition is satisfied. For any u ∈ N + we denote Proof. We only prove the first case. Let (u n ) n be a sequence in N + satisfying (5.3). Using that I λ (u n ) = ( 1p -1 p * )Φ λ (u n ), we can prove as in ((4) of Lemma 2.6 that the sequence is bounded. Let u be such that, up to a subsequence, u n u, strongly in L p (Ω), in L p (∂Ω) and a.e. Then we also have u ± n u ± , strongly in L p (Ω) and in L p (∂Ω). Let us assume by contradiction that, for and, since we are assuming that ∇u + n -∇u + p 0 , then ∇u + n -∇u + p ≥ (K 0 + ) -p * p(p * -p) + o(1).

T N + (u) = {v ∈ W 1,p (Ω); Φ λ (u ± ), v = B (u ± ), v }
(5.5)

Besides, using again Brézis-Lieb Lemma and (5.5) we have

I λ (u + n ) = I λ (u + n -u + ) + I λ (u + ) + o(1) = ( 1 p - 1 p * ) ∇u + n -∇u + p p + I λ (u + ) + o(1) ≥ ( 1 p - 1 p * )(K 0 + ) -p * p * -p + o(1), since I λ (u + ) ≥ 0. Also one has I λ (u - n ) ≥ C + λ . Finally we have the estimate I λ (u n ) = I λ (u + n ) + I λ (u - n ) ≥ ( 1 p - 1 p * )(K 0 + ) -p * p * -p + I λ (u - n ) + o(1) ≥ ( 1 p - 1 p * )(K 0 + ) -p * p * -p + C + λ + o(1).
Since by hypothesis (PS1) we have

I λ (u n ) → c < C + λ + ( 1 p -1 p * )K -p * p * -p 0
, we get a contradiction by choosing > 0 small enough.

As in the previous section, we need to assure that the infima D ± λ are not achieved for any u ∈ N ± satisfying either A(u + ) = 0 or A(u -) = 0. Let us introduce the values

η + λ def = inf{I λ (u); u ∈ N + , A(u + ) = 0, A(u -) ≥ 0}; η - λ def = inf{I λ (u); u ∈ N -, A(u + ) = 0, A(u -) ≤ 0}.
(5.6)

Clearly D ± λ ≤ η ± λ . Now, we prove that the infima D ± λ are achieved provided they are sufficiently smaller. Proposition 5.2. Let λ < λ 1 and assume that

D + λ < min η + λ , C + λ + 1 p - 1 p * K -p * p * -p 0 .
Then there exists u ∈ N + solution of problem (1.1) satisfying

I λ (u) = D + λ . Similarly, if λ > λ -1 and D - λ < min η - λ , C - λ + 1 p - 1 p * K -p * p * -p 0 then exists v ∈ N -solution of the problem (1.1) satisfying I λ (v) = D - λ .
Proof. First we are going to prove that we can find a minimizing sequence for the infimun D + λ that satisfies the hypothesis (P S1) and (P S2) of the previous proposition. The idea is to apply Ekeland's variational principle to the complete metric space X = N + inherited with the distance of W 1,p (Ω). Notice that

N + = {u ∈ W 1,p (Ω); u ± ∈ N , A(u ± ) ≥ 0} ∩ { u ± ≥ c and B(u ± ) ≥ c}
according to the estimates (2.7). For any > 0 let u ∈ N + such that I λ (u ) ≤ D + λ + 2 . We can assume that > 0 such that 0 < 2 < η + λ -D + λ . By Ekeland's variational principle (see [START_REF] Ekeland | On the variational principle[END_REF]) there exists v ∈ X such that

(E1) I λ (v ) < I λ (u ), (E2) dist (v , u ) < , (E3) I λ (v ) ≤ I λ (w) + v -w ∀w ∈ X, w = v .
Using the fact that D + λ < η + λ , we can assume that A(v ± ) > 0 otherwise we will have

D + λ + 2 ≥ I λ (v ) ≥ η + λ ,
which is a contradiction. For any w ∈ W 1,p (Ω) consider w t = v + tw for t small enough to have B(w ± t ) > 0 and A(w ± t ) > 0. Put

s 1 (t) = Φ λ (w + t ) B(w + t ) 1 p * -p ; s 2 (t) = Φ λ (w - t ) B(w - t ) 1 p * -p so s 1 (t)w + t -s 2 (t)w - t ∈ N + . Hence, using (E3), I λ (v ) -I λ (s 1 (t)w + t -s 2 (t)w - t ) t t v -s 1 (t)w + t -s 2 (t)w - t ≤ . (5.7) If we write h(t) = I λ (s 1 (t)w + t -s 2 (t)w - t ) then h(0) = I λ (v )

and by elementary computations

h (0) = I λ (v ), s 1 (0)v + -s 2 (0)v -+ w = I λ (v ), w , lim t→0 v -s 1 (t)(v + tw) + + s 2 (t)(v + tw) - t = -v + s 1 (0) + v -s 2 (0) -w , s 1 (0) = 1 p * -p Φ λ (v + ), w -B (v + ), w B(v + ) 1 p * -p -1 , s 2 (0) = 1 p * -p Φ λ (v -), w -B (v -), w B(v -) 1 p * -p -1 and therefore s 1 (0) = s 2 (0) = 0 if w ∈ T + N (v ). Letting t → 0 in (5.7) we get I λ (v ), w ≤ w ∀w ∈ T N + (v ).
Choosing = 1/n we have that v n = v 1/n provides a minimizing sequence in N + that satisfies both (P S1) and (P S2) of the previous proposition. Then there exists a converging subsequence and we will conclude from (E1) that D + λ is achieved at some u ∈ N + . Since the possibility that A(u ± ) = 0 is excluded from the hypothesis D + λ < η + λ the conclusion comes finally from Lemma 2.1. Proof. Let u 0 > 0 be a critical point of I λ with critical value C + λ . By hypothesis we assume that a > 0 on ∂Ω ∩ B r/4 (x 0 ) for some r > 0 satisfying furthermore ∂Ω\B r/4 (x 0 ) a|u 0 | p > 0.

Let u be defined as in (3.3) and define the map σ : [0, 1] 2 → W 1,p (Ω) by σ(s, t) = Kt(su 0 -(1 -s)u ) for some K > 0 to be fixed later.

First we claim that

D + λ ≤ max (s,t)∈[0,1] 2 I λ (σ(s, t)). (6.2)
To see that, consider the map : R 2 → R 2 defined as (s, t) = f λ (σ(s, t) + ) -f λ (σ(s, t) -), f λ (σ(s, t) + ) + f λ (σ(s, t) -) -2 , where

f λ (u) = 0 if u = 0, B(u) Φ λ (u) if u = 0.
Notice that the estimate (2.5) implies that f λ is a continuous map. Moreover we have f λ (σ(0, t) + ) -f λ (σ(0, t) -) ≤ 0 f λ (σ(1, t) + ) -f λ (σ(1, t) -) ≥ 0 f λ (σ(s, 0) + ) + f λ (σ(s, 0) -) -2 ≤ 0 and we choose K > 0 big enough to have f λ (σ(s, 1) + ) + f λ (σ(s, 1) -) -2 ≥ 0.

We can apply Miranda's theorem [START_REF] Miranda | Un'osservazione sur un teorema di Brouwer[END_REF] to get the existence of some (s, t) ∈ [0, 1] 2 such that (s, t) = (0, 0), i.e., f λ (σ(s, t) + ) = f λ (σ(s, t) -) = 1. (6.3)

Thus u = σ(s, t) is such that u ± ∈ N . It remains to proof that A(u ± ) > 0 to conclude (6.2). We have As a corollary of Proposition 5.2 and Proposition 6.1 we have the following existence result. In order to assure that the condition D ± λ < η ± λ is satisfied, we only consider now weights a with definite sign.

Proposition 3 . 1 .

 31 Let N ≥ 2p -1. Assume for convenience the hypothesis (3.2) and let u be as in (3.3). Then 1.

  ) To estimate ∂Ω b(x)|u | p * dσ we write ∂Ω b(x)|u | p * dσ = b(0)

∂Ω∩Br( 0 )

 0 |y| γ+1 |u | p * dσ = o( ) and the result follows. (4) First of all we use the fact that a ∈ C γ (∂Ω) and write a(y, ρ(y)) = a(0) + O(|y| γ ); |y| ≤ c so ∂Ω a(x)|u | p dσ = a(0) |u | p dσ and II := ∂Ω∩Br(0) |y| γ |u | p dσ.

6 .

 6 u p-1 p-1,∂Ω = O( N p -1 ).

(4. 4 )

 4 First case: If p > 2 and N > 2p -1 then the integrals (c) and (d) in Proposition 3.1 are o( ) and therefore

Theorem 4 . 2 .

 42 Let N ≥ 2p -1. Assume that there exists a point x 0 ∈ ∂Ω such that hypothesis (B) of (4.1) are satisfied. Let λ ∈ R satisfying λ + * < λ < λ 1 (4.5)

Proposition 5 . 1 ..

 51 the tangent subspace to N + at u. If L ∈ W -1,p (Ω), by L T N + (u) we mean the norm of the restriction of L to the subspace T N + (u) . Let λ < λ 1 and c ∈ R satisfy c < C + λ Then I λ satisfies the Palais-Smale condition at level c on N + , i.e., any sequence (u n ) ∈ A + satisfying(P S1) I λ (u n ) → c, (P S2) I λ (u n ) T N + (un) = o(1)(5.3)possesses a convergent subsequence.Similarly, if λ > λ -1 and c < C - satisfies the Palais-Smale condition at level c on N -.

Remark 5 . 3 . 6 .

 536 The condition D + λ < η + λ is needed here to avoid the minimizing sequences to converge to some u satisfying A(u + ) = 0 or A(u -) = 0 . Notice that we required the similar condition (i.e.C - λ < 1 p -1 p * γ p * p * -p a,b ) inorder to prove that C + λ is achieved and we have given in Proposition 2.7 a condition on λ to assure that C + λ < 1 p -1 p * γ p * p * -p a,b . We speculate that also D + λ < η + λ for λ close to λ 1 , but we have been unable to prove it. Notice that if a > 0 (or a < 0) then γ a,b = η ± λ = +∞. Existence of nodal solutions Proposition 6.1. Assume hypothesis (B) in (4.1), condition (4.6) and the additional constraint N > max{p 2 , 2p, in cases (1),(2) and (3) of Proposition 4.1.

A 4 ) 1 L 1 +K 2 1 L 1 Lp * - 1 L

 4112111 (u + ) = K p t p ∂Ω a|(su 0 -(1 -s)u ) + | p = = K p t p ∂Ω\B r/4 (x 0 ) a|su 0 | p + ∂Ω∩B r/4 (x 0 ) a + |u + | p > 0;and alsoA(u -) = K p t p ∂Ω∩B r/4 (x 0 ) a + |u -| p > 0,otherwise u ≥ 0 on ∂Ω and therefore B(u -) = 0, in contradiction with (6.3).(2) Next we prove thatmax (s,t)∈[0,1] 2 I λ (σ(s, t)) < C + λWe write for simplicity the functions u ∈ σ([0, 1] 2 ) as u = αu 0 + βu , with |α|, |β| ≤ K. Then using the inequality (2.13) we have for some positive constantsK 1 , K 2 , K 3 I λ (αu 0 + βu ) -I λ (|α|u 0 ) -I λ (|β|v ) ≤ +K 1 ∇αu 0 p-∞ (Br(0)) ∇βv 1 + ∇αu 0 L ∞ (Br(0)) ∇βv p-1 pαu 0 L ∞ (Br(0)) βv p-1 p-1 + αu 0 p-∞ (Br(0)) βv 1 +K 3 αu 0 L ∞ (∂ΩBr(0)) βv p-1 p-1,∂Ω + αu 0 p-∞ (∂ΩBr(0)) βv 1,∂Ω +K 4 αu 0 L ∞ (∂Ω∩Br(0)) βv p * -1 p * -1,∂Ω + αu 0 ∞ (∂Ω∩Br(0)) βv 1,∂Ω . (6.5) Using (1) of Lemma 2.6 and (4.3) in the proof of Proposition 4.1 we haveI λ (|α|u 0 ) ≤ I λ (u 0 ) = C + λ ;) defined in (4.4). Besides, the remaining terms in (6.5) are o( ) if (6.1) is satisfied. Indeed, notice that in the estimate of the norm ∇u 1 , all the powers of are > 1 if either p ≥ 2 and N > p 2 or 1 < p < 2 and N > max{2p, p p-1 }. The other terms ( ∇u p-1 p-1 , u 1 , • • • ) the power of is > 1 if p < N 2 .

Acknowledgments. This work was partilly carried out while the first author was visiting the IMSP of the Université d'Abomey Calavi (Porto-Novo) and also while the second author was visiting the Université du Littoral Côte d'Opale (ULCO). We would like to express our gratitude to those institutions.

instance, u + n → u + strongly in W 1,p (Ω). Let us denote, for each v ∈ W 1,p (Ω) and each u ∈ N , the real number

Using estimate (2.7) we have that the sequence B(u ± n ) is bounded away from 0 and therefore

and, because we also have that u + n u + , it comes also that

Using Brézis-Lieb identity we deduce

.

Using the fact that Φ λ (u + n ) = B(u + n ) and Brézis-Lieb Lemma we get

and letting n → ∞ we see that Φ λ (u + ) = B(u + ). In particular I λ (u + ) ≥ 0. Let now > 0. We have, by using Lemma 2.