Xin-Xin Qian

Yi-Cong Zhao

Claire-Lise Santini

Hong Miao Pan

Tian Xiao

Hai Tao Chen

Tao Song

Jin-Hua Hua Li

François Alberto

Sophie Brustlein

Qian Xin-Xin

Santini Claire-Lise

Xiao Tian

Chen Hai-Tao

Alberto Francois

Brustlein Sophie

W U Long-Fei

Qian

Al

Long - Fei

How light affect the magnetotactic behavior and reproduction of ellipsoidal multicellular magnetoglobules?

Keywords: Photo-response, magnetic alignment, coordinated swimming

Magnetotactic bacteria (MTB) synthesize intracellular magnetic organelles, magnetosomes, which consist of magnetic crystals that are enveloped in a membrane. Magnetosomes are organized into a chain(s) and confer on cells a magnetic dipolar moment. This magnetic property allows MTB cells to align and swim along geomagnetic field lines, a movement referred to as magnetotaxis. Some MTB species change their swim direction in response to illumination by UV, violet and blue light. Here we analyzed the polarity of morphology, magnetism and motion in Mediterranean multicellular magnetotactic prokaryotes, also called, magnetoglobules or MMP. The magnetoglobules were assembled from 60-80 cells into an asymmetric ellipsoidal morphology with a relative narrow and large end. They swam dominantly northward, parallel to the direction of the magnetic field, with the narrow-end as the leading side. In response to a reversal in the direction of the magnetic field they aligned quickly along the magnetic field lines and kept swimming northward. Interestingly, under constant illumination, 385 nm UV light, magnetoglobules changed their swimming direction southward anti-parallel to the direction of the magnetic field, with the large-end as the leading side. The change from a northward to southward direction, occurred along with an increase of swimming speed. A minimum of 35 mW/cm2 irradiance of UV light was sufficient to trigger the swimming re-orientation. UV radiation also triggered the unidirectional division of magnetoglobules. Together these results revealed a coordination of the polarity of magnetoglobule morphology, magnetic moment and swimming orientation, in response to magnetic and optical stimuli. The UV triggered the reversal of magnetotaxis and magnetoglobule division indicating the ecological significance of light for multicellular magnetotactic prokaryotes.

INTRODUCTION

Magnetotactic bacteria (MTB) are a group of heterogeneous Gram-negative bacteria that can produce single domain magnetite (Fe 3 O 4 ) or greigite (Fe 3 S 4 ) crystals in intracellular organelles called magnetosomes [START_REF] Bazylinski | Magnetosome formation in prokaryotes[END_REF]. The magnetosomes give bacteria the capacity of aligning to and swimming along geomagnetic field lines, this phenomenon is referred to as magnetotaxis. Anerobic or micro-aerobic MTB dwell at the oxic-anoxic interface (OAI) in chemically stratified sediments or water columns in freshwater and marine environments.

Interestingly, MTB in the Northern Hemisphere preferentially swim northward and parallel to the geomagnetic field lines (north-seeking, NS) [START_REF] Blakemore | Magnetotactic bacteria[END_REF]. The MTB in the Southern Hemisphere swim in the direction of the magnetic south pole anti-parallel to the geomagnetic field lines (south-seeking, SS) [START_REF] Blakemore R P | South-seeking magnetotactic bacteria in the Southern Hemisphere[END_REF]. The current geomagnetic field is inclined downward in the Northern Hemisphere and upward in the Southern Hemisphere, and its inclination increases from the equator to the poles. Therefore, it is believed that the magnetotactic effect guides the cells in each hemisphere down to the lower oxygen content of the aquatic habitat, where they may stop swimming until the conditions change [START_REF] Bazylinski | Magnetosome formation in prokaryotes[END_REF]. This behaviour has been referred to as magneto-aerotaxis. In previous studies we provided direct evidence to show that magnetotaxis is beneficial to magneto-cocci growth and even essential at low cell density (Zhang et al., 2014b).

MTBs display great morphologic diversity that includes coccoid to ovoid cells, rods, vibrios, and spirilla [START_REF] Bazylinski | The Prokaryotes: Prokaryotic Physiology and Biochemistry[END_REF]. Besides the most abundant unicellular MTB, a group of multicellular magnetotactic prokaryotes (MMPs in abbreviation, also called Magnetoglobules) in particular have attracted our attention because they display complex swimming behaviours such as photophobic response, bounce motion, axial magnetotaxis and magnetic photokinesis [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. Two morphotypes of magnetoglobules have been reported. The spherical mulberry-like magnetoglobules (sMMPs) are typically 3-12 μm in diameter and comprise of 10-40 cells arranged with helical symmetry (Keim et al., 2004b;[START_REF] Zhou | Adaptation of spherical multicellular magnetotactic prokaryotes to the geochemically variable habitat of an intertidal zone[END_REF]Zhang et al., 2014a). The other morphotype of ellipsoidal pineapple-like magnetoglobules (eMMPs) have been found from the China Sea and the Mediterranean Sea. They are typically comprise of 28-101 cells arranged in interlaced circles with 8-20 μm in length and 5-15 μm in width [START_REF] Zhou | A novel genus of multicellular magnetotactic prokaryotes from the Yellow Sea[END_REF][START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF][START_REF] Chen | Novel species and expanded distribution of ellipsoidal multicellular magnetotactic prokaryotes[END_REF][START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. Both morphotypes of magnetoglobules have peritrichous flagella (Keim et al., 2004a;Zhang, et al., 2014a;[START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF][START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF] and are capable of producing magnetite and / or greigite magnetosomes [START_REF] Zhou | Occurrence and microscopic analyses of multicellular magnetotactic prokaryotes from coastal sediments in the Yellow Sea[END_REF][START_REF] Zhou | A novel genus of multicellular magnetotactic prokaryotes from the Yellow Sea[END_REF][START_REF] Zhou | Adaptation of spherical multicellular magnetotactic prokaryotes to the geochemically variable habitat of an intertidal zone[END_REF]Zhang, et al., 2014a;[START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF][START_REF] Abreu | Culture-independent characterization of a novel magnetotactic member affiliated to the Beta class of the Proteobacteria phylum from an acidic lagoon[END_REF]. The ellipsoidal, oval shaped magnetoglobules are easier to recognise (from the side determination) than spherical magnetoglobules when the studying the relationship between magnetic polarity and swimming orientation as the narrow end indicates the north side of magnetoglobules.

Beside the aero-magnetotaxis observed for all MTB, the multicellular magnetoglobules exhibit two kinds of peculiar swimming behaviour. The first is escape motility or ping-pong motion, which was recently described as bounce motion [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. Magnetoglobules swim northward until they are blocked at the north edge of the droplets, where some magnetoglobules then swim southward with an increase of swimming speed. At variable distances, magnetoglobules change swimming direction northward and return to the north edge [START_REF] Rodgers | Intercellular structure in a many-celled magnetotactic prokaryote[END_REF][START_REF] Simmons | Spatiotemporal Distribution of Marine Magnetotactic Bacteria in a Seasonally Stratified Coastal Salt Pond[END_REF][START_REF] Zhou | A novel genus of multicellular magnetotactic prokaryotes from the Yellow Sea[END_REF][START_REF] Zhou | Adaptation of spherical multicellular magnetotactic prokaryotes to the geochemically variable habitat of an intertidal zone[END_REF]Zhang, et al., 2014a;[START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF][START_REF] Chen | Novel species and expanded distribution of ellipsoidal multicellular magnetotactic prokaryotes[END_REF][START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. When blocked by obstacles magnetoglobules display a similar bounce motion [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. In the second case, magnetoglobules accumulating at droplet edges swam in the opposite direction of magnetotaxis, towards the inner center of the droplet when they were illuminated by blue, violet, or UV light. This is named as photophobic response or is sometimes miscalled negative phototaxis [START_REF] Sobrinho | Geochemical characteristics related to the gregite-producing multicellular magnetotactic prokaryote Candidatus Magnetoglobus multicellularis in a hypersaline lagoon[END_REF][START_REF] Zhou | A novel genus of multicellular magnetotactic prokaryotes from the Yellow Sea[END_REF][START_REF] Zhou | Adaptation of spherical multicellular magnetotactic prokaryotes to the geochemically variable habitat of an intertidal zone[END_REF]Zhang, et al., 2014a;[START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF][START_REF] Chen | Novel species and expanded distribution of ellipsoidal multicellular magnetotactic prokaryotes[END_REF]. Recently we observed magnetotactic photokinesis of ellipsoidal magnetoglobules. The north-seeking eMMPs under constant illumination by UV light might suddenly change their swimming direction southward [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. The relationship between magnetic polarity and swimming direction in response to light stimuli is poorly understood. Fluorescence confocal microscopy and ultra-thin section images revealed peripherycore unilateral constriction of constituent cells and unidirectional binary fission of the ellipsoidal magnetoglobules [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. An "open space" was also found at one end of the ellipsoidal magnetoglobules. Is there a polarity that governs the division direction of the magnetoglobules? Here we carried out a detailed analysis of ellipsoidal magnetoglobule polarity and assessed the influence of light on swimming and division direction within the magnetic fields. We also discussed the potential ecological significance of photosensitive magnetotaxis.

Sample collection

Sediment samples of eMMPs were collected from the Mediterranean Sea, Southern France,Six-Four les Plages (43°07'N, 5°79'E). Freshly collected sediments were transferred into 1000 ml plastic bottles, with approximately 500 ml of in situ seawater. Bottles with sediments were transported to the laboratory and incubated at room temperature. MTBs were further enriched by magnetic collection [START_REF] Qian | Identification of novel species of marine magnetotactic bacteria affiliated with Nitrospirae phylum[END_REF][START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF].

Characterisation of swimming behaviour

Magnetotaxis was analyzed in droplets or microchannels with the Magnetodrome system (MAG Instruments UG, Germany) as previously described [START_REF] Zhang | Measurement of Free-Swimming Motility and Magnetotactic Behavior of Magnetococcus massalia Strain MO-1[END_REF][START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. In brief, the Magnetodrome consists of a Zeiss Axio Observer A1 microscope with a set of four electromagnets generating homogenous magnetic fields that guide the magnetoglobule swimming direction (Mag-Instruments, Germany). To characterize photosensitive magnetotactic behaviour, we modified the Zeiss Axio-Observer A1 microscope by replacing the HBO 100 light source with a Zeiss Colibri 7 (Zeiss, SAS, Germany) or CHROLIS-C2 6-Wavelength High-Power LED Source (Thorlabs, SAS, France). There are two light paths. The transmission halogen light was used for imaging and observation, whereas the epiillumination monochromatic LED light for triggering photophobic response or characterizing photokinesis.

The swimming of magnetoglobules was maintained within the UV spot in meddle of the microchannels by application of an alternate uniform magnetic field (10 Gs, 0.2 -0.4 Hz) in order to periodically reverse their swimming direction before they left the illumination area. To measure the light irradiance, a photodiode power sensor S120VC or S142c (Thorlabs SAS, France) was connected to a Powermeter PM100D (Thorlabs SAS, France). At the slide level with a 10 x objective (Zeiss A-Plan 10x/0.25) the irradiances were measured as 110 mW/cm 2 (for 385 nm UV LED), 100 mW/cm 2 (for 430 nm violet LED), 80 mW/cm 2 (for 475 nm blue LED), 40 mW/cm 2 (for 511 nm cyan LED), 20 mW/cm 2 (for 550 nm green LED), 12 mW/cm 2 (for 590 nm yellow LED) and 45 mW/cm 2 (for 630 nm red LED). The motility was recorded with Point_Grey_Grasshopper camera at 30 fps and analysis was conducted by MTrackJ Plugins of ImageJ choosing snap feature of bright centroid and snap range of 51 x 51 pixels [START_REF] Lins | Imaging faces of shadowed magnetite (Fe3O4) crystals from magnetotactic bacteria with energy-filtering transmission electron microscopy[END_REF].

Microscopy analyses

For laser confocal analysis, MMPs were firstly fixed with 1% paraformaldehyde for 1.5 hours at room temperature, and then stained membranes with 7.5 µg/ml FM4-64, and stained chromosomal DNA with 2µg/ml DAPI. The stained cells were observed with the Olympus FV1000 microscope with a laser set at 405 nm excitation and 425-475 nm emission for DAPI and 543 nm excitation and 555-655 nm emission for FM4-64. Images were taken from a series of 0.17µm intervals focal levels.

For scanning electron microscopy (SEM) observation, the MTB cells were fixed in 2.5% glutaraldehyde for 4h at 4℃, ethanol dehydrated, critical point-dried, coated with gold, and then observed using a Zeiss Ultra-55 field-emission gun SEM (Carl Zeiss, Germany) operating at 5 kV. General SEM morphology analyses of magnetoglobules were carried out using a KVKV-2800B microscope as described by Chen et al. [START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF].

RESULTS AND DISCUSSION

Basic swimming behaviour of ellipsoidal magnetoglobules

The ellipsoidal magnetoglobules isolated from the Mediterranean Sea sediments possess multiples chains of bullet-shaped magnetosomes that organize along the long body axis (Fig. 1A). The resulting magnetic dipolar moment confers on the magnetoglobules a north-seeking magnetotaxis with a velocity of 207.3 ± 60.4 µm/s (n=10). While translocating along the long body axis, the magnetoglobules also rotated around this axis with a frequency of 3.6 turn/s (Fig. 1B, Supplementary Information movie S1). Therefore, most flagellum-propelled bacteria magnetoglobules displayed a helical trajectory of swimming.

Polarities of ellipsoidal magnetoglobules

Morphology polarity

Fluorescence confocal microscopy inspection revealed that Mediterranean ellipsoidal magnetoglobules are assembled from 60-80 cells into a one-layer, hollow architecture (Fig. 2A and [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]). They had an oval morphology; one end was wider than the other. Phase contrast images consistently showed the oval morphology (Fig. 2B). Due to the relatively big size of magnetoglobules, 6 to 10 µm, the wide end can be distinguished from the narrow end under optical microscope especially when using magnification objectives 40x or higher. A ratio of the wide curvature radius over the narrow curvature radius was about 1.27 for the image in Fig. 2B.

Magnetic polarity and swimming orientation

Because of the multiple magnetosome chains magnetoglobules have an intrinsic magnetic polarity, which allows them to align along the magnetic field lines. To characterize the photosensitive magnetotaxis of magnetoglobules, two light-sources were used, i.e., a transmission halogen light for observation and imaging, and an epi-illumination (from bottom to top) with Colibri 7 or CHROLIS-C2 LED sources on a defined area for analyzing the photo-effect on swimming behaviour. The swimming of the magnetoglobules was maintained within the UV illumination zone by applying a uniform magnetic field (10 Gs) where the direction was alternately switched (0.4 Hz). When distinguishable the narrow end of the oval magnetoglobules often corresponded to the body north pole of the entity and a static magnetoglobule realigned, in response of the reversal of the field, within 4-5 frames at 30 fps (Fig. 2,C, panels 2 to 7, Supplementary Information movie S2). Under UV illumination magnetoglobules maintained for a while, the north-seeking motility and could change their swimming direction to south-seeking after a variable time of UV illumination (1 to 13 sec). When UV illumination was combined with the reversal of the magnetic field direction, magnetoglobules re-aligned their swimming direction to the novel north (Supplementary Information movie S3). Interestingly, the north-seeking magnetoglobules (Fig. 2,D, panels 1, 11 and 12, red spots) could change, within approximate 2 frames, their swimming direction southwards (Fig. 2,D, 13 and 14, green spots). The time required for the reversal of the swimming direction was less than half of that for magnetic realignment (Fig. 2, D, panels 23 to 29, magenta spots, Supplementary Information movie S4).

In addition, the southward swimming displayed an increase of speed and remained south-seeking for quite a long time (Fig. 2, D, panels 16 to 46; Supplementary Information movies 3 and 4). The change from northward to southward swimming might be the result of a reversal in the magnetosome magnetic polarity and subsequent realignment of magnetoglobules, or a change of swimming direction without magnetic realignment. The relatively big size and ellipsoidal morphology of the magnetoglobules enabled us to examine both possibilities. In fact, the magnetoglobules stayed alignment in the field but swarm southward (Fig. 2, D, panels 13 to 15, and Supplementary Information movie 4). Therefore, the UV irradiation might trigger the reversal of swimming from north-seeking to south-seeking through coordinated control of flagellar rotation.

We analyzed the wavelength and power required for changing the swimming orientation from northseeking to south-seeking. Minimal 35 mW/cm 2 UV (385 nm) irradiance was required for triggering the reversion of swimming orientation. From an average of 40 analyses each for more than 50 collected samples the proportion of magnetoglobules reacting to UV illumination ranged from 60 to 95% depending on sampling seasons and places, storage times of samples in laboratory and observation duration under the microscope. Notably magnetoglobules were micro-aerobic, sensitive to oxygen and lost progressively the motility during the observation. The violet light (430 nm) at maximal irradiance of 100 mW/cm 2 could lead to the reversal of less than 5% magnetoglobules whereas illumination by blue light (475 nm, 80 mW/cm 2 ), cyan (511 nm, 40 mW/cm 2 ), green (550 nm, 20 mW/cm 2 ), yellow (590 nm, 12 mW/cm 2 LED) and red (630 nm, 45 mW/cm 2 ) had no effect on the magnetotactic swimming of the magnetoglobules, perhaps because of insufficient photon energy and light intensity.

UV light triggered magnetoglobule division

As previously reported the ellipsoidal magnetoglobules divided unidirectionally from one side across the whole body to the other side during reproduction [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. Fluorescence confocal microscopy showed the division initiated at a position close to the narrow end of ellipsoidal magnetoglobules (Fig. 3,A). This position corresponded to an indentation site found at the narrow end on the SEM images of magnetoglobules (Fig. 3,B), which could suggest that the division may start at the narrow north end of the ellipsoidal magnetoglobules. Unidirectional division led to two offspring with similar ellipsoidal morphology (Fig. 3,C).

Following collection, sea sediment samples were stored in plastic sampling bottles, in the laboratory in dim light, at room temperature. During the first 3 days the number of magnetoglobules decreased, numbers then began to increase, reaching their maximal in about 7 days. Maximum size and division of magnetoglobules was observed in the 7-14-day samples. Interestingly when magnetoglobules under this state were observed under the microscope, UV radiation initiated their unidirectional division which was completed between 100 to 300 sec (Fig. 3,D). Therefore, UV irradiation not only results in the change in magnetotaxis direction but also triggers magnetoglobule division.

DISCUSSION

Here we report the UV-triggered reversal of swimming direction in MTB. Not all MTB change their magnetotactic swimming direction in response to light illumination. In addition, the swimming behaviour of photo-responsive magnetotaxis shows species specificity. The marine Magnetococcus marinus MC-1 strain observed in a capillary tube, swim away from high [O 2 ] toward low [O 2 ] in the same direction as magnetotaxis when illuminated with short-wavelength light (≤ 500 nm) [START_REF] Frankel | Magneto-aerotaxis in marine coccoid bacteria[END_REF]. Marine Magnetospira sp. strain QH-2 [START_REF] Zhu | Isolation and characterization of a marine magnetotactic spirillum axenic culture QH-2 from an intertidal zone of the China Sea[END_REF] and multicellular magnetoglobules [START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF] swim away from the edge of droplets in the opposite direction of magnetotaxis in response to illumination by blue (450-480 nm), violet (400-410 nm), and ultraviolet light (330-385 nm). The freshwater Magnetospirillum magneticum AMB-1 exhibits positive phototaxis behaviour, i.e. towards the light source, that is independent of wavelength and magnetotaxis [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF]Chen et al., 2020a). Photoreceptors have been identified only in the group of magnetospirillum among MTB. The extensively studied model strain is M. magneticum AMB-1 that has a cysteine-less light-oxygen-voltage (LOV) photoreceptor and bacteriophytochrome (MmBphPs) (Chen, et al., 2020a). Biochemistry and genetic analyses have revealed that red light, but not blue light, upregulated expression of genes involved in magnetosome synthesis and anti-oxidative stress, decreased intracellular ROS (reactive oxygen species) concentration, increased cell growth and phototaxis behavior of AMB-1 (Chen, et al., 2020a;Chen et al., 2020b). Ellipsoidal magnetoglobules showed a conspicuous photokinesis behaviour, they change the magnetotaxis direction with an increase of swimming speed upon constant illumination by short wavelength lights. Photokinesis behaviour has been reported for spherical magnetoglobules Candidatus Magnetoglobuls multicellularis(de [START_REF] De Azevedo | Effect of light wavelength on motility and magnetic sensibility of the magnetotactic multicellular prokaryote 'Candidatus Magnetoglobus multicellularis[END_REF][START_REF] De Azevedo | Photokinesis is magnetic field dependent in the multicellular magnetotactic prokaryote Candidatus Magnetoglobus multicellularis[END_REF]. In these studies, monochromatic LED lamp of blue (469 nm), green (517 nm), yellow (596 nm) or red (628 nm) was used for imaging and studying the effect of light on swimming speed. The spherical magnetoglobules were analyzed at edge of droplets [START_REF] De Azevedo | Effect of light wavelength on motility and magnetic sensibility of the magnetotactic multicellular prokaryote 'Candidatus Magnetoglobus multicellularis[END_REF][START_REF] De Azevedo | Photokinesis is magnetic field dependent in the multicellular magnetotactic prokaryote Candidatus Magnetoglobus multicellularis[END_REF]. In this area magnetoglobules display ping-pong or bounce motion, which consists of an original magnetotactic swimming with constant velocity to the edge, accumulation at the edge, and then a sudden swimming away from the edge against the original magnetotaxis direction with an increase of speed to variable distance and back again to the edge [START_REF] De Barros | Magnetotaxis[END_REF][START_REF] Rodgers | Intercellular structure in a many-celled magnetotactic prokaryote[END_REF][START_REF] Lins | Imaging faces of shadowed magnetite (Fe3O4) crystals from magnetotactic bacteria with energy-filtering transmission electron microscopy[END_REF]Keim, et al., 2004a;[START_REF] Greenberg | Observation of Magnetoreceptive Behavior in a Multicellular Magnetotactic Prokaryote in Higher than Geomagnetic Fields[END_REF][START_REF] Zhou | A novel genus of multicellular magnetotactic prokaryotes from the Yellow Sea[END_REF][START_REF] Chen | A novel species of ellipsoidal multicellular magnetotactic prokaryotes from Lake Yuehu in China[END_REF][START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF]. In the studies of spherical magnetoglobules photokinesis how the two distinct velocities of the ping-pong motion were considered is unclear. To avoid this edge effect, we studied the ellipsoidal magnetoglobules in microchannel and restricted their swimming in an area by application of an alternate uniform magnetic field.

What could be the mechanism that governs photosensitive magnetotaxis? At present magnetoglobules cannot be cultivated in laboratories. Environmental samples exhibit variable swimming velocities depending on the sample origin, collection season, time of enrichment in the laboratory and the duration of microscopic observation. Experimental data for photosensitive magnetotaxis show fluctuations due to the variation of cellular properties. Thought experiments can be proposed to speculate the mechanism of magnetic photokinesis and help guide future experimental analysis. UV light is harmful for cells because the photons of short wavelength are highly energetic and can break covalent bonds creating deleterious ROS that chemically alter and denature macromolecules. Magnetosomes are involved in the removal of ROS [START_REF] Chen | Phototaxis in the magnetotactic bacterium Magnetospirillum magneticum strain AMB-1 is independent of magnetic fields[END_REF][START_REF] Wang | Magnetic Fields and Reactive Oxygen Species[END_REF]. Therefore, the intracellular ROS concentration is controlled by the dynamic equilibrium between UV resulted production and magnetosome catalyzed reduction (Fig. 4).

When the accumulation reaches a certain threshold the concentration of ROS steers the magnetoglobules changing the swimming direction to escape from the deleterious light. The reaction time of 1 to 13 sec for magnetoglobules to change their swimming direction from north-seeking to south-seeking might indicate the physiological variation of the magnetoglobules. Application of magnetic field may affect intracellular ROS level depending on field type, intensity and frequency and model organisms analyzed [START_REF] Wang | Magnetic Fields and Reactive Oxygen Species[END_REF]. Reversal of the magnetic field interacts with the magnetosomes and applies a magnetic torque, which could affect the velocity of ROS removal by magnetosomes. As a consequence, the intracellular ROS concentration could quickly reach the toxic threshold, which is as indicated by the very narrow time delay, around 0.5 sec, for the magnetoglobules to change their magnetotaxis direction after the reversal of the direction of the magnetic field. An increase of swimming speed might require more energy.

Photoreceptor rhodopsin can convert photons into electrochemical potential of H + or Na + that can be used for driving flagellar motors. Metagenomic analysis can provide answers for this hypothesis. The ultimate solution is success in cultivating magnetoglobules, which would enable us to study the mechanisms for magnetic photokinesis at a molecular, biochemistry and cell biology level. Metagenomic analysis will provide metabolism information that is useful in the cultivation of magnetoglobules.

Fig. 4 Intracellular level of ROS might affect the swimming direction of magnetoglobules. The equilibrium of ROS level depends on magnetosome-catalyzed decrease of ROS (from big red ROS to small blue ROS, [START_REF] Li | Light irradiation helps magnetotactic bacteria eliminate intracellular reactive oxygen species[END_REF] and UV-generated ROS (from small blue ROS to big red ROS). At low level magnetoglobules swim northward, and at a level higher than the threshold magnetoglobules reverse the swimming direction to southward. Magnetic torque might decrease magnetosome capacity of reducing ROS, which results in sudden reversal of magnetoglobule swimming direction.

MTB dwell in chemically stratified sediments or water columns. They shuttle within and between layers to obtain appropriate nutrients. The nutrient layers exhibit daily vertical displacement in correlation with changing sunlight, temperature, and tide. Shapiro et al. have proposed that the behaviour of photophobic activate magnetoglobules to optimize its position and to adapt to circadian variation of chemical gradients and light intensity [START_REF] Shapiro | Multicellular photo-magnetotactic bacteria[END_REF]. Genes related to circadian rhythm control were found in spherical [START_REF] Abreu | Deciphering unusual uncultured magnetotactic multicellular prokaryotes through genomics[END_REF] and ellipsoidal [START_REF] Qian | Juxtaposed membranes underpin cellular adhesion and display unilateral cell division of multicellular magnetotactic prokaryotes[END_REF] magnetoglobules. Interestingly, Abreu et al. reported that Ca. M. multicellularis moved downward at night and the number of dividing microorganisms during this period was higher [START_REF] Abreu | Deciphering unusual uncultured magnetotactic multicellular prokaryotes through genomics[END_REF] . Here, we found that UV light triggered division of ellipsoidal magnetoglobules. There is no explanation for this difference at present. Cultures of magnetoglobules could provide detailed analysis on the effect of illumination rhythms on the life cycle and displacement of magnetoglobules.

Photophobic reactions of MTB to UV, violet and blue light is generally studied under fluorescence microscope with a mercury light source that is 10 to 100 times brighter than the monochromatic LED sources. The weaker irradiance provided by the LED source might explain why only UV and violet light in this study could trigger the magnetic photokinesis response. In order to characterize the magnetic photokinesis mechanism concerning the minimal photon energy and numbers of photons required we are designing and constructing a specific microscope for future investigations. It is based on an optical tweezer mounted with homogenous magnetic fields. MTB will be trapped with less harmful infrared light at 976 nm.

A second monochromatic beam at 385 nm (photon energy at 3.23 eV), 420 nm (2.96 eV), 475 nm (2.61 eV)

or 525 nm (2.36 eV) at different density will be used to illuminate the trapped MTB. The moving direction and force of illuminated MTB can then be characterized. The results obtained should contribute to a comprehensive understanding of the mechanism of the MTB reaction to light.

CONCLUSIONS

Light affects magnetotactic behaviour and magnetoglobule reproduction. Therefore, the impact of sunlight and circadian rhythm on the contribution of magnetoglobules to the biogeochemical cycles of iron and sulfur should be investigated.

DATA TRANSFER STATEMENT

The datasets generated during and/or analyzed during the current study are available from the corresponding author on request.

Fig. 1

 1 Fig. 1 Swimming behaviour of ellipsoidal Mediterranean magnetoglobules. (A) SEM shows magnetosomes (white

Fig. 2

 2 Fig. 2 Polarity of ellipsoidal magnetoglobules. Fluorescence confocal (A) and contrast (B) microscopy shows the oval

Fig. 3

 3 Fig. 3 UV triggered division of magnetoglobules. (A) images of fluorescence microscopy of a Mediterranean ellipsoidal

ACKNOWLEDGEMENTS

We are grateful to H. Le Guenno for assistance in fluorescence confocal microscopy analysis and P.

Kryczka for assistance in

Magnetodrome software development. This study was supported financially by the Excellence Initiative of Aix-Marseille University -A*Midex, a French "Investissements d'Avenir" programme, by the grant 2018YFC0309904 from National Key Research and Development Program of China, the grant 41920104009 from National Natural Science Foundation of China, and grants from CNRS for LIA-MagMC.


Louis Jeanjean
email: louis.jeanjean@univ-fcomte.fr

Sheng-Sen Lu
email: sslu@pku.edu.cn

On global minimizers for a mass constrained problem

Keywords: 35J60, 35Q55 Nonlinear scalar field equations, prescribed L 2 -norm solutions, constrained minimization problems, energy ground states versus least action solutions

In any dimension N ≥ 1, for given mass m > 0 and for the C 1 energy functional

we revisit the classical problem of finding conditions on F ∈ C 1 (R, R) insuring that I admits global minimizers on the mass constraint

Under assumptions that we believe to be nearly optimal, in particular without assuming that F is even, any such global minimizer, called energy ground state, proves to have constant sign and to be radially symmetric monotone with respect to some point in R N . Moreover, we manage to show that any energy ground state is a least action solution of the associated free functional. This last result settles, under general assumptions, a long standing open problem.

Introduction

Let N ≥ 1 and I : H 1 (R N ) → R be a C 1 energy functional defined by

I(u) := 1 2 R N |∇u| 2 dx - R N F (u)dx
where F (t) := t 0 f (τ )dτ for some function f ∈ C(R, R). In this paper we focus on the minimization problem

E m := inf u∈Sm I(u), (Inf m )
where m > 0 is prescribed and

S m := u ∈ H 1 (R N ) | u 2 L 2 (R N ) = m .
By a direct application of Lagrange multiplier's rule, if u ∈ S m solves (Inf m ) then there exists

µ = µ(u) ∈ R such that -∆u = f (u) -µu in H 1 (R N ). (1.1)
A minimizer of (Inf m ) is often called an energy ground state and E m the ground state energy.

The study of problem (Inf m ) naturally arises in the search of standing waves for nonlinear scalar field equations the form

iψ t + ∆ψ + f (ψ) = 0, ψ : R × R N → C. (1.2)
By standing waves, we mean solutions to (1.2) of the special form ψ(t, x) = e iµt u(x) with µ ∈ R and u ∈ H 1 (R N ). Clearly ψ(t, x) satisfies (1.2) if u(x) satisfies (1.1) for the corresponding µ ∈ R.

The study of such type of equations, which already saw major contributions forty years ago, [START_REF] Berestycki | Nonlinear scalar field equations I: Existence of a ground state[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF][START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case, part 1[END_REF][START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case, part 2[END_REF][START_REF] Stuart | Bifurcation for Dirichlet problems without eigenvalues[END_REF], now lies at the root of several models linked with current physical applications (such as nonlinear optics, the theory of water waves, ...). For these equations, finding solutions with a prescribed L 2 -norm is particularly relevant since this quantity is preserved along the time evolution. In addition, if the solutions correspond to energy ground states, then, in most situations, it is possible to prove that the associated standing waves are orbitally stable. This likely explains why the study of problem (Inf m ) is still the object of an intense activity. Among many others possible choices, we refer to [START_REF] Carles | On soliton (in)-stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF][START_REF] Carles | Orbital stability vs. scattering in the cubic-quintic Schrödinger equation[END_REF][START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Dovetta | Action versus energy ground states in nonlinear Schrödinger equations[END_REF][START_REF] Hajaiej | On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation[END_REF][START_REF] Hirata | Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches[END_REF][START_REF] Jeanjean | Nonradial normalized solutions for nonlinear scalar field equations[END_REF][START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF][START_REF] Shibata | Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term[END_REF][START_REF] Stefanov | On the normalized ground states of second order PDE's with mixed power nonlinearities[END_REF] and to the references therein.

Our first main result concerns the solvability of (Inf m ). It can be viewed as an extension of the one of [START_REF] Shibata | Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term[END_REF] already obtained in a very general setting. The following assumptions on f ∈ C(R, R) will be required. and the map m → E m is nonincreasing and continuous. Moreover,

(i) there exists a number m * ∈ [0, ∞) such that E m = 0 if 0 < m ≤ m * , E m < 0 when m > m * ; (ii) when m > m * , the global infimum E m is achieved and thus (Inf m ) has an energy ground state v ∈ S m with I(v) = E m < 0; (iii) when 0 < m < m * , E m = 0 is not achieved; (iv) m * = 0 if in addition lim t→0 F (t) |t| 2+ 4 N = +∞, (A.1)
and m * > 0 if in addition

lim sup t→0 F (t) |t| 2+ 4 N < +∞. (A.2)
Remark 1.2 (i) As it will be clear from the proof of Theorem 1.1 (ii), see also Remark 2.4, when m > m * we also show that any minimizing sequence for (Inf m ) is, up to a subsequence and up to translations in R N , strongly convergent.

(ii) When 0 < m < m * , it is proved in Theorem 1.1 (iii) that the global infimum E m = 0 is not achieved, but this does not mean that the constrained functional I |Sm may not admit critical points with positive energies, see the companion work [START_REF] Jeanjean | Normalized solutions with positive energies for a coercive problem[END_REF].

(iii) In the case m * > 0, studying existence and nonexistence of global minimizers with respect to E m * = 0 seems to be a delicate issue. Since it exceeds our scope of the present paper, we shall not explore further general conditions on f that ensure the existence or nonexistence but refer the interested reader to [START_REF] Jeanjean | Normalized solutions with positive energies for a coercive problem[END_REF] and [START_REF] Shibata | Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term[END_REF]Theorem 1.4] for some existence results.

(iv) For convenience of statement, we introduce the notation

m f m *
with the understanding that m ≥ m * if m * > 0 and E m * = 0 is achieved, and m > m * if otherwise. As one may observe, when m f m * and for any minimizer v ∈ S m of (Inf m ), the associated Lagrange multiplier µ = µ(v) is positive. Indeed, from the Pohozaev identity corresponding to (1.1), see [START_REF] Berestycki | Nonlinear scalar field equations I: Existence of a ground state[END_REF]Proposition 1],

P (v) := N -2 2N R N |∇v| 2 dx + 1 2 µ R N |v| 2 dx - R N F (v)dx = 0
and the fact that

I(v) = E m ≤ 0, we have 0 ≥ I(v) = I(v) -P (v) = 1 N R N |∇v| 2 dx - 1 2 µm
and hence µ > 0.

Remark 1.3 Let us give some examples of nonlinearities satisfying

(f 1) -(f 3). (i) f (t) = |t| p-2 t + A|t| q-2 t with A ∈ R and 2 < q < p < 2 + 4 N .
In particular, (A.1) and (A.2) hold when A ≥ 0 and when A < 0 respectively.

(ii) f (t) = |t| p-2 t -|t| q-2 t with    2 < p < q < ∞, if N = 1, 2, 2 < p < q ≤ 2N N -2 , if N ≥ 3.
In particular, (A.1) and (A.2) hold if p < 2 + 4 N and if p ≥ 2 + 4 N respectively, and when N = 1, 2, 3 we cover the so-called cubic-quintic nonlinearity

f (t) = |t| 2 t -|t| 4 t
which attracts much attention due to its physical relevance, see for example [START_REF] Carles | On soliton (in)-stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF][START_REF] Carles | Orbital stability vs. scattering in the cubic-quintic Schrödinger equation[END_REF][START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF][START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF][START_REF] Stefanov | On the normalized ground states of second order PDE's with mixed power nonlinearities[END_REF]. These examples are only some special cases, our Theorem 1.1 and the subsequent Theorems 1.4 and 1.6 apply to more general nonlinearities, in particular to those which are not a sum of powers.

Our next result shows that any energy ground state has constant sign and enjoys symmetry and monotonicity properties.

Theorem 1.4 Assume that N ≥ 1, f ∈ C(R, R) satisfies (f 1) -(f 3)
, and in addition f is locally Lipschitz continuous when N = 1. Let m f m * , where m * ≥ 0 is the number given by Theorem 1.1. Then any minimizer v ∈ S m of (Inf m ) satisfies the following properties:

(i) v has constant sign, (ii) v is radially symmetric up to a translation in R N ,
(iii) v is monotone with respect to the radial variable.

Our last theorem is the heart of the present paper, it answers for nonlinear scalar field equations a long standing open problem. To explain what is at stake, we need the following definition.

Definition 1.5 For given µ > 0, a nontrivial solution w ∈ H 1 (R N ) of the free problem

-∆u = f (u) -µu in R N , u ∈ H 1 (R N ), (Q µ )
is said to be a least action solution if it achieves the infimum of the C 1 action functional

J µ (u) := I(u) + 1 2 µ R N |u| 2 dx
among all the nontrivial solutions, namely

J µ (w) = A µ := inf{J µ (u) | u ∈ H 1 (R N ) \ {0}, J ′ µ (u) = 0}.
For future reference, the value A µ is called the least action of (Q µ ).

The open problem can be now formulated as follows.

Open Problem: We know that a ground state energy minimizer v ∈ S m is a nontrivial solution to (Q µ ), where µ = µ(v) > 0 is the associated Lagrange multiplier. Is the minimizer v ∈ S m a least action solution to (Q µ )? In other words, does an energy ground state is necessarily a least action solution?

For some odd f ∈ C(R, R) satisfying (f 1) -(f 3) it is known that there exists a unique positive solution to (Q µ ) and that it is a least action solution, see for example [START_REF] Carles | Orbital stability vs. scattering in the cubic-quintic Schrödinger equation[END_REF][START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF][START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF]. Thus, in such situations, Theorem 1.4 implies that any energy ground state is a least action solution. However, apart in some particular cases of this type where some uniqueness property was used, the Open Problem remained unsolved until recently. In 2020 a positive answer was given in [START_REF] Fernandez | Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation[END_REF] for a related problem of biharmonic type with a power nonlinearity, see [11, Proposition 3.9 and Theorem 1.3]. Also, very recently in [START_REF] Dovetta | Action versus energy ground states in nonlinear Schrödinger equations[END_REF], the authors answered positively the Open Problem for nonlinearities modeled on f (t) = |t| p-2 t, p ∈ (2, 2 * ) which allow to benefit from a Nehari manifold in the sense given in [START_REF] Szulkin | The method of Nehari manifold, Handbook of nonconvex analysis and applications[END_REF], see [START_REF] Dovetta | Action versus energy ground states in nonlinear Schrödinger equations[END_REF]Theorem 1.3] for more details. Note that the results of [START_REF] Dovetta | Action versus energy ground states in nonlinear Schrödinger equations[END_REF] also hold when the analog of problem (Q µ ) is set on an arbitrary domain Ω ⊂ R N . Finally, we mention [START_REF] Ilyasov | On orbital stability of the physical ground states of the NLS equations[END_REF] in which the Open Problem was claimed to be solved for a nonlinearity which is a sum of two powers.

Our result in that direction covers all the previous particular cases, at least when the associated equations are set on all the space R N .

Theorem 1.6 Assume that N ≥ 1, f ∈ C(R, R) satisfies (f 1) -(f 3)
, and in addition f is locally Lipschitz continuous when N = 1. Let m f m * and denote by µ(v) the Lagrange multiplier corresponding to an arbitrary minimizer v ∈ S m of (Inf m ), where m * ≥ 0 is the number given by Theorem 1.1. Then the following statements hold.

(i) Any minimizer v ∈ S m of (Inf m ) is a least action solution of (Q µ ) with µ = µ(v) > 0.
In particular,

A µ = E m + 1 2 µm. (ii) For given µ ∈ {µ(v) | v ∈ S m is a minimizer of (Inf m )}, any least action solution w ∈ H 1 (R N ) of (Q µ ) is a minimizer of (Inf m ), namely w 2 L 2 (R N ) = m and I(w) = E m . Remark 1.7 (i)
The conclusions of Theorem 1.6 (ii) were also observed in [START_REF] Dovetta | Action versus energy ground states in nonlinear Schrödinger equations[END_REF][START_REF] Fernandez | Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation[END_REF][START_REF] Ilyasov | On orbital stability of the physical ground states of the NLS equations[END_REF] in the corresponding frames.

(ii) For alternative variational characterizations of the energy ground states, in related problems, we refer to [START_REF] Cingolani | Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation[END_REF][START_REF] Cingolani | Ground state solutions for the nonlinear Choquard equation with prescribed mass, to appear on Geometric Properties for Parabolic and Elliptic PDE's[END_REF][START_REF] Hirata | Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches[END_REF]. Note that in [START_REF] Cingolani | Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation[END_REF][START_REF] Cingolani | Ground state solutions for the nonlinear Choquard equation with prescribed mass, to appear on Geometric Properties for Parabolic and Elliptic PDE's[END_REF], a variational characterization of the associated Lagrange multiplier is proposed, see also [START_REF] Dovetta | Action versus energy ground states in nonlinear Schrödinger equations[END_REF]Theorem 1.2] in that direction.

(iii) It is known, see for example [START_REF] Carles | On soliton (in)-stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF][START_REF] Jeanjean | Normalized solutions with positive energies for a coercive problem[END_REF][START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF], that under the assumptions (f 1) -(f 3) there may exist least action solutions which are not energy ground states.

The paper is organized as follows. In Section 2 we prove Theorem 1.1. Section 3 is devoted to the proof of Theorem 1.4. Finally, in Section 4 we prove Theorem 1.6.

Notations. Throughout this paper, for any

p ∈ [1, ∞), L p (R N ) is the usual Lebesgue space endowed with the norm u L p (R N ) := R N |u| p dx 1/p
, and H 1 (R N ) the usual Sobolev space endowed with the norm

u H 1 (R N ) := ∇u 2 L 2 (R N ) + u 2 L 2 (R N ) 1/2 .
Moreover, for given u ∈ H 1 (R N ) and any s ∈ R, we define the scaling function

s ⋄ u := e N s/2 u(e s •),
which remains in H 1 (R N ) and preserves the L 2 norm when s ∈ R varies.

This section aims to prove Theorem 1.1 and in particular we shall show the existence and nonexistence of minimizers of (Inf m ) for suitable range of the mass m > 0. As a necessary preparation, we have the following lemma the proof of which is standard.

Lemma 2.1 Assume that N ≥ 1 and f ∈ C(R, R) satisfies (f 1) -(f 2).
Then the following statements hold.

(i) For any bounded sequence {u n } in H 1 (R N ), lim n→∞ R N F (u n )dx = 0 if lim n→∞ u n L ∞ (R N ) = 0, and
lim sup n→∞ R N F (u n )dx ≤ 0 if lim n→∞ u n L 2+4/N (R N ) = 0. (ii) There exists C = C(f, N, m) > 0 such that I(u) ≥ 1 4 ∇u 2 L 2 (R N ) -C(f, N, m) for any u ∈ H 1 (R N ) satisfying u 2 L 2 (R N ) ≤ m.
In particular, I is coercive on S m .

To proceed further, we recall the global infimum

E m := inf u∈Sm I(u)
and make below a detailed study of its basic properties.

Lemma 2.2 Assume that N ≥ 1 and f ∈ C(R, R) satisfies (f 1) -(f 3). Then the following statements hold.

(i) -∞ < E m ≤ 0 for all m > 0.
(ii) There exists m 0 > 0 such that E m < 0 for any m > m 0 .

(iii) E m < 0 for all m > 0 if (A.1) holds, and E m = 0 for small m > 0 if (A.2) holds.

(iv) For any m > m ′ > 0 one has E m ≤ m m ′ E m ′ . (2.1)
If E m ′ is achieved then the inequality is strict.

(v) The function m → E m is nonincreasing and continuous.

Proof. (i) By Lemma 2.1 (ii), I is bounded from below on S m and thus

E m > -∞. For a fixed u ∈ S m ∩ L ∞ (R N ), we have ∇(s ⋄ u) L 2 (R N ) → 0 and s ⋄ u L ∞ (R N ) → 0 as s → -∞. In view of Lemma 2.1 (i), E m ≤ lim s→-∞ I(s ⋄ u) = 0. (ii) From (f 3) and Step 1 of [1, Proof of Theorem 2], there exists u ∈ H 1 (R N ) such that R N F (u)dx > 0. For any m > 0, set u m := u(m -1/N • u 2/N L 2 (R N ) • x) ∈ S m . Since I(u m ) = 1 2 R N |∇u m | 2 dx - R N F (u m )dx = m N-2 N 2 u 2(N -2)/N L 2 (R N ) R N |∇u| 2 dx - m u 2 L 2 (R N ) R N F (u)dx =: Am N-2 N -Bm =: g(m), it follows that E m ≤ I(u m ) = g(m) < 0 for any sufficiently large m > 0. (iii) When (A.1) holds, we choose u ∈ S m ∩ L ∞ (R N ). For D := R |∇u| 2 dx R N |u| 2+4/N dx > 0, by (A.1), there exists δ > 0 such that F (t) ≥ D|t| 2+4/N for all |t| ≤ δ. Since s ⋄ u L ∞ (R N ) ≤ δ for some s < 0, it is clear that E m ≤ I(s ⋄ u) ≤ 1 2 R |∇(s ⋄ u)| 2 dx -D R N |s ⋄ u| 2+4/N dx = 1 2 e 2s R |∇u| 2 dx -De 2s R N |u| 2+4/N dx = - 1 2 e 2s R |∇u| 2 dx < 0. When (A.2) is satisfied, there exists C f > 0 such that F (t) ≤ C f |t| 2+4/N for any t ∈ R. By the Gagliardo-Nirenberg inequality, R N F (u)dx ≤ C f C N m 2/N ∇u 2 L 2 (R N )
for all u ∈ S m .

For any m > 0 small enough such that C f C N m 2/N ≤ 1/4, we have

I(u) ≥ 1 4 ∇u 2 L 2 (R N ) > 0,
and thus E m ≥ 0. From Item (i), it follows that E m = 0 for m > 0 small.

(iv) Let t := m/m ′ > 1. For any ε > 0 there exists u ∈ S m ′ such that

I(u) ≤ E m ′ + ε. Clearly, v := u(t -1/N •) ∈ S m and then E m ≤ I(v) = tI(u) + 1 2 t N-2 N 1 -t 2 N R N |∇u| 2 dx < tI(u) ≤ m m ′ (E m ′ + ε). (2.2)
Since ε > 0 is arbitrary, we see that the inequality (2.1) holds. If E m ′ is achieved, for example, at some u ∈ S m ′ , then we can let ε = 0 in (2.2) and thus the strict inequality follows.

(v) By Item (i) and (2.1), it is clear that E m is nonincreasing. Our remaining task is to prove the continuity and this is equivalent to show that for given m > 0 and any positive sequence

{m k } such that m k → m as k → ∞, one has lim k→∞ E m k = E m . We first claim that lim sup k→∞ E m k ≤ E m . (2.3)
Indeed, for any u ∈ S m and each k ∈ N + , set

u k := m k /m • u ∈ S m k . Since u k → u strongly in H 1 (R N ), it is clear that lim k→∞ I(u k ) = I(u) and thus lim sup k→∞ E m k ≤ lim sup k→∞ I(u k ) = I(u).
By the arbitrariness of u ∈ S m , we conclude that (2.3) holds. To complete the proof of the continuity, we only need to show lim inf

k→∞ E m k ≥ E m . (2.4)
For each k ∈ N + , there exists v k ∈ S m k such that

I(v k ) ≤ E m k + 1 k . (2.5)
Setting

t k := (m/m k ) 1/N and ṽk := v k (•/t k ) ∈ S m , we have E m ≤ I(ṽ k ) ≤ I(v k ) + I(ṽ k ) -I(v k ) ≤ E m k + 1 k + I(ṽ k ) -I(v k ) =: E m k + 1 k + C(k),
where

C(k) ≤ 1 2 |t N -2 k -1| • R N |∇v k | 2 dx + |t N k -1| • R N |F (v k )|dx =: 1 2 |t N -2 k -1| • A(k) + |t N k -1| • B(k).
Since t k → 1, the proof of (2.4) can be reduced to showing that A(k) and B(k) are bounded.

To justify the boundedness, by (2.5) and (2.3), we have lim

sup k→∞ I(v k ) ≤ E m . Noting that v k ∈ S m k and m k → m, it follows from Lemma 2.1 (ii) that {v k } is bounded in H 1 (R N ). Since f ∈ C(R, R) satisfies (f 1
) and (f 2), it is clear that A(k) and B(k) are both bounded and thus the continuity is proved.

Remark 2.3 When N ≥ 2 the function m → E m is concave.
To see this we define for given u ∈ S 1 and any m > 0 the real function

Φ u (m) := I(u(m -1 N •)) = m 1-2 N ∇u 2 L 2 (R N ) -m R N F (u)dx.

It is clear that

E m = inf u∈S 1 Φ u (m).
Each function m → Φ u (m) being concave since

Φ ′′ u (m) = - 2 N 1 - 2 N m -2 N -1 ∇u 2 L 2 (R N ) ≤ 0,
we deduce that m → E m is a concave function. In particular, it is continuous.

Proof of Theorem 1.1. We define

m * := inf{m > 0 | E m < 0}.
It is easily seen from Lemma 2.2 that m * ∈ [0, ∞),

E m = 0 if 0 < m ≤ m * , E m < 0 when m > m * ; (2.6)
in particular, m * = 0 if (A.1) holds, and m * > 0 if (A.2) holds. Let us first show that if 0 < m < m * then E m = 0 is not achieved. Indeed, assuming by contradiction that E m = 0 is achieved for some m ∈ (0, m * ), we infer from Lemma 2.2 (iv) that

E m * < m * m E m =
n→∞ R N F (u n )dx ≤ 0 via Lemma 2.1 (i); noting that I(u n ) ≥ -R N F (u n )dx, we obtain a contradiction: 0 > E m = lim n→∞ I(u n ) ≥ -lim n→∞ R N F (u n )dx ≥ 0.
Since {u n } is non-vanishing, there exists a sequence {y n } ⊂ R N and a nontrivial element

v ∈ H 1 (R N ) such that up to a subsequence u n (• + y n ) ⇀ v in H 1 (R N ) and u n (• + y n ) → v almost everywhere on R N . Set m ′ := v 2 L 2 (R N ) ∈ (0, m] and w n := u n (• + y n ) -v. It is clear that lim n→∞ w n 2 L 2 (R N ) = m -m ′ (2.8)
and, using the splitting result [16, Lemma 2.6],

E m = lim n→∞ I(u n ) = lim n→∞ I(v + w n ) = I(v) + lim n→∞ I(w n ).
(2.9)

We shall prove below a claim and then conclude the whole proof.

Claim. lim n→∞ w n L 2 (R N ) = 0. In particular, m ′ = m by (2.8).

Let (2.8) gives that m ′ ∈ (0, m). In view of the definition of E tn and Lemma 2.2 (v), we obtain

t n := w n 2 L 2 (R N ) for each n ∈ N + . If lim n→∞ t n > 0, then
lim n→∞ I(w n ) ≥ lim n→∞ E tn = E m-m ′ .
From (2.9) and (2.1), it follows

E m ≥ I(v) + E m-m ′ ≥ E m ′ + E m-m ′ ≥ m ′ m E m + m -m ′ m E m = E m .
Thus necessarily I(v) = E m ′ and this shows that E m ′ is achieved at v ∈ S m ′ . But then still using (2.9) and (2.1), we obtain a contradiction:

E m ≥ E m ′ + E m-m ′ > m ′ m E m + m -m ′ m E m = E m ,
and so the claim is proved.

Conclusion.

Clearly, v ∈ S m by the above claim and thus I(v) ≥ E m . Since the claim and Lemma 2.1 (i) imply that

lim n→∞ R N F (w n )dx ≤ 0, (2.10)
we also have lim n→∞ I(w n ) ≥ 0. Therefore, by (2.9) we get E m ≥ I(v) and hence E m < 0 is achieved at v ∈ S m .

Remark 2.4 One can deduce further that u

n (• + y n ) → v in H 1 (R N )
. Indeed, from (2.9), (2.10) and the fact that

I(v) = E m , it follows ∇w n L 2 (R N ) → 0 as n → ∞.
Since lim n→∞ w n L 2 (R N ) = 0, we obtain the strong convergence.

Sign, symmetry and monotonicity

This section is devoted to the proof of Theorem 1.4. Unless otherwise noted, for given µ > 0 we use the notations

g µ (t) := -µt + f (t) and G µ (t) := - 1 2 µt 2 + F (t).
To deal with the case N = 1, a special treatment is required. To be more precise, we shall make use of the following classification result which is deduced by means of simple methods of ordinary differential equations.

Lemma 3.1 Assume that N = 1, f is a locally Lipschitz continuous function on R satisfying (f 1), and w ∈ H 1 (R) is a nontrivial critical point of J µ for some µ > 0. Then w have a sign. More precisely we have

(a) If w is negative somewhere, then ζ -:= sup{t < 0 | G µ (t) = 0} ∈ (-∞, 0), g µ (ζ -) < 0,
and after a suitable translation of the origin w satisfies

(a1) w(x) = w(-x) for any x ∈ R, (a2) w(x) < 0 for any x ∈ R, (a3) w(0) = ζ -, (a4) w ′ (x) > 0 for any x > 0. (b) If w is positive somewhere, then ζ + := inf{t > 0 | G µ (t) = 0} ∈ (0, ∞), g µ (ζ + ) > 0,
and after a suitable translation of the origin w satisfies

(b1) w(x) = w(-x) for any x ∈ R, (b2) w(x) > 0 for any x ∈ R, (b3) w(0) = ζ + , (b4) w ′ (x) < 0 for any x > 0.
In particular, w is a translation of the unique solution to the initial value problem -u ′′ = g µ (u) with u(0) = ζ -(or u(0) = ζ + ) and u ′ (0) = 0.

Proof. By regularity w ∈ C 2 (R, R) and thus

-w ′′ = g µ (w) in R. (3.1)
Since |w(x)| and |w ′ (x)| decay to zero exponentially as |x| → ∞, we have

1 2 |w ′ (x)| 2 + G µ (w(x)) = 0 for x ∈ R. (3.2)
Without loss of generality, we only consider the case when w is negative somewhere. By translating the point where w achieves its negative minimum to the origin, one may assume that w ′ (0) = 0. In view of (3.2),

G µ (w(0)) = 0 and ζ -> -∞.
Since (f 1) gives that G µ (t) < 0 for any t < 0 close enough to the origin, we also have ζ -< 0. Now assume by contradiction that

g µ (ζ -) ≥ 0. Since w(0) ≤ ζ -, there exists x * ∈ R such that w(x * ) = ζ -. Then w ′ (x * ) = 0 and w ′′ (x * ) = -g µ (ζ -) ≤ 0 via (3.
2) and (3.1) respectively. If g µ (ζ -) > 0, then since whenever w(x) = ζ -one also has w ′ (x) = 0 and w ′′ (x) < 0, w can never go above ζ -< 0, which is impossible. On the other hand, if g µ (ζ -) = 0, then by uniqueness the conditions

w(x * ) = ζ - and w ′ (x * ) = w ′′ (x * ) = 0 imply w ≡ ζ -, which is also impossible. With the desired conclusion g µ (ζ -) < 0 at hand, there exists ε > 0 such that G µ (t) > 0 for any t ∈ (ζ --ε, ζ -). If w(0) < ζ -, then w(x * ) ∈ (ζ --ε, ζ -) for some x * ∈ R and so 1 2 |w ′ (x * )| 2 + G µ (w(x * )) > 0.
This contradicts (3.2), and therefore w(0) = ζ -. Since w is the global solution of (3.1) with the initial conditions w(0) = ζ -and w ′ (0) = 0, the rest follows from a standard adaptation of some arguments in [1, Proof of Theorem 5].

Remark 3.2 Even though the nonlinearity f in Lemma 3.1 is locally Lipschitz continuous, the nontrivial critical points of J µ (if exist) are not necessarily unique up to a translation in R and up to a sign since we allow f to be not odd.

In the higher dimensional case N ≥ 2, the radial symmetry of minimizers will be obtained as a direct consequence of a general symmetry result in [START_REF] Mariş | On the symmetry of minimizers[END_REF], and the proof of the monotonicity relies on Lemma 3.3 below. We remark that the first part of Lemma 3.3 is well known and the second part is a simple corollary of [2, Lemma 3.2]. Lemma 3.3 Let v be a nonnegative measurable function defined on R N such that for any α > 0 the function (v -α) + belongs to H 1 (R N ) and has compact support, and denote by v * the Schwarz rearrangement of v. Then

R N |∇v * | 2 dx ≤ R N |∇v| 2 dx. (3.3)
Moreover, if the equality in (3.3) holds then the level set

χ α := {x ∈ R N | v(x) > α}
is equivalent to a ball for any α ∈ (0, ess sup(v)).

Proof of Theorem 1.4 By Lemma 3.1, the case N = 1 is proved. We treat below the case N ≥ 2. For given minimizer v ∈ S m of (Inf m ), we set v + := max{0, v} and v -:= min{0, v}.

If m ± := v ± 2 L 2 (R N ) = 0, then Lemma 2.2 (iv) gives that E m = I(v) = I(v + ) + I(v -) ≥ E m + + E m -≥ m + m E m + m - m E m = E m ,
and thus E m ± is achieved at v ± ∈ S m ± . Using Lemma 2.2 (iv) again, we obtain a contradiction:

E m ≥ E m + + E m -> m + m E m + m - m E m = E m .
Hence v has constant sign. Since any minimizer of (Inf m ) is a solution of (Q µ ) for some µ > 0 and then by regularity must be of class C 1 , we also deduce from [24, Theorem 2] that v is radially symmetric up to a translation in R N . To proceed further, without loss of generality, we may assume that v ≥ 0 and v(x) = v(|x|) for some one variable function v : [0, ∞) → [0, ∞). By the fact that v(x) → 0 as |x| → ∞, it can be seen that v is bounded and for any α > 0 the function (v -α) + belongs to H 1 (R N ) and has compact support. Since the Schwarz rearrangement v * satisfies v * ∈ S m and

R N F (v * )dx = R N F (v)dx, it follows from Lemma 3.3 that E m ≤ I(v * ) ≤ I(v) = E m and thus R N |∇v * | 2 dx = R N |∇v| 2 dx.
By Lemma 3.3 again, for any α ∈ (0, max(v)), the level set χ α is equivalent to a ball. We now assume by contradiction that v is not nonincreasing. Then v(r 2 ) > v(r 1 ) > 0 for some r 2 > r 1 ≥ 0.

Since v(r) → 0 as r → ∞, there exists r 3 > r 2 such that v(r 3 ) = v(r 1 ). Denoting a := v(r 1 ) and b := v(r 2 ), one may see that for any α ∈ (a, b) the level set χ α is nonempty but not equivalent to a ball. This gives a contradiction and thus v is nonincreasing with respect to the radial variable.

Least action characterization

In this section we show the least action characterization for any minimizer v ∈ S m of (Inf m ) by using a mountain pass characterization of nontrivial solutions of (Q µ ) with µ = µ(v) > 0. This mountain pass characterization, see Lemma 4.1 below, is the core of the proof of Theorem 1.6. It highlights the role of the L 2 mass and seems to have not been formulated before. Some of our arguments are motivated by [START_REF] Byeon | Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases[END_REF][START_REF] Jeanjean | A remark on least energy solutions in R N[END_REF][START_REF] Jeanjean | A note on a mountain pass characterization of least energy solutions[END_REF]. and suppose in addition that f is locally Lipschitz continuous when N = 1. Then for any nontrivial critical point w ∈ H 1 (R N ) of J µ , any δ > 0 and any M > 0, there exist a constant T = T (w, δ, M ) > 0 and a continuous path γ :

[0, T ] → H 1 (R N ) satisfying (i) γ(0) = 0, J µ (γ(T )) < -1, max t∈[0,T ] J µ (γ(t)) = J µ (w);
(ii) γ(τ ) = w for some τ ∈ (0, T ), and

J µ (γ(t)) < J µ (w) for any t ∈ [0, T ] such that γ(t) -w L 2 (R N ) ≥ δ; (iii) m(t) := γ(t) 2 L 2 (R N ) is a strictly increasing continuous function with m(T ) > M .
Proof. When N ≥ 3 and for the given w ∈ H 1 (R N ), we set

γ(t) :=    w • t , for t > 0, 0, for t = 0. Note that m(t) := γ(t) 2 L 2 (R N ) = t N w 2 L 2 (R N )
and by the Pohozaev identity

J µ (γ(t)) = 1 2 t N -2 R N |∇w| 2 dx -t N R N G µ (w)dx = 1 2 t N -2 - N -2 N t N R N |∇w| 2 dx.
For any δ > 0 and M > 0, we can thus choose a large constant T = T (w, M ) > 0 such that the continuous path γ : [0, T ] → H 1 (R N ) satisfies Items (i) -(iii) of Lemma 4.1.

In the case of N = 1, without loss of generality, we only consider the situation when the given w ∈ H 1 (R) is negative somewhere. Then the statement (a) of Lemma 3.1 holds and we can define a negative continuous function W : R → R by

W (x) =      w(x), for x ≥ 0, ζ --x 4 , for x ∈ [-ε, 0), ζ --ε 4 , for x < -ε.
Here ε > 0 is a chosen small constant such that

1 2 |W ′ (x)| 2 -G µ (W (x)) = 8x 6 -G µ (ζ --x 4 ) < 0 for x ∈ [-ε, 0), (4.1)
and it follows from G µ (ζ -) = 0 and g µ (ζ -) < 0. Setting

γ(t) := W (| • | -ln t), for t > 0, 0, for t = 0, one may see that the path γ : [0, ∞) → H 1 (R) is continuous, m(t) := γ(t) 2 L 2 (R) =                w 2 L 2 (R) - -ln t ln t |w(x)| 2 dx, for t ∈ (0, 1), w 2 L 2 (R) , for t = 1, w 2 L 2 (R) + 2 0 -ln t |W (x)| 2 dx, for t > 1,
and

J µ (γ(t)) =                J µ (w) - -ln t ln t 1 2 |w ′ (x)| 2 -G µ (w(x)) dx, for t ∈ (0, 1), J µ (w), for t = 1, J µ (w) + 2 0 -ln t 1 2 |W ′ (x)| 2 -G µ (W (x)) dx, for t > 1.
By the fact that G µ (w(x)) < 0 for x = 0 and (4.1), we have

J µ (γ(t)) < J µ (w) for t = 1 and J µ (γ(t)) < J µ (w) -2G µ (ζ --ε 4 ) • (ln t -ε) → -∞ as t → ∞.
Noting also that m(t) is strictly increasing and m(t) → ∞ as t → ∞, for any δ > 0 and M > 0 there exists a large constant T = T (w, M ) > 0 such that γ : [0, T ] → H 1 (R) is a desired continuous path of Lemma 4.1 when N = 1.

To cope with the remaining case N = 2, we adapt some arguments from [3, Proposition 2]. For the given w ∈ H 1 (R 2 ), we define Ψ : [0, ∞) × (0, ∞) → R by Ψ(θ, s) := J µ θw

• s = 1 2 θ 2 R 2 |∇w| 2 dx -s 2 R 2
G µ (θw)dx.

It can be easily seen that

Ψ θ (θ, s) = θ R 2 |∇w| 2 dx -s 2 R 2 g µ (θw)wdx, Ψ s (θ, s) = -2s R 2 G µ (θw)dx, d dθ R 2 G µ (θw)dx = R 2
g µ (θw)wdx.

Since the Nehari and Pohozaev identities give respectively

→ (1, 1 + ε) → (1 + θ * , 1 + ε) → (1 + θ * , ∞).
Here θ * = θ * (w, δ) ∈ (0, 1) is a chosen constant satisfying

θ * ≤ min{1 -θ 1 , θ 2 -1, ϑ 1-ε , ϑ 1+ε },
and each segment is horizontal or vertical. Let 0 =: t 0 < t 1 < • • • < t 6 < t 7 := ∞ be such that for each k = 0, 1, • • • , 7, the element η(t k ) ∈ R 2 is an end point of a linear segment of the piecewise linear curve η. We define γ(t) := θ(t)w • s(t) , t ≥ 0.

(f 1 ) 3 )

 13 lim t→0 f (t)/t = 0. (f 2) When N ≥ 3e αt 2 = 0 ∀α > 0;and also for any N ≥ 1There exists ζ = 0 such that F (ζ) > 0.Theorem 1.1 Assume that N ≥ 1, f ∈ C(R, R) satisfies (f 1) -(f 3). Then E m := inf u∈Sm I(u) > -∞

Lemma 4 . 1 when N = 2 ,

 412 Assume that N ≥ 1, µ > 0 and f ∈ C(R, R) satisfies the conditions (f 1) and (f 2) ′ when N ≥ 3, lim sup t→∞ |f (for any α > 0 lim t→∞ f (t) e αt 2 = 0,

2 G

 2 µ (w)dx = 0, there exist two positive constantsθ 1 < 1 < θ 2 such that d dθ R 2 G µ (θw)dx > 0 for θ ∈ [θ 1 , θ 2 ], for θ ∈ (1, θ 2 ].

  for (θ, s) ∈ [θ 1 , 1) × (0, ∞), = 0, for (θ, s) ∈ {1} × (0, ∞), < 0, for (θ, s) ∈ (1, θ 2 ] × (0, ∞).

(4. 3 ) 2 |∇w| 2 dx -s 2 R 2 g 2 |∇w| 2 2 |∇w| 2 dx -s 2 R 2 h

 32222222222 On the other hand, noting thatΨ θ (1, s) = R µ (w)wdx = (1 -s 2 ) R dx,for any s = 1 there exists ϑ s ∈ (0, 1) such thatΨ θ (θ, s) > 0, for (θ, s) ∈ [1 -ϑ s , 1 + ϑ s ] × (0, 1), < 0, for (θ, s) ∈ [1 -ϑ s , 1 + ϑ s ] × (1, ∞). a small constant s * ∈ (0, 1) such that Ψ θ (θ, s) = θ R (θw)w 2 dx > 0 for (θ, s) ∈ (0, 1] × (0, s * ]. (4.5)Now for any δ > 0 we fix a small constant ε = ε(δ) > 0 such that 1 -ε > s * andw • s -w L 2 (R 2 ) < δ for s ∈ [1 -ε, 1 + ε],and denote by η(t) = (θ(t), s(t)) : [0, ∞) → R 2 the piecewise linear curve joining(0, s * ) → (1 -θ * , s * ) → (1 -θ * , 1 -ε) → (1, 1 -ε) → (1, 1)

  It is clear that {u n } is bounded in H 1 (R N ) by Lemma 2.1 (ii) and then one may assume that up to a subsequence lim n→∞ R N |∇u n | 2 dx and lim n→∞ R N F (u n )dx exist. Since E m < 0 by (2.6), we deduce that {u n } is non-vanishing, namely

	lim n→∞	sup y∈R N B(y,1)	|u n | 2 dx > 0.	(2.7)

0 and this leads a contradiction since E m * = 0 by (2.6). The rest is to prove that the global infimum E m is achieved when m > m * . Fix m > m * and let {u n } ⊂ S m be any minimizing sequence with respect to E m . Indeed, if (2.7) were not true, then u n → 0 in L 2+4/N (R N ) by Lions Lemma [23, Lemma I.1] and thus lim

Acknowledgements

The authors thank Prof. Thierry Cazenave for useful remarks on a preliminary version of this work. Sheng-Sen Lu acknowledges the support of the China Postdoctoral Science Foundation (No. 2020M680174) and the National Natural Science Foundation of China (Nos. 11771324 and 11831009).

Then the function J µ (γ(t)) = Ψ(η(t)) is strictly increasing on (t 0 , t 1 ), (t 1 , t 2 ) and (t 2 , t 3 ) by (4.5), (4.3) and (4.4) respectively. One may also see that J µ (γ(t)) is constant on (t 3 , t 4 ), (t 4 , t 5 ) by (4.3), and strictly decreasing on (t 5 , t 6 ) and (t 6 , t 7 ) via (4.4) and (4.3) respectively. Moreover, using (4.2),

Finally we observe that the mass function m(t

is strictly increasing and

Since for any M > 0 we can deduce from (4.6) and (4.7) the existence of a large constant

) is a desired one and this completes the proof of Lemma 4.1.

Proof of Theorem 1.6. To prove Item (i), denoting by w ∈ H 1 (R N ) an arbitrary nontrivial critical point of J µ , we only need to show that

For a fixed δ > 0 and M := m > 0, let γ : [0, T ] → H 1 (R N ) be the continuous path given by Lemma 4.1. In view of Lemma 4.1 (i) and (iii), there exists t 0 ∈ (0, T ) such that

and thus

We now prove Item (ii). In view of Item (i), an arbitrary least action solution w ∈ H

Assume by contradiction that w 2 L 2 (R N ) = m. Then, for

we have the continuous path γ : [0, T ] → H 1 (R N ) given by Lemma 4.1. Noting that by Lemma 4.1 (iii) there exists t 0 ∈ (0, T ) such that

it follows from Lemma 4.1 (ii) a contradiction:

Since we have proved w 2 L 2 (R N ) = m, it is easy to see further that I(w) = E m by (4.8).