Jean Dolbeault

Robert Stańczy

Non-existence and uniqueness results for supercritical semilinear elliptic equations

Keywords: Mathematics Subject Classification (2000). Primary: 35A05, Secondary: 35J60, 35J65 Semi-linear elliptic equation, uniqueness, Gelfand's problem, Pohožaev's method, Rellich-Pohožaev identities, Brezis-Nirenberg problem, bifurcation, branches of solutions, non-existence results, critical explosion parameter, nonlinear eigenvalue problem, non-local constraints

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

Introduction

This paper is devoted to non-existence and uniqueness results for various supercritical semilinear elliptic equations depending on a bifurcation parameter, in a star-shaped domain in R d . We shall distinguish the multiplicative case when the equation can be written as ∆u + λ f (u) = 0 [START_REF] Adams | Sobolev spaces[END_REF] and the additive case for which the equation is

∆u + f (u + µ) = 0 . (2)
We shall also distinguish two sub-cases for each equation: the local case when λ and µ are the bifurcation parameters, and the non-local case when λ and µ are determined by a non-local condition, respectively

λ Ω f (u) dx = κ and Ω f (u + µ) dx = M.
In the multiplicative non-local case, the equation is

∆u + κ f (u) Ω f (u) dx = 0 . (3)
In many applications, the term f (u)/ Ω f (u) dx is interpreted as a probability measure and κ is a coupling parameter. Such a parameter arises from physical constants after a proper adimensionalization. In the additive non-local case, the problem to solve is

∆u + f (u + µ) = 0 , M = Ω f (u + µ) dx . (4)
The parameter M is typically a mass and, in a variational setting, µ can be interpreted as a Lagrange multiplier associated with the mass constraint, that is, a chemical potential from the point of view of physics. We shall consider the four problems, (1), ( 2), ( 3) and ( 4), and prove that if the domain Ω is star-shaped, with boundary ∂Ω in C 2,γ , γ ∈ (0, 1), and if f is a non-decreasing nonlinearity with supercritical growth at infinity, such that f (0) > 0 in the case of ( 1) or ( 3), or such that f > 0 on (μ, ∞) and lim µ→μ f (µ) = 0 for some μ ∈ [-∞, ∞) in the case of ( 2) or ( 4), then solutions are unique in L ∞ ∩ H 1 0 (Ω) in a certain range of the parameters λ, µ, κ or M , while no solution exists for large enough values of the same parameters. Typical nonlinearities are the exponential function f (u) = e u and the power law nonlinearity f (u) = (1 + u) p , for some p > (d + 2)/(d -2), d ≥ 3. In the exponential case, [START_REF] Adams | Sobolev spaces[END_REF] is the well known Gelfand equation, cf. [START_REF] Gel | Some problems in the theory of quasilinear equations[END_REF].

Our approach is based on Pohožaev's estimate, see [START_REF] Pohožaev | On the eigenfunctions of the equation ∆u + λf (u) = 0[END_REF], which is obtained by multiplying the equations by (x • ∇u), integrating over Ω and then integrating by parts. Also see [START_REF] Rellich | Darstellung der Eigenwerte von ∆u + λu = 0 durch ein Randintegral[END_REF] for an earlier result based on the local dilation invariance in a linear setting. In this paper, we shall only consider solutions in L ∞ ∩ H 1 0 (Ω), which are therefore classical solutions, so that multiplying the equation by u or by (x • ∇u) is allowed. Some results can be extended to the H 1 0 (Ω) framework, but some care is then required. This paper is organized as follows. In Section 2, we consider the multiplicative local and non-local bifurcation problems, respectively [START_REF] Adams | Sobolev spaces[END_REF] and [START_REF] Bellout | On some singular solutions of the equations ∆u = -λe u[END_REF]. In Section 3, we study the additive local and non-local bifurcation problems, respectively [START_REF] Bandle | The Brézis-Nirenberg problem on S 3[END_REF] and [START_REF] Benguria | Classification of the solutions of semilinear elliptic problems in a ball[END_REF]. In all cases, we establish non-existence and uniqueness results, and give some indications on how to construct the branches of solutions, although this is not our main purpose.

Before giving the details of our results, let us give a brief review of the literature. Concerning [START_REF] Adams | Sobolev spaces[END_REF], we primarily refer to the contributions of K. Schmitt in [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF] and R. Schaaf in [START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF], which cover even more general cases than ours and will be discussed more thoroughly later in this section.

The parameter λ in (1) can be seen as a bifurcation parameter. Equation ( 1) is sometimes called a nonlinear eigenvalue problem. It is well known that for certain values of λ, multiplicity of solutions can occur, see for instance [START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources[END_REF]. In some cases there are infinitely many positive solutions, even in the radial case, when Ω is a ball. Radial solutions have been intensively studied. We refer for instance to [START_REF] Castro | Nonlinear eigenvalue problems with semipositone structure[END_REF] for a review of problems with positone structure, i.e. for which f (0) < 0 and f changes sign once on R + . A detailed analysis of bifurcation diagrams can be found in [START_REF] Ouyang | Exact multiplicity of positive solutions for a class of semilinear problems[END_REF][START_REF]Exact multiplicity of positive solutions for a class of semilinear problem. II[END_REF]. Also see [START_REF] Lions | On the existence of positive solutions of semilinear elliptic equations[END_REF] for earlier and more qualitative results. Positive bounded solutions of such a nonlinear scalar field equation are often called ground states and can be characterized in many problems as minimizers of a semi-bounded coercive energy functional. They are relevant in many cases of practical interest in physics, chemistry, mathematical biology, etc.

When Ω is a ball, all bounded positive solutions are radial under rather weak conditions on the nonlinearity f , according to [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF] and subsequent papers. Lots of efforts have been devoted to uniqueness issues for the solutions of the corresponding ODE and slightly more general problems like quasilinear elliptic ones, see, e.g., [START_REF] Erbe | Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball[END_REF]. Several other results also cover the case Ω = R d , see [START_REF] Serrin | Uniqueness of ground states for quasilinear elliptic equations[END_REF]. There are also numerous papers in case of more general nonlinearities, including, for instance, functions of x, u, and ∇u, or more general bifurcation problems than the ones considered in this paper. It is out of the scope of this introduction to review all of them. In a ball, the set of bounded solutions can often be parametrized. The corresponding bifurcation diagrams have the following properties. For nonlinearities with subcritical growth, for instance for f (u) = (1 + u) p , p < (d + 2)/(d -2), d ≥ 3, multiple positive solutions may exist when λ is positive, small, while for supercritical growths, for example f (u) = (1 + u) p with p > (d + 2)/(d -2), d ≥ 3, or f (u) = e u and d = 3, there is one branch of positive solutions which oscillates around some positive, limiting value of λ and solutions are unique only for λ positive, small. See [START_REF] Benguria | Classification of the solutions of semilinear elliptic problems in a ball[END_REF][START_REF] Del Pino | Multiple bubbling for the exponential nonlinearity in the slightly supercritical case[END_REF][START_REF] Dolbeault | Geometry of phase space and solutions of semilinear elliptic equations in a ball[END_REF][START_REF] Lions | On the existence of positive solutions of semilinear elliptic equations[END_REF][START_REF] Ouyang | Exact multiplicity of positive solutions for a class of semilinear problems[END_REF][START_REF]Exact multiplicity of positive solutions for a class of semilinear problem. II[END_REF][START_REF] Suzuki | Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity[END_REF] for more details.

Another well known fact is that, at least for star-shaped domains, Pohožaev's method allows to discriminate between super-and subcritical regimes. This approach has been used mostly to prove the non-existence of non-trivial solutions, see [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF][START_REF] Pucci | A general variational identity[END_REF][START_REF]Variational methods for nonlinear elliptic eigenvalue problems[END_REF], and [START_REF] Pohožaev | On the eigenfunctions of the equation ∆u + λf (u) = 0[END_REF][START_REF] Rellich | Darstellung der Eigenwerte von ∆u + λu = 0 durch ein Randintegral[END_REF] for historical references. Such a method is for instance at the basis of the result of [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] on the Brezis-Nirenberg problem. Also see [START_REF] Benguria | Classification of the solutions of semilinear elliptic problems in a ball[END_REF] and references therein for more details. The identity in Pohožaev's method amounts to consider the effect of a dilation on an energy associated to the solution and therefore carries some important information on the problem, see, e.g., [START_REF] Dolbeault | The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions[END_REF]. In this context, stereographic projection and connections between euclidean spaces and spheres are natural, as was already noted in [START_REF] Bandle | The Brézis-Nirenberg problem on S 3[END_REF] by C. Bandle and R. Benguria. In this paper we are going to study first the regime corresponding to λ small and show that Pohožaev's method provides a uniqueness result also in cases for which a non-trivial solution exists. The existence of a branch of positive solutions of (1) is a widely studied issue, see for instance [START_REF] Crandall | Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems[END_REF][START_REF] Rabinowitz | Some global results for nonlinear eigenvalue problems[END_REF]. Also see [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF] for a review, and references therein. As already said, our two basic examples are based on the power law case, f (u) = (1 + u) p , and the exponential nonlinearity, f (u) = e u , for which useful informations and additional references can be found in [START_REF] Del Pino | Multiple bubbling for the exponential nonlinearity in the slightly supercritical case[END_REF][START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources[END_REF][START_REF] Mignot | Quelques résultats sur un problème elliptique avec non linéarité exponentielle[END_REF][START_REF] Senba | Applied analysis[END_REF][START_REF]Free energy and self-interacting particles[END_REF]. We shall also consider a third example, with a nonlinearity corresponding to the case of Fermi-Dirac statistics, which behaves like a power law for large, positive values of u, and like an exponential function for large, negative values of u.

The functional framework of bounded solutions and a bootstrap argument imply that we work with classical solutions. Apart from the condition that the domain is star-shaped and satisfies the Poincaré inequality, e.g., is bounded in one direction, we will assume no other geometrical condition. In the local multiplicative case, several uniqueness results are known for small λ > 0, including in the case of Gelfand's equation, see [START_REF] Mcgough | Pohozaev obstructions on non-starlike domains[END_REF][START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF][START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF]. One should note that in the framework of the larger space H 1 0 (Ω), if the boundedness assumption is relaxed, it is not even known if all solutions are radial when Ω is a ball. The results of [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF] and subsequent papers almost always rely on the assumption that the solutions are continuous or at least bounded on Ω. Notice that, according to [START_REF] Matano | Solutions of a nonlinear elliptic equation with a prescribed singularity[END_REF][START_REF]Solutions of semilinear elliptic equations with one isolated singularity[END_REF], even for a ball, it is possible to prescribe a given isolated singularity which is not centered. In [START_REF]Solutions of semilinear elliptic equations with one isolated singularity[END_REF], the case of our two basic examples, f (u) = e u and f (u) = (1 + u) p , with d+2 d-2 < p < d+1 d-3 , d > 3, has been studied and then generalized to several singularities in [START_REF] Rébaï | Solutions of semilinear elliptic equations with many isolated singularities: the unstable case[END_REF]. Also see [START_REF] Pacard | Solutions de ∆u = -λe u ayant des singularités ponctuelles prescrites[END_REF] for an earlier result. These singularities are in H 1 0 (Ω) and, for a given value of a parameter λ set apart from zero, they are located at an a priori given set of points. Similar problems on manifolds were considered in [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF].

We refer to [START_REF] Bellout | On some singular solutions of the equations ∆u = -λe u[END_REF][START_REF] Gallouët | Quelques résultats sur le problème -∆u = λe u[END_REF] for bounds on the solutions to Gelfand's problem, which have been established earlier than uniqueness results but are actually a key tool. Also see [START_REF] Mcgough | Applications of variational identities to quasilinear elliptic differential equations[END_REF] for a more recent contribution. Concerning the uniqueness of the solutions to Gelfand's problem for d ≥ 3 and λ > 0, small, we refer to [START_REF] Mcgough | Pohozaev obstructions on non-starlike domains[END_REF][START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF][START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF]. In the case of a ball, this is even known since the paper of D.D. Joseph and T.S. Lundgren, [START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources[END_REF], when combined with the symmetry result of [START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF].

The local multiplicative case corresponding to Problem (1) is the subject of Section 2.1. The literature on such semilinear elliptic problems and associated biffurcation problems is huge. The results of non-existence of non-trivial solutions are well known, see [START_REF] Dancer | Uniqueness of solutions for some elliptic equations and systems in nearly star-shaped domains[END_REF][START_REF] Pucci | A general variational identity[END_REF][START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF] and references therein. Also see [START_REF] Mcgough | Nonexistence and uniqueness in semilinear elliptic systems[END_REF] for extension to systems. Concerning the uniqueness result on non-trivial solutions, the method was apparently discovered independently by several people including F. Mignot and J.-P. Puel, and X. Cabré and P. Majer, [15], but it seems that the first published reference on uniqueness results by Rellich-Pohožaev type estimates is due to K. Schmitt [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF] and later, to R. Schaaf [START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF]. A more general result for the multiplicative case has been obtained in [START_REF] Biler | Parabolic-elliptic systems with general density-pressure relations[END_REF] to the price of more intricate reasoning.

Numerous papers have been devoted to the understanding of the role of the geometry and they extend the standard results, mostly the non-existence results, to the case of non strictly star-shaped domains: see for instance [START_REF] Dancer | Uniqueness of solutions for some elliptic equations and systems in nearly star-shaped domains[END_REF][START_REF] Pucci | A general variational identity[END_REF][START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF] and several papers of J. McGough et al., see [START_REF] Mcgough | Pohozaev obstructions on non-starlike domains[END_REF][START_REF] Mcgough | Domain geometry and the Pohozaev identity[END_REF][START_REF] Mcgough | Applications of variational identities to quasilinear elliptic differential equations[END_REF], which are, as far as we know, the most up-to-date results on such issues. As already mentioned above, Problem (1) has been studied by K. Schmitt in [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF] and R. Schaaf in [START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF]. In [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF]Theorem 2.6.7], it is proved that if one replaces f (u) in ( 1) by a more general function f

(x, u) in C 2 (Ω × R + ) satisfying (i) f (x, u) > 0, f u (x, u) > 0, u ≥ 0, x ∈ Ω , (ii) lim sup u→∞ sup x∈Ω 2 d F (x, u) (d -2) u f (x, u) < 1 , (iii) [∇ x F (x, u + 1) -∇ x F (x, u) -u ∇ x f (x, u)] • x ≤ 0 for u ≫ 1 , x ∈ Ω ,
then uniqueness holds for a star-shaped domain Ω. A survey on the existence and continuation results for linear and superlinear (sub-and supercritical) growth of the nonlinear term f in (1) can also be found in [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF], as well as a study of the influence of the geometry, topology and dimension of the domain, which is of interest for our purpose.

In [START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF], R. Schaaf studies uniqueness results for the semilinear elliptic problem (1) under the asymptotic condition lim sup u→∞

F (u) u f (u) < 1 2 -M (Ω)
where M (Ω) = 1/d for star-shaped domains. In general M (Ω) is some number in the interval (0, 1/d]. In the autonomous case, the above asymptotic condition is equivalent to the assumption (ii) made by K. Schmitt in [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF] or to our assumption [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF], to be found below. Our contribution to the question of the uniqueness for (1) relies on a simplification of the proof in [START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF][START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF].

Imposing a non-local constraint dramatically changes the picture. For instance, in case of Maxwell-Boltzmann statistics, f (u) = e u , in a ball of R 2 , the solution of (1) has two solutions for any λ ∈ (0, λ * ) and no solution for λ > λ * , while uniqueness holds in (3) in terms of M , for any M for which a solution exists, see [START_REF] Biler | Steady states for Streater's energy-transport models of self-gravitating particles[END_REF][START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources[END_REF]. Non-local constraints are motivated by considerations arising from physics. In the case of the exponential nonlinearity with a mass normalization constraint, a considerable effort has been done in the two-dimensional case for understanding the statistical properties of the so-called Onsager solutions of the Euler equation, see [START_REF] Caglioti | A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description[END_REF][START_REF]A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description[END_REF][START_REF] Neri | Statistical mechanics of the N -point vortex system with random intensities on a bounded domain[END_REF]. The same model, but rather in dimension d = 3, is relevant in astrophysical models for systems of gravitating particles, see [START_REF] Biler | Parabolic-elliptic systems with general density-pressure relations[END_REF].

Other standard examples are the polytropic distributions, with f (u) = u p , and Bose-Einstein or Fermi-Dirac distributions which result in nonlinearities involving special functions. Existence and non-existence results were obtained for instance in [START_REF] Biler | Steady states for Streater's energy-transport models of self-gravitating particles[END_REF] and [START_REF] Stańczy | Self-attracting Fermi-Dirac particles in canonical and microcanonical setting[END_REF][START_REF]Steady states for a system describing self-gravitating Fermi-Dirac particles[END_REF], respectively for Maxwell-Boltzmann and Fermi-Dirac statistics.

An evolution model compatible with Fermi-Dirac statistics and the convergence of its solutions towards steady states has been thoroughly examined in [START_REF] Biler | On an evolution system describing self-gravitating Fermi-Dirac particles[END_REF], while the steady state problem was considered by R. Stańczy in [START_REF] Stańczy | Self-attracting Fermi-Dirac particles in canonical and microcanonical setting[END_REF][START_REF]Steady states for a system describing self-gravitating Fermi-Dirac particles[END_REF][START_REF]The existence of equilibria of many-particle systems[END_REF]. See [START_REF] Chavanis | Thermodynamics of self-gravitating systems[END_REF] and references therein for a model improved with respect to thermodynamics, [START_REF]Steady states for a system describing self-gravitating Fermi-Dirac particles[END_REF] and references therein for more elaborate models, and [START_REF] Chavanis | Statistical mechanics of twodimensional vortices and collisionless stellar systems[END_REF] for a derivation of an evolution equation involving a mean field term, which also provides a relevant, stationary model studied in [START_REF] Biler | Parabolic-elliptic systems with general density-pressure relations[END_REF][START_REF]On some parabolic-elliptic system with self-similar pressure term[END_REF]. Also see [START_REF] Chavanis | Chapman-Enskog derivation of the generalized Smoluchowski equation[END_REF][START_REF] Dolbeault | Non linear diffusions as limit of kinetic equations with relaxation collision kernels[END_REF] for an alternative, phenomenological derivation of drift-diffusion equations and their stationary counterparts, and [START_REF]The existence of equilibria of many-particle systems[END_REF] for the existence of radial solutions by fixed point methods in weighted function spaces, under nonlocal constraints. The case of a decoupled, external potential goes back to the work of Smoluchowski, see [START_REF] Chandrasekhar | Marian Smoluchowski: his life and scientific work[END_REF][START_REF] Smoluchowski | Drei vortrage über diffusion, brownsche molekularbewegung und koagulation von kolloidteilchen[END_REF]. For this reason, the evolution model is often referred to as the Smoluchowski-Poisson equation.

Our purpose is not to study the above mentioned evolution equations, but only to emphasize that for the corresponding steady states, non-local constraints are very natural, since they correspond to quantities which are conserved along evolution. Hence, to identify the asymptotic state of the solutions to the evolution equation, we have to solve a semilinear elliptic equation with a non-local constraint, which corresponds, for instance, to mass conservation.

The multiplicative case 2.1. The local bifurcation problem

We consider Problem (1) on a domain Ω in R d . Our first assumption is the geometrical condition that a Poincaré inequality holds:

Ω |u| 2 dx ≤ C P Ω |∇u| 2 dx (5)
for any u ∈ H 1 0 (Ω) and some positive constant C P > 0. Such an inequality holds for instance if Ω is bounded in one direction. See [69, Proposition 2.1] for more details. Inequality ( 5) is called Friedrichs' inequality in some areas of analysis (see [START_REF] Friedrichs | Die Randwert-und Eigenwertprobleme aus der Theorie der elastischen Platten. (Anwendung der direkten Methoden der Variationsrechnung[END_REF][START_REF] Poincaré | Sur la théorie analytique de la chaleur[END_REF] for historical references; we also refer to [START_REF] Jakszto | On a sharp Poincaré's inequality and On a Poincaré inequality in some unbounded sets[END_REF]). We shall further require that

∃ u ∈ H 1 0 (Ω) such that u > 0 and Ω |u| 2 dx = C P Ω |∇u| 2 dx . (6)
This is straightforward in some cases, for instance if Ω is bounded, simply connected, with a Lipschitz boundary, or if Ω is unbounded, simply connected and such that the embedding H 1 0 (Ω) ֒→ L 2 (Ω) is compact. For such a compactness property, see for instance [START_REF] Berger | Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains[END_REF]Theorem 2.8] and [START_REF] Adams | Sobolev spaces[END_REF]Theorems 6.16 and 6.19].

The goal of this section is to state a non-existence result for large values of λ and give sufficient conditions on f ≥ 0 such that, for some λ 0 > 0, Equation (1) has a unique solution in L ∞ ∩ H 1 0 (Ω) for any λ ∈ (0, λ 0 ). We assume that f is of class C 2 . By standard elliptic bootstraping arguments, a bounded solution is then a classical one.

Next we assume that for some λ * > 0, there exists a branch of positive minimal solutions (λ, u λ ) λ∈(0,λ * ) originating from (0, 0) and such that

lim λ→0+ u λ L ∞ (Ω) + ∇u λ L ∞ (Ω) = 0 . (7)
Sufficient conditions for such a property to hold can be found in various papers. We can for instance quote the following result.

Lemma 1. Assume that Ω is bounded with smooth, i.e. C 2,γ for some γ ∈ (0, 1), boundary, f ∈ C 2 is positive on [0, ∞) and inf u>0 f (u)/u > 0. Then (7) holds.

We refer for instance to [START_REF] Schmitt | Positive solutions of semilinear elliptic boundary value problems[END_REF] for a proof. The solutions satisfying [START_REF] Biler | Steady states for Streater's energy-transport models of self-gravitating particles[END_REF] can be characterized as a branch of minimal solutions, using sub-and super-solutions. Although this is standard, for the sake of completeness let us state a non-existence result for values of the parameter λ large enough.

Proposition 2. Assume that (5) and (6) hold. If Λ := inf u>0 f (u)/u > 0, then there exists λ * > 0 such that (1) has no non-trivial nonnegative solution in

H 1 0 (Ω) if λ > λ * .
The lowest possible value of λ * is usually called the critical explosion parameter.

Proof. Let ϕ 1 be a positive eigenfunction associated with the first eigenvalue

λ 1 = 1/C P of -∆ in H 1 0 (Ω): -∆ϕ 1 = λ 1 ϕ 1 .
By multiplying this equation by u and (1) by ϕ 1 , we get

λ 1 Ω u ϕ 1 dx = Ω ∇u • ∇ϕ 1 dx = λ Ω f (u) ϕ 1 dx ≥ Λ λ Ω u ϕ 1 dx ,
thus proving that there are no non-trivial nonnegative solutions if λ > λ 1 /Λ.

Next we present a simplified version of the proof of a uniqueness result stated in [START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF], under slightly more restrictive hypotheses. We assume that d ≥ 3 and that f has a supercritical growth at infinity, i.e., f is such that

lim sup u→∞ F (u) u f (u) = η < d -2 2 d , (8)
where F (u) := u 0 f (s) ds. Notice that, in Proposition 2, Λ > 0 if [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF] holds and if we assume that f is positive.

Theorem 3. Assume that Ω is a bounded star-shaped domain in R d , d ≥ 3, with C 2,γ boundary, such that (5) holds for some C P > 0. If f (z) is positive for large values of z, of class C 2 and satisfies [START_REF] Biler | Steady states for Streater's energy-transport models of self-gravitating particles[END_REF] and [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF], then there exists a positive constant λ 0 such that Equation (1) has at most one solution in L ∞ ∩ H 1 0 (Ω) for any λ ∈ (0, λ 0 ).

Proof. We follow the lines of the proof of [START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF] with some minor simplifications. Up to a translation, we can assume that Ω is star-shaped with respect to the origin. Assume that (1) has two solutions, u and u+v. With no restriction, we can assume that u is a minimal solution and satisfies [START_REF] Biler | Steady states for Streater's energy-transport models of self-gravitating particles[END_REF]. As a consequence, v is nonnegative and satisfies ∆v + λ f

(u + v) -f (u) = 0 . (9)
If we multiply (9) by v and integrate with respect to x ∈ Ω, we get

Ω |∇v| 2 dx = λ Ω v f (u + v) -f (u) dx . (10)
Multiply ( 9) by x • ∇v and integrate with respect to x ∈ Ω to get

d -2 2 Ω |∇v| 2 dx + 1 2 ∂Ω |∇v| 2 (x • ν(x)) dσ = d λ Ω [F (u + v) -F (u) -F ′ (u) v] dx + λ Ω (x • ∇u) [f (u + v) -f (u) -f ′ (u) v] dx ( 11
)
where dσ is the measure induced by Lebesgue's measure on ∂Ω. Recall that F is a primitive of f such that F (0) = 0. Take

η 1 ∈ (η, (d -2)/(2 d))
where η is defined in Assumption [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF]. Since u = u λ is a minimal solution and therefore uniformly small as λ → 0 + , for any ε > 0, we obtain |x • ∇u| ≤ ε for any x ∈ Ω, provided λ > 0 is small enough. Define h ε by

h ε (u, v) := d F (u + v) -F (u) -F ′ (u) v + ε |f (u + v) -f (u) -f ′ (u) v| -d η 1 v f (u + v) -f (u) .
Because of the smoothness of f and by Assumption (8), the function h ε (u, v)/v 2 is bounded from above by some constant H, uniformly in ε > 0, small enough. By the assumption of star-shapedeness of the domain Ω, x • ν(x) ≥ 0 for any x ∈ ∂Ω. From ( 10) and [START_REF]Existence and nonexistence of solutions for a model of gravitational interaction of particles. I[END_REF], it follows that d -2 2

Ω |∇v| 2 dx ≤ d λ H Ω |v| 2 dx + d η 1 Ω |∇v| 2 dx .
Due to the Poincaré inequality (5), the condition

λ < 1 C P H d -2 2 d -η 1
implies v = 0 and the uniqueness follows. . Also see [START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources[END_REF] for more details. Similarly in the same range of parameters for f (u) = u p we only get the trivial, zero solution.

Examples 1. If f (u) = e u ,

The Fermi-Dirac distribution

f (u) = f δ (u) := ∞ 0 t δ 1 + e t-u dt (12)
behaves like 1 δ+1 u δ+1 as u → ∞. Condition (8) holds if and only if δ

+ 1 > (d + 2)/(d -2). The physically relevant examples require that δ = d/2 -1, that is d > 2 (1 + √ 2) ≈ 4.83.
For more properties of these functions see, e.g., [START_REF] Biler | On an evolution system describing self-gravitating Fermi-Dirac particles[END_REF][START_REF] Biler | Nonisothermal systems of self-attracting Fermi-Dirac particles[END_REF].

The non-local bifurcation problem

In this section we address, in L ∞ ∩ H 1 0 (Ω), the non-local boundary value problem (3) with parameter κ > 0. Here Ω is a bounded domain in R d , d ≥ 3, with C 1 boundary.

We start with a non-existence result. Computations are similar to the ones of Section 2.1 and rely on Pohožaev's method. First multiply (3) by u to get

Ω |∇u| 2 dx = κ Ω u f (u) dx Ω f (u) dx . ( 13
)
Multiplying ( 3) by (x • ∇u), we also get

d -2 2 Ω |∇u| 2 dx + 1 2 ∂Ω |∇u| 2 (x • ν) dσ = d κ Ω F (u) dx Ω f (u) dx ( 14
)
where F is the primitive of f chosen so that F (0) = 0 and dσ is the measure induced by Lebesgue's measure on ∂Ω. A simple integration of (3) gives

κ = - Ω ∆u dx = - ∂Ω ∇u • ν dσ .
By the Cauchy-Schwarz inequality,

κ 2 = ∂Ω ∇u • ν dσ 2 ≤ |∂Ω| ∂Ω |∇u • ν| 2 dσ = |∂Ω| ∂Ω |∇u| 2 dσ ,
where the last equality holds because of the boundary conditions. Assume that Ω is strictly star-shaped with respect to the origin:

α := inf x∈∂Ω (x • ν(x)) > 0 . ( 15
)
Because of the invariance by translation of the problem, this is equivalent to assume that Ω is strictly star-shaped with respect to any other point in R d . Hence

∂Ω |∇u| 2 (x • ν) dσ ≥ α ∂Ω |∇u| 2 dσ ≥ α κ 2 |∂Ω| .
Collecting this estimate with ( 13) and ( 14), we obtain

Ω 2 d F (u) -(d -2) u f (u) dx ≥ α κ |∂Ω| Ω f (u) dx .
As a straightforward consequence, we obtain the following result.

Theorem 4. Assume that Ω is a bounded domain in R d , d ≥ 3, with C 1 boundary satisfying (15) for some α > 0. If f is a C 1 function such that for some C > 0, 2 d F (u) ≤ (d -2) u f (u) + C f (u) ( 16
)
for any u ≥ 0, then (3) has no solution in

L ∞ ∩ H 1 0 (Ω) if κ > C |∂Ω|/α.
Standard examples, for which Condition ( 16) is satisfied, are: 1. Exponential case: f (u) = e u with C = 2 d, cf. [START_REF] Biler | Steady states for Streater's energy-transport models of self-gravitating particles[END_REF]. A sharper estimate can be easily achieved as follows. The function h(u) := C e u +(d-2) u e u -2 d (e u -1) is nonnegative if C is such that 0 = h ′ (u) = h(u) for some u ≥ 0. After eliminating u, we find

C = d + 2 + (d -2) log d -2 2 d . (17)
2. Pure power law case: If f (u) = u p , the result holds with p ≥ d+2 d-2 and C = 0, cf. [START_REF] Gel | Some problems in the theory of quasilinear equations[END_REF][START_REF]Semilinear elliptic equations[END_REF]. There are no non-trivial solutions.

Power law case:

If f (u) = (1 + u) p with p ≥ d+2 d-2 , then (16) holds with C = d -2.
Uniqueness results in the non-local case follow from Section 2.1, when the coupling constant κ is positive, small. In case of nonlinearities of exponential type, as far as we know, uniqueness results were guaranteed only under some additional assumptions, see [START_REF] Biler | A class of nonlocal parabolic problems occurring in statistical mechanics[END_REF][START_REF]Existence and nonexistence of solutions for a model of gravitational interaction of particles. I[END_REF]. We are now going to extend such uniqueness results to more general nonlinearities satisfying ( 7) and ( 8) by comparing Problems ( 1) and ( 3).

Denote by u λ the solutions of (1). For λ > 0, small, a branch of solutions of (3) can be parametrized by λ → κ(λ) 3) is also a solution of ( 1) with

:= λ Ω f (u λ ) dx , u λ . Reciprocally, if Ω is bounded and 0 < β := inf u≥0 f (u) , then any solution u ∈ L ∞ ∩ H 1 0 (Ω) of (
λ = κ Ω f (u) dx ≤ κ β |Ω| .
This implies that λ is small for small κ and, as a consequence, for small values of κ, all solutions to (3) are located somewhere on the local branch originating from (0, 0). Moreover, as κ → 0 + , the solution of (3) also converges to (0, 0). To prove the uniqueness in L ∞ ∩H 1 0 (Ω) of the solutions of (3), it is therefore sufficient to establish the monotonicity of λ → κ(λ) for small values of λ. Assume that f (0) > 0 and f is monotone non-decreasing on R + . [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF] Under this assumption, we observe that β = f (0). Let u 1 and u 2 be two solutions of (1) with λ 1 < λ 2 and let v := u 2 -u 1 . Then for some function θ on Ω, with values in [0, 1], we have

-∆v -λ 1 f ′ (u 1 + θ v) v = (λ 2 -λ 1 ) f (u 2 ) ≥ 0 ,
so that, by the Maximum Principle, v is nonnegative. Notice indeed that for λ 2 small enough, u 1 and u 2 are uniformly small since they lie on the local branch, close to the point (0, 0) and therefore

λ 1 f ′ (u 1 + θ v) < 1/C P . It follows that Ω f (u 2 ) dx = Ω f (u 1 + v) dx ≥ Ω f (u 1 ) dx , thus proving that κ(λ 2 ) = λ 2 Ω f (u 2 ) dx > λ 1 Ω f (u 1 ) dx = κ(λ 1 ).
Corollary 5. Under the assumptions of Theorem 3, if moreover f satisfies [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF], then there exists a positive constant κ 0 such that Equation (3) has at most one solution in L ∞ ∩ H 1 0 (Ω) for any κ ∈ (0, κ 0 ).

The additive case

The local bifurcation problem

Consider in L ∞ ∩ H 1 0 (Ω) the equation ( 2). In the two standard examples of this paper the problem can be reduced to (1) as follows.

1. 2) is equivalent to (1) with λ = e µ and the limit λ → 0 + corresponds to µ → -∞. 2) is equivalent to (1) with λ = (1+µ) p-1 and the limit λ → 0

Exponential case: If f (u) = e u , (

Power law case:

If f (u) = (1+u) p , (
+ corresponds to µ → -1 + . If u is solution of ∆u + (1 + u + µ) p = 0, one can indeed observe that v such that 1 + u + µ = (1 + µ)(1 + v) solves ∆v + λ (1 + v) p = 0 with λ = (1 + µ) p-1 .
Equation ( 2) is however not completely equivalent to [START_REF] Adams | Sobolev spaces[END_REF]. To obtain a nonexistence result for large values of µ, we impose the assumption that reads

lim u→∞ f (u) u = +∞ . ( 19
)
Proposition 6. Assume that (5), ( 6) and ( 19) hold. There exists µ * > 0 such that (2) has no positive, bounded solution in

H 1 0 (Ω) if µ > µ * . Proof.
The proof is similar to the one of Proposition 2. Let ϕ 1 be a positive eigenfunction associated with the first eigenvalue λ 1 = 1/C P of -∆ in H 1 0 (Ω). For any µ ≥ 0,

λ 1 Ω u ϕ 1 dx = Ω f (u + µ) ϕ 1 dx ≥ Λ(µ) Ω (u + µ) ϕ 1 dx ≥ Λ(µ) Ω u ϕ 1 dx ,
where Λ(µ) := inf s≥µ f (s)/s, thus proving that there are no nonnegative solutions if Λ(µ) > λ 1 .

Let us make a few comments on the existence of a branch of solutions, although this is out of the main scope of this paper. Let f be a positive function of class C 2 on (μ, ∞), for some μ ∈ [-∞, ∞), with lim µ→μ+ f (µ) = 0. We shall assume that there is a branch of minimal solutions (µ, u µ ) originating from (μ, 0) and such that lim

µ→μ u µ L ∞ (Ω) + ∇u µ L ∞ (Ω) = 0 . (20)
This can be guaranteed if Ω is bounded and if we additionally require that the function f is increasing, as in [START_REF] Stańczy | Self-attracting Fermi-Dirac particles in canonical and microcanonical setting[END_REF] for the Fermi-Dirac model. This is also true for exponential and power-like nonlinearities. At least at a formal level, this can easily be understood by taking ζ = f ′ (µ) as a bifurcation parameter. A solution of ( 2)

is then a zero of F (ζ, u) = u -(-∆) -1 f (u + (f ′ ) -1 (ζ))
and it is therefore easy to find a branch issued from (ζ, u) = (0, 0) by applying the implicit function theorem at (ζ, u) = (0, 0) with F (0, 0) = 0, even if μ = -∞. Using comparison arguments, one can prove that this branch is a branch of minimal solutions.

We shall now address the uniqueness issues. We assume that (8) holds:

∀ η 1 ∈ η, d -2 2 d , lim sup u→∞ F (u) -η 1 u f (u) u f (u) = η -η 1 < 0 .
As a consequence, for any µ > μ,

F (v + µ) -F (µ) -F ′ (µ) v -η 1 v f (v + µ) -f (µ)
is negative for large v, and the function H(v, µ, η 1 ) defined by

v 2 H(v, µ, η 1 ) = F (v + µ) -F (µ) -F ′ (µ) v -η 1 v f (v + µ) -f (µ)
achieves a maximum for some finite value of v. With H(µ, η 1 ) = sup v>0 H(v, µ, η 1 ), we have

F (v + µ) -F (µ) -F ′ (µ) v -η 1 v f (v + µ) -f (µ) ≤ H(µ, η 1 ) v 2 . (21)
Next we assume that, for some η 1 ∈ η, d-2 2 d , we have

C P H(µ, η 1 ) < d -2 2 d -η 1 , (22)
where C P is the Poincaré constant. This condition is non-trivial. It relates H(µ, η 1 ), a quantity attached to the nonlinearity, to C P which has to do only with Ω. It is satisfied for all our basic examples.

1. Exponential case: If f (u) = e u , we take µ negative, with |µ| big enough. Indeed, using first the homogeneity, one obtains H(v, µ, η 1 ) = e µ H(v, 0, η 1 ). Since lim v→0+ H(v, 0, η 1 ) = (1 -2 η 1 )/2 and H(v, 0, η 1 ) becomes negative as v → +∞, as a function of v ∈ R + H(v, 0, η 1 ) admits a maximum value. To get a more explicit bound, we take a Taylor expansion at second order, namely e θ v (1 -2 η 1 -η 1 θ v)/2 for some intermediate number θ ∈ (0, 1). An upper bound is given by η 1 e 1/η1-3 /2, which corresponds to the above expression evaluated at θ v = 1/η 1 -3. According to [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF], η = 0: taking η 1 small enough guarantees [START_REF] Chavanis | Thermodynamics of self-gravitating systems[END_REF].

Power law case:

If f (u) = (1+u) p , we have H(v, µ, η 1 ) = (1+µ) p+1 H(w, 0, η 1 )
where w = v/(µ+1). Since lim v→0+ H(v, 0, η 1 ) = p (1-2 η 1 )/2 and H(v, 0, η 1 ) becomes negative as v → +∞, H achieves a positive maximum.

Fermi-Dirac distribution case:

If f (u) = f d/2-1 (u), we observe that lim sup u→∞ f ′ (u) u f ′′ (u) + 2 f ′ (u) = η < d -2 2 d (23) if d > 2 (1 + √ 2)
, which is stronger than Assumption (8), as can easily be recovered by integrating

f ′ (u) -η u f ′′ (u) + 2 f ′ (u) twice, for large values of u. Take η 1 ∈ (η, (d -2)/(2 d)). A Taylor expansion shows that H(v, µ, η 1 ) = f ′ (u) -η 1 u f ′′ (u) + 2 f ′ (u) + µ η 1 f ′′ (u) = a f ′ (u) -η+η1 2 u f ′′ (u) + 2 f ′ (u) + (µ -b u) η 1 f ′′ (u) with a = 1-2 η1 1-η-η1 , b = η1-η 2 η1(1-η-η1
) and u = µ + θ v for some θ ∈ (0, 1). Both terms in the above right hand side are negative for u large enough, which proves the existence of a constant H(µ, η 1 ) such that [START_REF] Chavanis | Chapman-Enskog derivation of the generalized Smoluchowski equation[END_REF] holds. Notice that by [START_REF] Biler | Nonisothermal systems of self-attracting Fermi-Dirac particles[END_REF]Appendix], f and its derivatives behave like exponentials for u < 0, |u| large. Under the additional assumption d ≥ 6, a tedious but elementary computation shows that, as µ → -∞, the maximum of

u → a f ′ (u) -η+η1 2 u f ′′ (u) + 2 f ′ (u) + (µ -b u) η 1 f ′′ (u)
is achieved at some u = o(µ), which proves that lim µ→-∞ H(µ, η 1 ) = 0. Moreover, for any d > 2(1 + √ 2) one can still show that this maximum value behaves like exp(µ) and thus can be made arbitrarily small for negative µ with |µ| large enough.

Assume that (2) has two solutions, u and u + v, with v ≥ 0, and let us write the equation for the difference v as

∆v + f (u + v + µ) -f (u + µ) = 0 . ( 24
)
The method is the same as in Section 2. Multiply ( 24) by x • ∇v and integrate with respect to x ∈ Ω. If F is a primitive of f such that F (μ) = 0, then

d -2 2 Ω |∇v| 2 dx + 1 2 ∂Ω |∇v| 2 (x • ν(x)) dσ = d Ω F (u + v + µ) -F (u + µ) -F ′ (u + µ) v dx + Ω (x • ∇u) f (u + v + µ) -f (u + µ) -f ′ (u + µ) v dx .
Assume that [START_REF] Chavanis | Thermodynamics of self-gravitating systems[END_REF] holds for some η 1 . If Ω is bounded, |x • ∇u| is uniformly small as µ → μ+ , and we may assume that for any ε > 0, arbitrarily small, there exists µ 0 > μ, sufficiently close to μ (that is, µ 0μ > 0, small if μ > -∞, or µ 0 < 0,

|µ 0 | big enough if μ = -∞), such that |x • ∇u| ≤ ε for any x ∈ Ω if µ ∈ (μ, µ 0 ). Next we define h ε (v) := d F (z + v) -F (z) -F ′ (z) v + ε | f (z + v) -f (z) -f ′ (z) v | -d η 1 v f (z + v) -f (z) ,
where z = u + µ. Using the star-shapedeness of the domain Ω, we have

d -2 2 Ω |∇v| 2 dx ≤ Ω h ε (v) dx + d η 1 Ω v f (z + v) -f (z) dx .
Up to a small change of η 1 , so that Condition [START_REF] Chavanis | Thermodynamics of self-gravitating systems[END_REF] still holds, for ε > 0, small enough, we get

1 d h ε (v) ≤ F (z + v) -F (z) -F ′ (z) v -η 1 v f (z + v) -f (z) .
As ε → 0 + , z converges to µ uniformly and the above right hand side is equivalent to

F (v + µ) -F (µ) -F ′ (µ) v -η 1 v f (v + µ) -f (µ) .
For some δ > 0, arbitrarily small, we obtain

1 d h ε (v) ≤ (H(µ, η 1 ) + δ) v 2 .
From ( 24) multiplied by v, after an integration by parts we obtain

Ω |∇v| 2 dx = Ω v f (z + v) -f (z) dx .
Hence we have shown that

d -2 2 d -η 1 Ω |∇v| 2 dx ≤ (H(µ, η 1 ) + δ) Ω |v| 2 dx ,
By the Poincaré inequality ( 5), the left hand side is bounded from below by

d -2 2 d -η 1 Ω |∇v| 2 dx ≥ 1 C P d -2 2 d -η 1 Ω |v| 2 dx .
Summarizing, we have proved that, if Ω |v| 2 dx = 0, then, for an arbitrarily small δ > 0, 1

C P d -2 2 d -η 1 ≤ H(µ, η 1 ) + δ if µ -μ > 0 is small if μ > -∞, or µ < 0, |µ| big enough if μ = -∞.
This contradicts [START_REF] Chavanis | Thermodynamics of self-gravitating systems[END_REF] unless v ≡ 0.

Theorem 7. Assume that Ω is a bounded star-shaped domain in R d , with C 2,γ boundary, γ ∈ (0, 1), such that (5) holds. If f ∈ C 2 satisfies (8) and [START_REF] Chavanis | Thermodynamics of self-gravitating systems[END_REF], if lim µ→μ f (µ) = 0, then there exists a µ 0 ∈ (μ, ∞) such that Equation (2) has at most one solution in L ∞ ∩ H 1 0 (Ω) for any µ ∈ (μ, µ 0 ). In cases of practical interest for applications, one often has to deal with the equation ∆u + f (x, u + µ) = 0. Our method can be adapted in many cases, that we omit here for simplicity. The necessary adaptations are left to the reader.

The non-local bifurcation problem

In this section we address problem (4) with parameter M > 0, in a bounded starshaped domain Ω in R d . Consider in L ∞ ∩ H 1 0 (Ω) the positive solutions of (4), that is of ∆u + f (u + µ) = 0 (25) where µ is determined by the non-local normalization condition

M = Ω f (u + µ) dx . (26)
We observe that in the exponential case, f (u) = e u , ( 4) is equivalent to the nonlocal multiplicative case, (3). The condition ( 26) is indeed explicitly solved by e µ Ω e u dx = M = κ. Non-existence results for large values of M can be achieved by the same method as in the multiplicative non-local case. If we multiply (25) by u and (x•∇u), we get

Ω |∇u| 2 dx = Ω u f (u + µ) dx , d -2 2 Ω |∇u| 2 dx + 1 2 ∂Ω |∇u| 2 (x • ν) dσ = d Ω F (u + µ) -F (µ) dx . The elimination of Ω |∇u| 2 dx gives Ω 2 d F (u + µ) -F (µ) -(d -2) u f (u + µ) dx ≥ ∂Ω |∇u| 2 (x • ν) dσ .
By the Cauchy-Schwarz inequality, we know that

M 2 = ∂Ω ∇u • ν dσ 2 ≤ |∂Ω| ∂Ω |∇u| 2 dσ .
If (15) holds, then, as in Section 2.2,

α M 2 ≤ |∂Ω| ∂Ω |∇u| 2 (x • ν) dσ .
Summarizing we have found that

Ω 2 d F (u + µ) -F (µ) -(d -2) u f (u + µ) dx ≥ α M 2 |∂Ω| . (27)
This suggests a condition similar to the one in the multiplicative case, [START_REF] Caglioti | A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description[END_REF]. Define

G(µ) := sup z>µ 2 d F (z) -F (µ) -(d -2) f (z) (z -µ) / f (z) .
If f is supercritical in the sense of (8), G is well defined, but in some cases, it also makes sense for d = 2. For simplicity, we shall assume that G is a nondecreasing function of µ. As a consequence, we can state the following theorem, which generalizes known results on exponential and Fermi-Dirac distributions, cf. [START_REF] Biler | Steady states for Streater's energy-transport models of self-gravitating particles[END_REF] and [START_REF] Stańczy | Self-attracting Fermi-Dirac particles in canonical and microcanonical setting[END_REF][START_REF]Steady states for a system describing self-gravitating Fermi-Dirac particles[END_REF], respectively.

(Ω) if M > |∂Ω| α (G • f -1 ) M |Ω| .
Here by f -1 one has to understand the generalized inverse given by f -1 (t) := sup {s ∈ R : f (s) ≤ t}.

Proof. From the above definitions and computations, we have

α M 2 |∂Ω| ≤ G(µ) M .
Since f is non-decreasing and the solution u of ( 25) is positive, while M = Ω f (u+ µ) dx ≥ f (µ) |Ω|, this completes the proof. 4) has no bounded solution if M > |∂Ω| G/α. We recover here the condition corresponding to [START_REF]A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description[END_REF] and Theorem 4.

Power law case

: if f (u) = u p with p ≥ d+2 d-2 , then G(µ) = µ G(1). Using µ ≤ (M/|Ω|) 1/p , it follows that (4) has no bounded solution if M p-1 p > G(1) α |∂Ω| |Ω| 1/p . 3. Fermi-Dirac distribution case: If f (u) = f δ (u) where f δ is the Fermi-Dirac distribution defined by (12) with δ = d/2 -1 and d > 2 (1 + √ 2), then f is increasing, F = 2 d f d/2 is the primitive of f such that lim u→-∞ F (u) = 0, G d := sup z∈R 4 f d/2 (z) -(d -2) z f d/2-1 (z) = sup z∈R [2 d F (z) -(d -2) z f (z)]
is finite according to [START_REF] Biler | Nonisothermal systems of self-attracting Fermi-Dirac particles[END_REF]Appendix] For a similar approach, one can refer to [START_REF]Steady states for a system describing self-gravitating Fermi-Dirac particles[END_REF].

Denote by u µ a branch of solutions of (2) satisfying [START_REF] Chandrasekhar | Marian Smoluchowski: his life and scientific work[END_REF]. For µμ > 0, small if μ > -∞, or µ < 0, |µ| big enough if μ = -∞, a branch of solutions of (4) can be parametrized by µ → M (µ) := Ω f (u µ + µ) dx , u µ . Reciprocally, if Ω is bounded, then any solution u ∈ L ∞ ∩H 1 0 (Ω) of ( 4) is of course a solution of (2) with µ = µ(M ) determined by [START_REF] Dancer | Uniqueness of solutions for some elliptic equations and systems in nearly star-shaped domains[END_REF]. If f is monotone increasing, we additionally know that μ < µ < f -1 (M/|Ω|). To prove the uniqueness in L ∞ ∩H 1 0 (Ω) of the solutions of (4), it is therefore sufficient to establish the monotonicity of µ → M (µ). Assume that lim µ→μ f (µ) = lim µ→μ f ′ (µ) = 0 and f is monotone increasing on (μ, ∞) . (

The function v := du µ /dµ is a solution in H 1 0 (Ω) of ∆v + f ′ (u µ + µ) (1 + v) = 0 .

As in the proof of Corollary 5, by the Maximum Principle, v is nonnegative when µ is in a right neighborhood of μ, thus proving that dM dµ

= Ω f ′ (u µ + µ) (1 + v) dx
is nonnegative. Using Theorem 7, we obtain the following result.

Theorem 9. Assume that Ω is a bounded star-shaped domain in R d with C 2,γ boundary. If f ∈ C 2 is nonnegative, increasing, satisfies (5), ( 8), [START_REF] Chavanis | Thermodynamics of self-gravitating systems[END_REF], and (28), then there exists M 0 > 0 such that (4) has at most one solution in L ∞ ∩ H 1 0 (Ω) for any M ∈ (0, M 0 ).

Concluding remarks

Uniqueness issues in nonlinear elliptic problems are difficult questions when no symmetry assumption is made on the domain. In this paper we have considered only a few simple cases, which illustrate the efficiency of the approach based on Pohožaev's method when dealing with bifurcation problems. Our main contribution is to extend what has been done in the local multiplicative case to the additive case, and then to problem with non-local terms or constraints.

The key point is that Pohožaev's method, which is well known to provide non-existence results in supercritical problems, also gives uniqueness results. One can incidentally notice that non-existence results in many cases, for instance supercritical pure power law, are more precisely non-existence results of non-trivial solutions. The trivial solution is then the unique solution.

The strength of the method is that minimal geometrical assumptions have to be done, and the result holds true even if no symmetry can be expected. As a non-trivial byproduct of our results, when the domain Ω presents some special symmetry, for instance with respect to an hyperplane, then it follows from the uniqueness result that the solution also has the corresponding symmetry.

Theorem 8 can be illustrated by the following examples. 1 .

 1 Exponential case: if f (u) = e u and d ≥ 3, then G(µ) ≡ d+2+(d-2) log( d-2 2 d ) does not depend on µ. If d = 2, G(µ) ≡ 4. In both cases (

  and depends only on the dimension d.It is indeed known that f ′ δ = δ f δ-1 , f δ (z) ∼ Γ(δ + 1) e z as z → -∞ and f δ (z) ∼ u δ+1 /(δ + 1) as z → +∞. From (27), we deduce that α M 2 |∂Ω| ≤ Ω 2 d F (z) -F (µ) -(d -2) z f (z) dx + (d -2) Ω µ f (z) dxwith z := u + µ. By dropping the term F (µ), we see that the first integral in the right hand side is bounded by G d |Ω|, and the second one by (d -2)µ M . Since f is increasing and u positive, f (µ) |Ω| ≤ Ω f (z) dx = M and therefore µ ≤ f -1 (M/|Ω|)). As a consequence, (4) has no bounded solution if α M 2 > |∂Ω| G d |Ω| + (d -2) M f -1 M |Ω| .

Acknowledgments. J.D. thanks J.-P. Puel for explaining him the method in the local, multiplicative case, for the exponential nonlinearity, f (u) = e u , and X. Cabré for pointing to him [65]. The authors thank a referee for pointing them an important missing assumption and M. Jakszto for pointing them references on the Poincaré / Friedrichs inequality. This work has been partially supported by the EU financed network HPRN-CT-2002-00282 and the Polonium contract no. 13886SG. It was also partially supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge "Harmonic Analysis, Nonlinear Analysis and Probability" MTKD-CT-2004-013389 and by the Polish Ministry of Science project N201 022 32/0902. c 2009 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.