title: Orbital stability of ground states for a Sobolev critical Schrödinger equation
authors: Louis Jeanjean Jacek Jendrej Thanh Trung Le Nicola Visciglia
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      [Figures/Orbital stability of ground states for a Sobolev critical  Schrödinger equation]Orbital stability of ground states 
       for a Sobolev critical  Schrödinger equationL. Jeanjean, J. Jendrej, T.T. Le and N. Visciglia]Louis Jeanjean, Jacek Jendrej, Thanh Trung Le and Nicola Visciglia
       Louis Jeanjean Laboratoire de Mathématiques (CNRS UMR 6623), Université de Bourgogne Franche-Comté, Besançon 25030, Francelouis.jeanjean@univ-fcomte.fr
       Jacek Jendrej CNRS and LAGA (CNRS UMR 7539), Université Sorbonne Paris Nord, Villetaneuse 93430, Francejendrej@math.univ-paris13.fr
       Thanh Trung Le  Laboratoire de Mathématiques (CNRS UMR 6623), Université de Bourgogne Franche-Comté, Besançon 25030, Francethanh_trung.le@univ-fcomte.fr
       Nicola Visciglia  Dipartimento di Matematica, Università Degli Studi di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italynicola.visciglia@unipi.itJ. Jendrej is supported by ANR-18-CE40-0028 project ESSED and Chilean projects FONDECYT 1170164 and France-Chile ECOS-Sud C18E06 project. N.V. is supported  by PRIN grant 2020XB3EFL and  by the Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituzione Nazionale di Alta Matematica (INDAM)[2010][
          [
          
      =====
      
       Dedicated to the memory of Professor Jean Ginibre 
      	We study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schrödinger equation with mixed power nonlinearities
      	
          i ∂_t v + Δ v + μ v |v|^q-2 + v |v|^2^* - 2 = 0,    (t, x) ∈×,
      
      	where N ≥ 3, v: ×→, μ > 0, 2 < q < 2 + 4/N and 2^* = 2N/(N-2) is the critical Sobolev exponent. We show that all ground states correspond to local minima of the associated Energy functional. Next, despite the fact that the nonlinearity is Sobolev critical, 
      we show that the set of ground states is orbitally stable.
      Our results settle a question raised by N. Soave <cit.>.
       
      
      
      § INTRODUCTION
      
      
      In this paper, we study the existence and orbital stability of ground state standing waves of prescribed mass for the nonlinear Schrödinger equation with mixed power nonlinearities
      
          i ∂_t v + Δ v + μ v |v|^q-2 + v |v|^2^* - 2 = 0,    (t, x) ∈×,
      
      where N ≥ 3, v: ×→, μ > 0, 2 < q < 2 + 4N and 2^* = 2NN-2. 
      
      The nonlinear Schrödinger equation (NLS) with pure and mixed power nonlinearities has attracted much attention in the last decades. The local existence result for the pure power energy critical NLS
      has been established in <cit.>. The corresponding global existence and scattering
      for defocusing quintic NLS in dimension N=3 
      has been established
      in the papers <cit.> respectively 
      in the radial and non-radial case.
      We also quote the concentration-compactness/rigidity approach introduced
      in <cit.> in order to study global existence and scattering in the focusing energy critical NLS
      below the ground state.
      Concerning the case of NLS with mixed nonlinearities let us quote 
      
      <cit.>.
      
      We recall that standing waves to (<ref>) are solutions of the form v(t,x) = e^-iλ tu(x), λ∈. Then the function u(x) satisfies the equation
      
          - u - λ u - μu^q-2 u - u^2^*-2 u = 0   .
      
      
      When looking for solutions to (<ref>) a possible choice is to consider λ∈ fixed and to search for solutions as critical points of the action functional
      
          𝒜_λ, μ(u) :=  12 u2^2 - λ2u2^2    -
          μquq^q - 12^*u2^*^2^*.
      
      In this case one usually focuses on the existence of minimal action solutions, namely of solutions minimizing 𝒜_λ, μ among all non-trivial solutions. In that direction, we refer to <cit.> where, relying on the pioneering work of Brezis-Nirenberg <cit.>, the existence of positive real solutions for equations of the type of (<ref>) is  addressed in a very general setting; to <cit.> which concerns the case where q > 2 + 4/N and μ >0; to <cit.> where the fixed λ∈ problem is analyzed for q = 2 + 4/N and μ <0; 
      
      see also <cit.> and the reference therein. 
      
      Alternatively, one can search for solutions to (<ref>) having a prescribed 𝐿^2-norm. Defining on H:= H^1(^N, ) the Energy functional
      
          F_μ(u) := 12 u2^2 - μquq^q - 12^*u2^*^2^*
      
      it is standard to check that F_μ is of class C^1 and that a critical point of F_μ restricted to the (mass) constraint
      
          S(c) := {u ∈ H: u2^2 = c}
      
      gives rise to a solution to (<ref>), satisfying u2^2 = c.
      
      In this approach the parameter λ∈ arises as a Lagrange multiplier. In particular, λ∈ does depend on the solution and is not a priori given.  This approach, that we shall follow here, is relevant from the physical point of view, in particular, since the L^2 norm is a preserved quantity of the evolution and  since the variational characterization of such solutions is often a strong help to analyze their orbital stability, see for example, <cit.>. 
      
      We shall focus on the existence of ground state solutions. 
      
      
      	We say that  u_c ∈ S(c) is a ground state solution to (<ref>) if it is a solution having minimal Energy among all the solutions which belong to S(c). Namely, if
      	
          F_μ(u_c) =  inf{F_μ(u), u ∈ S(c), (F_μ|_S(c))'(u) = 0 }.
      
      Note that this definition keeps a meaning even in situations where the Energy F_μ is unbounded from below on S(c). Implicit in <cit.>, this definition was formally introduced, on a related model, in 
      <cit.> and is now becoming standard. 
      
      It is well-known that the study of problems with mixed nonlinearities and the type of results one can expect, depend on the behavior of the nonlinearities  at infinity, namely on the value of the various power exponents. In particular, this behavior determines whether the functionnal  is bounded from below on S(c). One speaks of a mass subcritical case if it is bounded from below on S(c) for any c>0, and of a mass supercritical case if the functional is unbounded from below on S(c)  for any c>0.  One also refers to  a mass critical case when the boundedness from below does depend on the value c>0. To be more precise, consider an equation of the form
      
          i ∂_t v + Δ v + μ v |v|^p_1-2 + v |v|^p_2 - 2 = 0,    (t, x) ∈×,
      
      where it is assumed that 2< p_1 ≤ p_2 ≤ 2^*. The threshold exponent is the so-called L^2-critical exponent
      
          p_c = 2 + 4/N.
      
      A very complete analysis of the various cases that may happen for (<ref>), depending on the values of (p_1,p_2), has been provided recently in <cit.>. Let us just recall here some rough elements. If both p_1 and p_2 are strictly less than p_c then the associated Energy functional is bounded from below on S(c) and to find a ground state one looks for a global minimum on S(c). The problem then directly falls into the setting covered by the Compactness by Concentration Principle introduced by P.L. Lions <cit.> which, for more complicated equations, in particular non autonomous ones, is still a very active field. Such solutions are expected to be orbitally stable, see <ref> below.  If p_c ≤ p_1 ≤ p_2 ≤ 2^*, then the  Energy functional is unbounded from below on S(c) but it is possible to show that a ground state exists. This ground state is characterized as a critical point of  mountain-pass type and it lies at a strictly positive level of the Energy functional. Such ground states are expected to be orbitally unstable. We refer, for the link between the variational characterization of a solution and its instability, to the classical paper <cit.>, and to <cit.> for more recent developments. 
      
      In the case we consider here : 2 < p_1 < p_c <p_2 = 2^*, the Energy functional  is thus unbounded from below on S(c) but, as we shall see, the presence of the lower order, mass subcritical term - μuq^q creates, for sufficiently small values of c>0, a geometry of local minima on S(c). The presence of such geometry, in problems which are mass supercritical, had already been observed in several related situations. In <cit.> for related scalar problems, in 
      <cit.> in the case of a system or <cit.> for an evolution problem set on a bounded domain.  Actually, it was already observed on (<ref>) in <cit.>.
      
      Precisely, for any fixed μ >0, we shall find an explicit value c_0 = c_0(μ) >0 such that, for any c ∈ (0, c_0), there exists a set V(c) ⊂ S(c) having the property that
      
          m(c) := inf_u ∈ V(c) F_μ(u) < 0 <  inf_u ∈∂ V(c)F_μ(u).
      
      The sets V(c) and ∂ V(c) are given by
      
          V(c) := { u ∈ S(c) :  u2^2  < ρ_0},     ∂ V(c) := { u ∈ S(c) :  u2^2  = ρ_0}
      
      for a suitable ρ_0 >0, depending only on c_0 >0 but not on c ∈ (0,c_0).
      We also introduce the set
      
          ℳ_c := {u ∈ V(c) : F_μ(u) = m(c)}.
      
      Our first result is,
      
      	Let N ≥ 3, 2 < q < 2 + 4/N. For any μ >0 there exists a c_0 = c_0(μ) >0 such that, for any c ∈ (0, c_0),  F_μ restricted to S(c)
      	has a ground state. This ground state is a (local) minimizer of F_μ in the set V(c) and any ground state for F_μ on S(c) is a local minimizer of F_μ on V(c). In addition, if (u_n) ⊂ V(c) is such that F_μ(u_n) → m(c) then, up to translation,  u_n → u ∈ℳ_c in 𝐻^1(,). 
      
      	The value of c_0 = c_0(μ) >0 is explicit and is given in (<ref>)-(<ref>). In particular c_0 >0 can be taken arbitrary large by taking μ >0 small enough.
      
        (i) If u ∈ S(c) is a ground state then the associated Lagrange multiplier λ∈ in (<ref>) satisfies λ < 0. This follows directly combining that u being a solution to (<ref>) it satisfies u2^2 - λu2^2    -
      μuq^q - u2^*^2^* =0 with the fact that F_μ(u) = m(c) <0.
        (ii) There exists a ground state which is a real valued, positive, radially symmetric decreasing function. Indeed if u ∈ S(c) is a ground state then its Schwartz symmetrization is clearly also a ground state.
      
        (iii) More globally, under the assumption of <ref> it can be proved that, for any c ∈ (0, c_0), ℳ_c has the following structure:
      	
          ℳ_c = { e^i θu, θ∈, u ∈ℳ̃_c, u >0 },
      
      	where
      	
          ℳ̃_c = { u ∈ S(c) ∩ H^1(^N, ), F_μ(u) = m(c)}.
      
      	Indeed, this description directly follows from the convergence, up to translation, of the minimizing sequences of F_μ restricted 
      	to V(c), applying the argument of <cit.>. We leave the details to the interested reader.
      
      
      We shall now focus on the (orbital) stability of the set ℳ_c. 
      Following the terminology of <cit.>, see also <cit.>, we give the following definition.
      Z ⊂ is stable if : Z ≠∅ and for any v ∈ Z and any ε >0, there exists a δ >0 such that if φ∈ satisfies ||φ- v||_ < δ then 
      	 u_φ(t) is globally defined and inf_z ∈ Z ||u_φ(t) - z||_ < ε for all 
      	t ∈, where u_φ(t) is the solution to (<ref>) corresponding to the initial condition φ.
      
      
      Notice that the orbital stability of the set Z implies the global existence 
      of solutions to (<ref>) for initial datum φ close enough to the set Z. We underline that this fact 
      is non trivial due to the critical exponent that appears in (<ref>), even if the H norm of the solution
      is uniformly bounded on the lifespan of the solution.
      
      
      The fact that ground states are characterized as local minima suggests, despite the problem being mass supercritical, that the set ℳ_c could be orbitally stable. Actually, such orbital stability results have now been proved, on related problems (but always Sobolev subcritical) in several recent papers <cit.>.  Along this line we now present the main result of this paper.
      
      	Let N ≥ 3, 2 < q < 2 + 4/N,  μ >0 and  c_0 = c_0(μ) >0 be given in <ref>. Then, for any c ∈ (0, c_0), the set ℳ_c is compact, up to translation, and it is orbitally stable.
      
      
      
      
      
      In <cit.>, Soave studied  equation (<ref>) and derived, for any small c>0 depending on μ>0, an existence result which is very similar to the one contained in <ref>, see <cit.>. Actually, the motivation of our study originated from <cit.> where what is now our <ref> was proposed as an open problem.
      However, it does not seem possible to use <cit.> as a starting point to prove <ref>. The existence of a ground state in <cit.> is obtained through the study of one particular (locally) minimizing sequence which is radially symmetric. As already explained in <cit.>, to obtain the orbital stability of the set ℳ_c, following the classical approach laid down in <cit.>, two ingredients are essential. First, the relative compactness, up to translation, of all minimizing sequences for F_μ on V(c), as guaranteed by our <ref>. Secondly, the global existence of solutions to (<ref>) for initial data close to ℳ_c.
      
      
      To obtain the relative compactness of all minimizing sequences, the fact that one minimizes only on a subset of S(c), in contrast to a global minimization on all S(c), increases the difficulty to rule out a possible  dichotomy. Different strategies have been recently implemented to deal with this issue <cit.>, all relying on a suitable choice of the set where the local minima are searched. In the presence of a Sobolev critical term an additional difficulty arises. In a Sobolev subcritical setting, if a sequence (v_n) ⊂ S(c) is  vanishing then applying <cit.> one would immediately get 
      
          lim inf_n →∞ F_μ(v_n) = lim inf_n →∞1/2||v_n||_2^2 ≥ 0.
      
      Thus the  vanishing can directly be ruled out knowing that m(c) <0. Here <cit.> does not apply anymore; the term ||v_n||_2^* may not go to 0 if (v_n) is  vanishing. Thus we need a better understanding of this possible loss of compactness and this leads to our definition of the set V(c).
      
      As to the global existence of solutions to (<ref>), it is also affected by the presence of the Sobolev critical exponent. In Sobolev subcritical cases, it is well known <cit.> that if, for an initial datum φ∈ H, the maximum time of existence T_φ^max>0 is finite then necessarily the corresponding solution v satisfies ||∇ v(t)||_2 → + ∞ as t → T_φ^max. Thus, a uniform a priori bound on ||∇ v(t)||_2 yields global existence. Note that, by conservation of the Mass and Energy, in view of (<ref>),
      for an initial datum in V(c) ∩{u ∈ S(c) : F_μ(u) <0}, the evolution takes place in the (bounded) set V(c). Thus, in a subcritical setting, the global existence would follow directly. However, in our case it is unknown if the previous blow-up alternative holds and hence, we cannot deduce global existence just since the evolution takes place in V(c), see <cit.> or <cit.> for more details.  To overcome this difficulty, building on the pioneering work of Cazenave-Weissler <cit.>, see also <cit.> , we first derive an upper bound on the propagator e^it Δ which provides a kind of uniform local existence result, see <ref>. Next, using the information that all minimizing sequences are, up to translation, compact and also specifically and crucially that ℳ_c is compact, up to translation, we manage to show that, for initial data sufficiently close to the set ℳ_c the global existence holds and this leads to the orbital stability of ℳ_c, proving <ref>.  
      
      
      We point out that, in order to prove <ref>, we have only established the global existence of solutions for initial data  close to ℳ_c. We believe it would be interesting to inquire if the global existence holds  away from ℳ_c, typically for any initial data in V(c) ∩{u ∈ S(c) : F_μ(u) <0}. If so, investigating the long time behavior of these solutions would be worth to. Our guess is that these solutions evolve toward the sum of an element of ℳ_c and a part which scatter. However, so far nothing is known in that direction. 
      
      The paper is organized as follows. <ref>, is devoted to clarifying the local minima structure and to establish the convergence, up to translation, of all minimizing sequences for F_μ on V(c). The proof of <ref> is then given. In <ref>, we establish <ref>. Finally, in <ref>, we prove <ref> which states the orbital stability of the set ℳ_c.  Notation : We write  for 𝐻^1(,).
      
      For p ≥ 1, the 𝐿^p-norm of u ∈ (or of u ∈) is denoted by up.
      	
      	
      	
      
      
      
      
      
      
      
      § THE VARIATIONAL PROBLEM
      
      
      
      
      We shall make use of the following classical inequalities :
      For any N ≥ 3  there exists an optimal constant
      𝒮 > 0 depending only on N, such that
      
          𝒮f_2^*^2≤ f_2^2 ,     ∀ f ∈ H,
      
      see <cit.>. 
      If
      N ≥ 2 and p ∈ [2, 2N/N-2) then
      
          f_p≤ C_N,p f_2^βf_2^(1-β),     β = N12 - 1p
      
      for all f ∈ H, see <cit.>.
      Now, letting
      
          α_0 := N(q-2)2 -2,     α_1:= 2N - q(N-2)2,     α_2: = 4N-2,
      
      we consider the function f(c,ρ) defined on (0, ∞) × (0, ∞) by
      
          f(c,ρ) = 1/2 - μ/q C_N,q^q ρ^α_0/2 c^α_1/2 - 1/2^*1𝒮^2^*/2ρ^α_2/2,
      
      and, for each c ∈ (0, ∞), its restriction g_c(ρ) defined on (0, ∞) by ρ↦ g_c(ρ) := f(c, ρ).
      
      For future reference, note that for any N ≥ 3,  α_0 ∈ (-2,0), 
      α_1 ∈
          4N, 2andα_2 ∈ (0,4].For each c > 0, the function g_c(ρ) has a unique global maximum and the maximum value satisfies
          	
              max_ρ > 0 g_c(ρ) > 0      c < c_0,
              max_ρ > 0 g_c(ρ) = 0      c = c_0,
              max_ρ > 0 g_c(ρ) < 0      c > c_0,
          
          	where
          	
              c_0 := 12K^N/2 > 0,
          
          	with
          	
              K := μ/q C_N,q^q 
                  - α_0α_2μ C_N,q^q 2^* 𝒮^2^*/2/q
              ^α_0/α_2 - α_0 + 1/2^*1𝒮^2^*/2
                  - α_0α_2μ C_N,q^q 2^* 𝒮^2^*/2/q
              ^α_2/α_2 - α_0 > 0.
          
          	By definition of g_c(ρ), we have that
          	
              g_c'(ρ) = - α_02μ/q C_N,q^q ρ^α_0/2 -1 c^α_1/2
              	- α_221/2^*1𝒮^2^*/2ρ^α_2/2 -1 .
          
          	Hence, the equation g_c'(ρ) = 0 has a unique solution given by
          	
              ρ_c = 
                  - α_0α_2μ C_N,q^q 2^* 𝒮^2^*/2/q
              ^2/α_2 - α_0  c^α_1/α_2 - α_0.
          
          	Taking into account that g_c(ρ) → -∞ as ρ→ 0 and g_c(ρ) → -∞ as ρ→∞, we obtain that ρ_c is the unique global maximum point of g_c(ρ) and the maximum value is
          	
              max_ρ > 0 g_c(ρ)    = 1/2 - μ/q C_N,q^q 
                  - α_0α_2μ C_N,q^q 2^* 𝒮^2^*/2/q
              ^α_0/α_2 - α_0  c^α_0α_1/2(α_2 - α_0) c^α_1/2 
              	 - 1/2^*1𝒮^2^*/2
                  - α_0α_2μ C_N,q^q 2^* 𝒮^2^*/2/q
              ^α_2/α_2 - α_0  c^α_1α_2/2(α_2 - α_0)
                 = 1/2 - μ/q C_N,q^q 
                  - α_0α_2μ C_N,q^q 2^* 𝒮^2^*/2/q
              ^α_0/α_2 - α_0  c^α_1α_2/2(α_2 - α_0) 
              	 - 1/2^*1𝒮^2^*/2
                  - α_0α_2μ C_N,q^q 2^* 𝒮^2^*/2/q
              ^α_2/α_2 - α_0  c^α_1α_2/2(α_2 - α_0)
                 = 12 - K c^2/N.
          
          	By the definition of c_0, we have that max_ρ > 0 g_c_0(ρ) = 0, and hence the lemma follows.
          
          	Let (c_1, ρ_1) ∈  (0, ∞) × (0, ∞) be such that f(c_1, ρ_1) ≥ 0. Then for any c_2 ∈ (0,c_1], we have that
          	
              f(c_2, ρ_2) ≥ 0     ρ_2 ∈
                  c_2/c_1ρ_1, ρ_1
              .
          
          	Since c → f(·, ρ) is a non-increasing  function we clearly have that 
          	
              f(c_2, ρ_1) ≥ f(c_1, ρ_1) ≥ 0.
          
          	Now taking into account that α_0 + α_1 = q-2 > 0 we have, by direct calculations, that 
          	
              fc_2, c_2c_1ρ_1≥ f(c_1, ρ_1) ≥ 0.
          
          	We observe that if g_c_2(ρ') ≥ 0 and g_c_2(ρ”) ≥ 0 then
          	
              f(c_2, ρ) = g_c_2(ρ) ≥ 0      ρ∈ [ρ', ρ”].
          
          	Indeed, if g_c_2(ρ) < 0 for some ρ∈ (ρ', ρ”) then there exists a local minimum point on (ρ_1, ρ_2) and this contradicts the fact that the function g_c_2(ρ)  has a unique  critical point
          	which has to coincide necessarily with its unique global maximum  (see <ref>).
          	By (<ref>), (<ref>), we can choose ρ' =  (c_2/c_1) ρ_1 and ρ” = ρ_1, and  (<ref>) implies the lemma.
           For any u ∈ S(c),  we have that
          	
              F_μ(u) ≥ u2^2 f(c,  u2^2).
          
          	Applying the Gagliardo-Nirenberg inequality (<ref>)
          	and the Sobolev inequality (<ref>) we obtain that, for any u ∈ S(c),
          	
              F_μ(u)    = 1/2 u2^2 - μ/quq^q - 1/2^*u2^*^2^*≥1/2 u2^2 - μ/q C_N,q^q  u_2^α_0+2u_2^α_1 - 1/2^*1𝒮^2^*/2 u2^2^*
                 =  u2^2 
                  1/2 - μ/q C_N,q^q  u_2^α_0 u_2^α_1 - 1/2^*1𝒮^2^*/2 u2^α_2
              
              	=  u2^2 f( u_2^2,  u2^2).
          
          	The lemma is proved.
          
          
          
          Now letc_0>0be given by (<ref>)  andρ_0 := ρ_c_0 >0being determined by (<ref>). Note that by <ref> and <ref>, we have thatf(c_0, ρ_0) = 0andf(c, ρ_0) > 0for allc ∈ (0, c_0). 
          We define
          
              B_ρ_0 := {u ∈:  u2^2 < ρ_0}     V(c) := S(c) ∩ B_ρ_0.
          
          We shall now consider the following local minimization problem:  for anyc ∈ (0, c_0),
          
              m(c) := inf_u ∈ V(c) F_μ(u).
           For any c ∈ (0, c_0), the following properties hold, 
          	
            * m(c) = inf_u ∈ V(c) F_μ(u) < 0 <  inf_u ∈∂ V(c)F_μ(u).
            *  If m(c) is reached, then any ground state is contained in V(c).
          	
          	(<ref>) For any u ∈∂ V(c) we have u2^2 = ρ_0. Thus, using <ref>, we get
          	
              F_μ(u)  ≥ u2^2 f( u_2^2,  u2^2)
              	= ρ_0 f(c, ρ_0) >0.
          
          	Now let u ∈ S(c) be arbitrary but fixed. For s ∈ (0, ∞) we set
          	
              u_s(x) := s^N/2 u(s x).
          
          	Clearly u_s ∈ S(c) for any s ∈ (0, ∞). We define on (0, ∞) the map,
          	
              ψ_u(s) := F_μ (u_s) = s^22 u2^2 - μq s^N(q-2)/2uq^q - s^2^*2^*u2^*^2^*.
          
          	Taking into account that
          	
              N(q-2)2 < 2          2^* > 2,
          
          	we see that ψ_u(s) → 0^-, as s → 0. Therefore, there exists s_0 >0 small enough such that (u_s_0)2^2 = s_0^2  u2^2 < ρ_0 and F_μ (u_s_0) = ψ_μ(s_0) < 0. This implies that m(c) < 0.
          	
          	(<ref>) It is well known, see for example <cit.>, that all critical points of F_μ restricted to S(c) belong to the Pohozaev's type set 
          	𝒫_c := { u ∈ S(c) : 𝒫(u)=0 }
          	where
          	
              𝒫(u):=  u2^2 - μ N(q-2)2quq^q - u2^*^2^*.
          
          	Also a direct calculation shows that, for any v ∈ S(c) and any s ∈ (0, ∞), 
          	
              ψ_v'(s) = 1/s𝒫(v_s).
          
          	Here ψ_v' denotes the derivative of ψ_v with respect to s ∈ (0, ∞). Finally, observe that any u ∈ S(c) can be written as u = v_s with v ∈ S(c), ||∇ v||_2 =1 and s ∈ (0, ∞).
          	
          Since the set 𝒫_c contains all the ground states (if any), we deduce from (<ref>) that if w ∈ S(c) is a ground state there exists a v ∈ S(c), ||∇ v||_2^2=1 and a s_0 ∈ (0, ∞) such that w= v_s_0, F_μ(w) = ψ_v(s_0) and ψ_v'(s_0)=0. Namely, s_0 ∈ (0, ∞) is a zero of the function ψ_v'.
          	
          Now, since ψ_v(s) → 0^-, ||∇ v_s||_2 → 0, as s → 0 and ψ_v(s) = F_μ(v_s) ≥ 0 when v_s ∈∂ V(c) = {u ∈ S(c) : ||∇ u||_2^2 = ρ_0 }, necessarily ψ_v' has a first zero 
          s_1 >0 corresponding to a local minima. In particular,  v_s_1∈ V(c) and F(v_s_1) = ψ_v(s_1) <0. Also, from ψ_v(s_1) <0,ψ_v(s) ≥  0 when v_s ∈∂ V(c) and ψ_v(s) → - ∞ as s →∞,  ψ_v has a second zero s_2 >s_1 corresponding to a local maxima of ψ_v. Since v_s_2 satisfies F(v_s_2)= ψ_v(s_2) ≥ 0, we have that m(c) ≤ F(v_s_1) < F(v_s_2). In particular, since m(c) is reached, v_s_2 cannot be a ground state.
          
          To conclude the proof of (ii) it then just suffices to show that ψ_v' has at most two zeros, since this will imply s_0 = s_1 and w = v_s_0 = v_s_1∈ V(c). However, this is equivalent to showing that the function
          	s ↦ψ'_u(s)/s
          	has at most two zeros. We have
          	θ(s) := ψ'_u(s)/s =  u2^2 - μ N(q-2)2q s^α_0uq^q - s^α_2u2^*^2^*
          	and
          	
              θ'(s) = - α_0 μ N(q-2)2q s^α_0 - 1uq^q - α_2 s^α_2-1u2^*^2^*.
          
          	Since α_0 <0 and α_2 >0, the equation θ'(s) =0 has a unique solution, and θ(s) has indeed at most two zeros.
          
          
          We now introduce the set
          
              ℳ_c := {u ∈ V(c) : F_μ(u) = m(c)}.
          
           The main aim of this section is the following result.
          
          	For any c ∈ (0, c_0), if (u_n) ⊂ B_ρ_0 is such that u_n2^2 → c and F_μ(u_n) → m(c) then, up to translation,  u_n  H→ u ∈ℳ_c. In particular the set ℳ_c is compact in H, up to translation.
          <ref> will both imply the existence of a ground state but also, as it may be expected, will be a crucial step to derive the orbital stability of the setℳ_c.
          
          
          
          
          In order to prove <ref> we collect some properties ofm(c)defined in (<ref>).
          
          	It holds that
          	
            * c ∈ (0,c_0) ↦ m(c) is a continuous mapping.
          		
            *  Let c ∈ (0,c_0). We have for all α∈ (0,c) : m(c) ≤ m(α) + m(c-α) and if m(α) or m(c-α) is reached  then  the inequality is strict.
          	
          	(<ref>) Let c ∈ (0, c_0) be arbitrary and (c_n) ⊂ (0, c_0) be such that c_n → c. From the definition of m(c_n) and since m(c_n) <0, see <ref> (<ref>),
          	for any ε >0 sufficiently small, there exists u_n ∈ V(c_n) such that
          	
              F_μ(u_n) ≤ m(c_n) + ε     F_μ(u_n) <0.
          
          	We set y_n := √(c/c_n) u_n and hence y_n ∈ S(c). We have that y_n ∈ V(c). Indeed, if c_n ≥ c, then
          	
              y_n2^2 = c/c_n u_n2^2 ≤ u_n2^2 < ρ_0.
          
          	If c_n < c, by <ref>, we have f(c_n, ρ) ≥ 0 for any ρ∈
              c_ncρ_0, ρ_0. Hence, we deduce from <ref> and (<ref>) that f(c_n, ∇ u_n_2^2) < 0, thus ∇ u_n_2^2 < c_n/cρ_0 and
          	
              y_n2^2 = c/c_n u_n2^2 < c/c_nc_ncρ_0 = ρ_0.
          
          	Since y_n ∈ V(c) we can write
          	
              m(c) ≤ F_μ(y_n) = F_μ(u_n) + [F_μ(y_n) - F_μ(u_n)]
          
          	where
          	
              F_μ(y_n) - F_μ(u_n) = - 1/2(c/c_n-1) u_n2^2 - μ/q[ (c/c_n)^q/2 - 1 ]  u_nq^q -  1/2^* [ (c/c_n)^2^*/2 - 1 ] u_n2^*^2^*.
          
          	Since u_n2^2 < ρ_0, also u_nq^q and u_n2^*^2^* are uniformly bounded. 
          	Thus, as n →∞ we have
          	
              m(c) ≤ F_μ(y_n) = F_μ(u_n) + o_n(1).
          
          	Combining (<ref>) and (<ref>), we get
          	
              m(c) ≤ m(c_n) + ε + o_n(1).
          
          	Now, let u ∈ V(c) be such that
          	
              F_μ(u) ≤ m(c) + ε     F_μ(u) <0.
          
          	Set u_n := √(c_n/c) u and hence u_n ∈ S(c_n).
          	Clearly, ∇ u_2^2 < ρ_0 and c_n → c imply ∇ u_n_2^2 < ρ_0 for n large enough,
          	so that u_n ∈ V(c_n). Also, F_μ(u_n) → F_μ(u). We thus have
          	
              m(c_n) ≤ F_μ(u_n) = F_μ(u) + [F_μ(u_n) - F_μ(u)] ≤ m(c) + ε + o_n(1).
          
          
          	Therefore, since ε > 0 is arbitrary, we deduce that m(c_n) → m(c). The point (<ref>) follows.
          	
          	(<ref>) Note that, fixed α∈ (0,c), it is sufficient to prove that the following holds
          	
              ∀θ∈1, c/α :  m(θα) ≤θ m(α)
              	
              	and that, if m(α) is reached, the inequality is strict. Indeed, if (<ref>) holds then we have
              	
                  m(c)=c-α/cm(c)+α/cm(c)=c-α/cm( c/c-α(c-α) )+α/cm( c/αα) ≤ m(c-α)+m(α),
              
              	with a strict inequality if m(α) is reached. To prove that (<ref>) holds, note that in view of <ref> (<ref>), for any ε >0 sufficiently small, there exists a u ∈ V(α) such that
              	
                  F_μ(u) ≤ m(α) + ε     F_μ(u) <0.
              
              	In view of <ref>,  f(α, ρ) ≥ 0 for any ρ∈
                  αcρ_0, ρ_0.
              	Hence, we can deduce from <ref> and (<ref>) that
              	
                  ||∇ u||_2^2 < α/cρ_0.
               
              	Consider now v = √(θ) u.  We first note that ||v||_2^2 = θ ||u||_2^2 = θα and also, because of (<ref>),
              	||∇ v||_2^2 = θ ||∇ u||_2^2 < ρ_0.
              	Thus v ∈ V(θα) and we can write
              	
                  m(θα)    ≤ F_μ(v) = 12θ u2^2 - μ/qθ^q/2uq^q - 12^*θ^2^*/2u2^*^2^*  
                  	< 12θ u2^2 - μ/qθuq^q - 12^*θu2^*^2^*
                     = θ12 u2^2 - μ/quq^q - 12^*u2^*^2^* 
                  	= θ F_μ(u) ≤θ (m(α) + ε).
              
              	Since ε > 0 is arbitrary, we have that  m(θα) ≤θ m(α). If m(α) is reached then we can let ε = 0 in (<ref>) and thus the strict inequality follows.
              
              	Let (v_n) ⊂ B_ρ_0 be such that v_nq→ 0. Then there exists a β_0 > 0 such that
              	
                  F_μ(v_n) ≥β_0 ||∇ v_n||_2^2 + o_n(1).
              
              	Indeed, using the Sobolev inequality (<ref>), we obtain that
              	
                  F_μ(v_n)    = 1/2 v_n2^2  - 1/2^*v_n2^*^2^* + o_n(1)  
                  	≥1/2 v_n2^2  - 1/2^*1𝒮^2^*/2 v_n2^2^* + o_n(1) 
                     =  v_n2^2 
                      1/2  - 1/2^*1𝒮^2^*/2 v_n2^α_2
                   + o_n(1) 
                  	 ≥ v_n2^2 
                      1/2  - 1/2^*1𝒮^2^*/2ρ_0^α_2/2
                   + o_n(1).
              
              	Now, since f(c_0,ρ_0) = 0, we have that
              	
                  β_0 := 
                      1/2  - 1/2^*1𝒮^2^*/2ρ_0^α_2/2
                   = μ/q C_N,q^q ρ_0^α_0/2 c_0^α_1/2 > 0.
              
              	For any c ∈ (0, c_0), let (u_n) ⊂ B_ρ_0 be such that u_n2^2 → c and F_μ(u_n) → m(c). Then, there exist a β_1 > 0 and a sequence (y_n) ⊂^N such that
              	
                  ∫_B(y_n, R)u_n^2 dx ≥β_1 > 0,     for some  R > 0.
              
              	We assume by contradiction that (<ref>) does not hold. Since (u_n) ⊂ B_ρ_0 and u_n2^2 → c, the sequence (u_n) is bounded in H. From <cit.> and since  2 < q < 2^*, we deduce that u_nq→ 0, as n →∞.
              	At this point, <ref> implies that F_μ (u_n) ≥ o_n(1). This contradict the fact that m(c) <0 and the lemma follows.
              
              	We know from <ref>  and Rellich compactness theorem that there exists a sequence (y_n) ⊂^N such that 
              	
                  u_n(x - y_n)  u_c ≠ 0   in .
              
              	Our aim is to prove that w_n(x) := u_n(x - y_n) - u_c(x) → 0 in . Clearly
              	
                  u_n2^2    = u_n(x-y_n)2^2
                  	= u_n(x-y_n) - u_c(x)2^2 + u_c2^2 + o_n(1)
                     =w_n2^2 + u_c2^2 + o_n(1).
              
              	Thus, we have
              	
                  w_n2^2 = u_n2^2 -  u_c2^2 + o_n(1) = c - u_c2^2 + o_n(1).
              
              	By a similar argument, 
              	
                  w_n2^2 =  u_n2^2 -   u_c2^2 + o_n(1).
              
              	More generally, taking into account that any term in F_μ fulfills the splitting properties of Brezis-Lieb <cit.>, we have
              	
                  F_μ(w_n ) + F_μ(u_c) =F_μ(u_n(x-y_n)) + o_n(1),
              
              	and, by the translational invariance, we obtain
              	
                  F_μ(u_n) =F_μ(u_n(x-y_n))
                  	=F_μ(w_n) + F_μ(u_c) + o_n(1).
              
              	Now, we claim that
              	
                  w_n2^2 → 0.
              
              	In order to prove this, let us denote c_1:= u_c2^2 > 0. By (<ref>), if we show that c_1 = c then the claim follows. We assume by contradiction that c_1 < c.
              	In view of (<ref>) and (<ref>), for n large enough, we have w_n2^2 ≤ c and w_n2^2 ≤ u_n2^2 < ρ_0. Hence, we obtain that w_n ∈ V(w_n2^2) and F_μ(w_n) ≥ mw_n2^2.
              	Recording that F_μ(u_n) → m(c), in view of (<ref>), we have
              	
                  m(c) = F_μ(w_n) + F_μ(u_c) + o_n(1) ≥ mw_n2^2 + F_μ(u_c) + o_n(1).
              	
              	Since the map c ↦ m(c) is continuous (see <ref>(<ref>)) and in view of (<ref>), we deduce that
              	
                  m(c) ≥ m(c-c_1) + F_μ(u_c).
              
              	We also have that u_c ∈ V(c_1) by the weak limit. This implies that F_μ(u_c) ≥ m(c_1). If F_μ(u_c) > m(c_1), then it follows from (<ref>) and <ref>(<ref>) that
              	
                  m(c) > m(c-c_1) + m(c_1) ≥ m(c -c_1 + c_1) = m(c),
              
              	which is impossible. Hence, we have F_μ(u_c) = m(c_1), namely u_c is a local minimizer on V(c_1). So,  using <ref>(<ref>) with the strict inequality, we deduce from (<ref>) that
              	
                  m(c) ≥ m(c-c_1) + F_μ(u_c) = m(c-c_1) + m(c_1) > m(c -c_1 + c_1) = m(c),
              
              	which is impossible. Thus, the claim (<ref>) follows and from (<ref>) we deduce that
              	u_c2^2 = c.
              	
              	Let us now show that ||∇ w_n||_2^2 → 0. This will prove that w_n → 0 in  and completes the proof.
              	In this aim first observe that 
              	 in view of (<ref>) and since u_c ≠ 0,  we have w_n2^2 ≤ u_n2^2 < ρ_0, for n large enough.
              	Hence (w_n) ⊂ B_ρ_0 and in particular it is bounded in H. Then by using the Gagliardo-Nirenberg inequality (<ref>), and by recalling ||w_n||_2^2 → 0 we also have ||w_n||_q^q → 0.
              	Thus  
              	<ref> implies that 
              	
                  F_μ (w_n) ≥β_0  w_n2^2+o_n(1)   β_0 > 0.
              
              	  Now we remember that
              	
                  F_μ(u_n) = F_μ(u_c) + F_μ(w_n) + o_n(1) → m(c).
               
              	Since u_c ∈ V(c) by weak limit, we have that  F_μ(u_c) ≥ m(c) and hence
              	F_μ(w_n) ≤ o_n(1). In view of (<ref>), we then conclude that w_n2^2 → 0. 
              
              	
              	
              
              
              We end this section with,
              
              The fact that if (u_n) ⊂ V(c) is such that F_μ(u_n) → m(c) then, up to translation, u_n → u ∈ℳ_c in H follows from <ref>.  In particular, it insures the existence of a minimizer for F_μ on V(c). The fact that this minimizer is a ground state and that any ground state for F_μ on S(c) belongs to V(c) was proved in <ref>.
              
              
              § A CONDITION FOR A UNIFORM LOCAL EXISTENCE RESULT
              
              In this section we focus  on the local existence of solutions to the following Cauchy problem 
              
                  i ∂_t u + Δ u + μ u |u|^q-2 + u |u|^2^* - 2 = 0,    (t, x) ∈×,    N≥ 3 
                  
                  u(0, x) = φ(x) ∈.
              
              Denoting g: → by g(u) := μ u |u|^q-2 + u |u|^2^* - 2,  (<ref>) reads as
              
                  i∂_t u + Δ u + g(u) = 0.
              
              
               Next we give the notion of integral equation associated with (<ref>).
              In order to do that first we give another definition.
              
              	If N≥ 3 the pair (p, r) is said to be (Schrödinger) admissible if
              	
                  2p + Nr = N2,      p, r ∈ [2, ∞].
              
              We shall work with two particular admissible pairs (see <ref>):
              
                  (p_1, r_1) := 4q(q - 2)(N-2), Nqq + N -2,
              
              and
              
                  (p_2, r_2) := 4 × 2^*(2^* - 2)(N-2), N × 2^*2^* + N -2.
              
              Along with those couples we introduce the spaces Y_T:= Y_p_1, r_1, T∩ Y_p_2, r_2, T and X_T:= X_p_1, r_1, T∩ X_p_2, r_2, T equipped with the following norms:
              
                  w_Y_T = w_Y_p_1, r_1, T + w_Y_p_2, r_2, T,     w_X_T = w_X_p_1, r_1, T + w_X_p_2, r_2, T.
              
              where for a generic function w(t,x) defined on the time-space strip [0, T)× we have defined:
              
                  ||w(t, x)||_Y_p,r,T = ∫_0^Tw(t, ·)r^p dt^1/p 
                  and 
                  
                  ||w(t, x)||_X_p,r,T = ∫_0^T ||w(t, ·)||_W^1,r()^p dt^1/p.
              
              	Let T >0.
              	We say that u(t,x) is an integral solution of the Cauchy problem (<ref>) on the time interval [0, T] if:
              	
                * u ∈𝒞([0, T], ) ∩ X_T;
              		
                *  for all t ∈ (0, T) it holds
              		u(t) = e^itΔφ - i∫_0^t e^i(t-s)Δg(u(s))ds.
              
              
              The main result of this section is the following local existence result. We do not claim a real originality here, related versions already exist in the literature, see for example<cit.>. However, we believe convenient to the reader to provide a version specifically adapted to our problem and to give a proof of this result as self-contained as possible.
              
              
              	There exists γ_0  > 0 such that if φ∈ and  T ∈ (0, 1] satisfy
              	
                  e^itΔφ_X_T≤γ_0,
              
              	then there exists a unique integral solution u(t,x) to (<ref>) on the time interval [0, T]. Moreover
              	u(t,x)∈ X_p,r, T for every admissible couple (p,r) and satisfies the following conservation laws:
              	
                  F_μ(u(t)) =  F_μ(φ),   u(t)2 = φ2,   t ∈ [0, T].
              
              
              
              In order to prove <ref> we need some preliminary results. 
              
              
              
              
              Let us recall Strichartz's estimates that will be useful in the sequel (see for example <cit.> and <cit.> for the endpoint estimates). 
                Let N≥ 3 then 
              	for every admissible pairs (p, r) and (p̃, r̃), there exists a constant C> 0 such that for every T>0, the following properties hold:
              	
                (i) For every φ∈2, the function t ↦ e^itΔφ belongs to Y_p, r, T∩𝒞([0,T], 2) and
              		
                  *e^itΔφ_Y_p,r,T≤ C φ_2.
              
                (ii) Let F ∈ Y_p̃ ', r̃', T, where we use a prime to denote conjugate indices. Then the function
              		
                  t ↦Φ_F(t):= ∫_0^te^i(t-s)ΔF(s) ds
              
              		belongs to Y_p, r, T∩𝒞([0,T], 2) and 
              		
                  *Φ_F_Y_p,r,T≤ C F_Y_p̃ ', r̃', T.
              
                (iii) For every φ∈, the function t ↦ e^itΔφ belongs to X_p, r, T∩𝒞([0,T], ) and
              		
                  *e^itΔφ_X_p,r,T≤ C φ_.
              
              
              The following result will be useful in the sequel.
              
              	Let N≥ 3 and  2 < α≤ 2^* be given. Then the couple (p,r) defined as follows
              	
                  p := 4α(α - 2)(N-2)         r:= Nαα + N -2
              
              	 is admissible. 
              	Moreover for every admissible couple (p̃, r̃) there exists a constant C > 0 
              	such that for every T >0 the following inequalities hold:
              	
                  *∫_0^te^i(t-s)Δ [∇ g_α(u(s)) ]ds_Y_p̃,r̃,T   ≤ C T^μ∇ u_Y_p,r,T^α-1, 
                  *∫_0^te^i(t-s)Δ [g_α(u(s)) - g_α(v(s))] ds_Y_p̃,r̃,T   ≤ C T^μ (∇ u_Y_p,r,T^α-2 + ∇ v_Y_p,r,T^α-2)  u-v_Y_p,r,T,
              
              	where g_α(u) := u|u|^α - 2 and
              	μ := (N-2)(2^*-α)4≥ 0.	
              	By direct calculations, one can check that
              	
                  2p + Nr = N2     p, r ≥ 2.
              
              	Hence, (p, r) is an admissible pair. Also it is easy to check that there exists a C > 0 such that :
              	
                  |g_α'(u)|    ≤ C|u|^α - 2, 
                  
                  	|g_α(u) - g_α(v)|    ≤ C|u - v|(|u|^α - 2 + |v|^α - 2).
              
              	Combining (<ref>) and the Chain Rule, gives
              	
                  |∇ g_α(u) | = |g_α'(u)∇ u| ≤ C |∇ u| |u|^α - 2.
              
              	Using Hölder's inequality,  we obtain that
              	
                  ∇ g_α(u)r'≤ C |∇ u| |u|^α - 2r'≤ C |∇ u|rur^*^α - 2≤ C ∇ u ^α - 1_r,
              
              	where we also used the Sobolev embedding of W^1,r(^N) into L^r^*(^N) with r^* := NrN-r, see <cit.>.
              	Hence, using  Hölder's inequality,
              	
                  ∇ g_α(u)_Y_p', r', T   = ∫_0^T∇ g_α(u)r'^p' dt^1/p'≤ C∫_0^T  || ∇ u ||^(α - 1)p'_r dt^1/p'
                     ≤ C T^(α-1)1/(α - 1) p' - 1/p∫_0^T  || ∇ u ||^p_r dt^α - 1/p
                  	= C T^μ∇ u_Y_p,r,T^α-1.
              
              	At this point (<ref>) follows by applying <ref> (ii). To establish (<ref>) note that by   (<ref>) and the Hölder's inequality, we have
              	
                  g_α(u) - g_α(v)r'≤ C |u - v|(|u|^α - 2 + |v|^α - 2)r'≤ C u-vr|u| + |v|r^*^α -2.
              
              	Hence, we can deduce that
              	
                  g_α(u) - g_α(v)_Y_p', r', T   = ∫_0^Tg_α(u) - g_α(v)r'^p' dt^1/p'≤ C ∫_0^Tu-vr^p'|u| + |v|r^*^(α -2)p' dt^1/p'
                     ≤ C ∫_0^Tu-vr^p dt^1/p∫_0^T|u| + |v|r^*^(α -2)pp'/p-p' dt^p-p'/pp'≤ C T^μu-v_Y_p,r,T∫_0^T|u| + |v|r^*^p dt^α-2/p
                     = C T^μ|u| + |v|_Y_p,r^*,T^α-2u-v_Y_p,r,T≤ C T^μu_Y_p,r^*,T + v_Y_p,r^*,T^α-2u-v_Y_p,r,T.
              
              	The inequality (<ref>) follows by applying the previous Sobolev embedding and <ref> (ii).
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              
              In order to prove <ref> we shall need two lemmas from Functional Analysis.
              
              	For all 1 < p, r < ∞, X_p, r, T is a separable reflexive Banach space.
              
              	This is a direct consequence of Phillips' theorem, see <cit.>.
              
              	For all R, T > 0 the metric space (B_R,T, d), where
              	
                  B_R,T := {u ∈ X_T: u_X_T≤ R},
              
              	and
              	
                  d(u, v) := u - v_Y_T
              
              	is complete.
              
              	Let (u_n) be a Cauchy sequence. Since Y_T is a Banach space, there exists u ∈ Y_T such that
              	
                  lim_n→∞u_n - u_Y_T = 0.
              
              	It remains to show that u ∈ B_R,T.
              	
              	By taking a subsequence, we can assume that l_1 := lim_n→∞u_n_X_p_1, r_1, T
              	and l_2 := lim_n→∞u_n_X_p_2, r_2, T exist.
              	By <ref>, there exists a subsequence of (u_n) which converges weakly in X_p_1, r_1, T.
              	In particular, this sequence converges in the sense of distributions and hence
              	the limit equals u. Thus,
              	
                  u_X_p_1, r_1, T≤ l_1.
              
              	Similarly,
              	
                  u_X_p_2, r_2, T≤ l_2.
              
              	Taking the sum, we get u_X_T≤ l_1 + l_2 ≤ R.	
              Step 1. Existence and uniqueness in B_2γ_0,T for γ_0 small enough.
              	For any u ∈ X_T and t ∈ [0, T], we define
              	
                  Φ(u)(t) := e^itΔφ + i∫_0^te^i(t-s)Δg(u(s))ds.
              
              	We claim that, if γ_0 >0 is small enough, then Φ defines a contraction on 
              	the metric space (B_2γ_0,T, d) (see  <ref>).
              	
              	Let u ∈ B_2γ_0, T and consider any admissible pair (p̃, r̃). Let T ∈ (0,1] and apply <ref>. We deduce from (<ref>) and (<ref>) that
              	
                  ∇Φ(u)-e^it∇φ_Y_p̃, r̃,T≤ C ∇ u_Y_p_1,r_1,T^q-1 + C ∇ u_Y_p_2,r_2,T^2^*-1≤ C 2^q γ_0^q - 1,   ∀ u∈ 
                  	B_2γ_0, T .
              
              	Similarly, we deduce from (<ref>) (applied with v = 0) that
              	
                  Φ(u)- e^itφ_Y_p̃, r̃,T≤ C ∇ u_Y_p_1,r_1,T^q-2u_Y_p_1,r_1,T + C ∇ u_Y_p_2,r_2,T^2^*-2u_Y_p_2,r_2,T≤ C 2^q γ_0^q-1,   ∀ u∈ 
                  	B_2γ_0, T .
              
              	In particular if we choose (p̃, r̃)=(p_1, r_1) and (p̃, r̃)=(p_2, r_2) then
              	
                  Φ(u)_X_T≤γ_0 + C 2^q γ_0^q-1
              
              	and hence if γ_0 >0 is small enough in such a way that 
              	C 2^q+2γ_0^q-1≤γ_0, then B_2γ_0,T is an invariant set of Φ.
              	
              	Now, let u, v ∈ B_2γ_0, T. By (<ref>), we have for every admissible pair (p̃, r̃)
                  Φ(u) - Φ(v)_Y_p̃, r̃, T   ≤ C(∇ u_Y_p_1,r_1,T^q-2 + ∇ v_Y_p_1,r_1,T^q-2)u-v_Y_p_1, r_1, T 
                  	+ C(∇ u_Y_p_2,r_2,T^2^*-2+ ∇ v_Y_p_2,r_2,T^2^*-2)u-v_Y_p_2, r_2, T
                     ≤ C 2^q γ_0^q - 2(u-v_Y_p_1, r_1, T + u-v_Y_p_2, r_2, T),   ∀ u,v ∈ 
                  	B_2γ_0, T.
              
              	In particular if we choose (p̃, r̃)=(p_1, r_1) and (p̃, r̃)=(p_2, r_2) then
              	
                  Φ(u) - Φ(v)_Y_T≤ C 2^q+1γ_0^q - 2u - v_Y_T
              
              	and if we choose γ_0 >0 small enough in such a way that C 2^q+1γ_0^q - 2<1/2
              	then Φ is a contraction on (B_2γ_0, T, d). In particular Φ has one unique
              	fixed point in this space.
              	The property  u ∈ C([0, T], ) and u∈ X_p,r,T for every admissible couple (p,r) is straightforward and     follows 
                       by Strichartz estimates. 
                       
              	Step 2. Uniqueness in X_T. Assume u_1(t,x) and u_2(t,x) are two fixed points
              of Φ in the space X_T. 
              	We define 
              	T_0=sup{T̅∈ [0,T]| sup_i u_i(t,x)_X_T̅≤ 2γ_0}.
              	It is easy to show that T_0∈ (0, T̅] and arguing as in step 1 the operator Φ is a contraction on (B_2γ_0, T_0, d). Hence by uniqueness of the fixed point in this space necessarily 
              	u_1(t,x)=u_2(t,x) in X_T_0.  Moreover since u_i(t,x)∈𝒞([0, T_0]; H)
              	we have u_1(T_0,x)=u_2(T_0,x)=ψ(x). Hence at time T_0 the solutions coincide and starting from T_0
              	(that we can also identify with T_0=0 by using the traslation invariance w.r.t. to time of the equation),
              	we can apply again the step 1 in the ball (B_2γ_0, T̃, d) with initial condition
              	ψ(x), where 
              	T̃>0 is such that e^itΔψ_X_T̃≤γ_0. Again by uniqueness
              	of the fixed point of Φ in the space (B_2γ_0, T̃, d) we deduce that
              	u_1(t,x)=u_2(t,x) in X_T_0+T̃, hence contradicting the definition of T_0 unless T_0=T. 
              	
              	
              	
              	
              	
              	
              	
              	Step 3. Conservation laws. The proof of (<ref>) is 
              	rather classical. In particular 
              	it follows by Proposition 1 and Proposition 2 in <cit.>. Another possibility is to follow the proof of
              	Propositions 5.3 and 5.4 in <cit.>, that
              	 can be repeated  mutatis mutandis 
              	in the context of (<ref>). The minor modification compared with  <cit.> is that we use the end-point Strichartz   
              	estimate in order to treat the Sobolev critical nonlinearity.    
              
              
              § ORBITAL STABILITY
              
              
              We shall prove in this section that the set ℳ_c defined in  (<ref>) is orbitally stable. 
              In particular a nontrivial point concerns the fact that the local solutions, whose existence has been established in <ref>, 
              can be extended to global solutions provided that the initial datum is close to ℳ_c.
              The main difficulty is related to the criticality of the nonlinearity in (<ref>), which implies that
              an a priori bound on the Mass and the Energy is not sufficient
              to exclude a finite-time blow-up.
              We will overcome this issue by deducing
              from the uniform local well-posedness (Proposition <ref>)
              a uniform lower bound on the time of existence of the solution
              corresponding to initial data close to a set which is compact up to translations, see Theorem <ref>.
              
              
              
              
              
              To simplify the next statement we denote by u_φ(t) the integral solution associated with (<ref>) and we denote by T_φ^max its maximal time of existence.
              
              
              
              	Let v ∈ℳ_c. Then, for  every ε > 0 there exists δ > 0 such that:
              	
                  ∀φ∈ ||φ - v||_ < δ⟹sup_t ∈ [0, T_φ^max)u_φ(t)ℳ_c < ε.
              
              	In particular we have
              	
                  u_φ(t) = m_c(t) + r(t),   ∀ t ∈ [0, T_φ^max),  m_c(t) ∈ℳ_c,  r(t)_ < ε.
              
              	Suppose the theorem is false. Then there exists (δ_n) ⊂^+  a decreasing sequence converging to 0 and (φ_n)  ⊂ satisfying
              	
                  ||φ_n - v||_ < δ_n
              
              	and
              	
                  sup_t ∈ [0, T_φ_n^max)u_φ_n(t)ℳ_c > ε_0,
              
              	for some ε_0 >0.  We observe that ||φ_n||_2^2 → c and, by continuity of F_μ, F_μ(φ_n) → m(c). By conservation laws, for n ∈ large enough, u_φ_n will remains inside of B_ρ_0 for all t ∈ [0, T_φ_n^max).
              	Indeed, if for some time t>0||∇ u_φ_n(t)||_2^2 = ρ_0 then, in view of <ref> (<ref>) we have that F_μ(u_φ_n(t)) ≥ 0 in contradiction with m(c) <0. Now let t_n>0 be the first time such that
              	u_φ_n(t_n)ℳ_c = ε_0 and set u_n := u_φ_n(t_n).
              	By conservation laws, 
              	(u_n) ⊂ B_ρ_0 satisfies u_n2^2 → c and F_μ(u_n) → m(c) and thus, in view of <ref>, it converges, up to translation, to an element of ℳ_c. Since ℳ_c is invariant under translation this contradicts the equality 
              	u_nℳ_c = ε_0 >0.
              
              
              The rest of this section is devoted to showing that T_φ^max=∞ and it will conclude the proof of
              <ref>.
              
              
              
              	Let 𝒦⊂∖{0} be compact up to translation and assume that (p, r) is an admissible pair with p ≠∞. Then, for every γ > 0 there exists ε = ε(γ) > 0 and T = T(γ) > 0 such that 
              	
                  sup_{φ∈| φ𝒦 < ε}*e^itΔφ_X_p,r,T < γ.
              
              	We first claim, for every γ > 0, the existence of a T > 0 such that
              	
                  sup_φ∈𝒦*e^itΔφ_X_p,r,T < γ2.
              
              	If it is not true then there exists sequences (φ_n) ⊂𝒦 and (T_n) ⊂^+
              	such that T_n → 0 and
              	
                  *e^itΔφ_n_X_p,r,T_n≥γ
              
              	for a suitable γ>0. Since 𝒦 is compact up to translation, passing to a subsequence, there exists a sequence (x_n) ⊂^N such that
              	
                  φ̃_̃ñ(·) := φ_n(· - x_n)  H →φ(·)
              
              	for a φ∈. By continuity (induced by Strichartz's estimates) we have, for every T̅>0,
              	
                  *e^itΔφ̃_n_X_p,r,T̅→*e^itΔφ_X_p,r,T̅.
              	
              	Also, recording the translation invariance of Strichartz's estimates we get from (<ref>) that
              	
                  *e^itΔφ̃_n_X_p,r,T_n  = *e^itΔφ_n_X_p,r,T_n≥γ.
              
              	Now, by <ref> (iii), we have e^itΔφ∈ X_p,r, 1, namely the function 
              	
                  [0,1]∋ t→  g(t):=||e^itΔφ||_W^1,r()^p
              
              	belongs to L^1([0,1]). Then by 
              	the Dominated Convergence Theorem we get χ_[0, T̃](t) g(t)_L^1([0,1])→ 0
              	as T̃→ 0,
              	namely *e^itΔφ_X_p,r,T̃^p 
              	→ 0 as T̃→ 0. Hence, we can choose T̅ >0 such that
              	
                  *e^itΔφ_X_p,r,T̅ < γ.
              
              	At this point gathering (<ref>)- (<ref>) we get a contradiction and the claim holds.
              	Now, fix a T>0 such that (<ref>) holds.  By <ref> (iii), we have
              	
                  *e^itΔη_X_p,r,T≤ C ||η||_,     ∀η∈.
              
              	Thus, assuming  that ||η||_ < γ/2C := ε, we obtain that
              	
                  *e^itΔη_X_p,r,T < γ/2.
              
              	Summarizing, we get that, for all φ∈𝒦 and all η∈ such that ||η||_ < ε,
              	
                  *e^itΔ (φ + η)_X_p,r,T≤*e^itΔφ_X_p,r,T + *e^itΔη_X_p,r,T < γ.
              
              	This implies the proposition.
              
              	Let 𝒦⊂∖{0} be compact up to translation. Then, for every γ > 0 there exists ε = ε(γ) > 0 and T = T(γ) > 0 such that 
              	
                  sup_{φ∈| φ𝒦 < ε}*e^itΔφ_X_T < γ.
              
              	We apply <ref> twice with the admissible pairs (p_1, r_1) and (p_2, r_2). Then, the proposition follows from the definition of the norm X_T given in (<ref>).
              
              	Let  𝒦⊂∖{0} be compact up to translation. Then there exist ε_0 > 0 and T_0>0 such that the Cauchy problem (<ref>), where φ satisfies φ𝒦 < ε_0, has a unique solution on the time interval [0, T_0] in the sense of <ref>.
              
              	We apply <ref> where γ = γ_0 is given in <ref>. Then <ref> guarantees that the theorem holds for ε_0 = ε (γ_0) >0 and T_0 = min{ T(γ_0),1 } >0. 
              
              	Let ℳ_c be defined in (<ref>). Then there exists a δ_0 > 0 such that, if φ∈ satisfies φℳ_c < δ_0 the corresponding solution to (<ref>) satisfies T_φ^max = ∞.
              
              	We make use of <ref> where we choose 𝒦 = ℳ_c. By <ref>, we can choose a δ_0 >0 such that (<ref>) and (<ref>) holds for ε = ε_0 where ε_0 >0 is given in <ref>. Then <ref> guarantees that the solution 
              	u_φ(t) where φℳ_c < δ_0 satisfies u_φ(t)ℳ_c < ε_0 up to the maximum time of existence T_φ^max≥ T_0. Since, at any time in (0, T_φ^max) we can apply again <ref> that guarantees an uniform additional time of existence T_0 >0, this contradicts the definition of T_φ^max if T_φ^max < ∞.
              
              
              
              
              
              At this point we can give,
              
              	The fact that ℳ_c is compact, up to translation, was established in <ref>. The orbital stability of ℳ_c, in the sense of 
              	<ref> follows from <ref> and <ref>. 	
              plain