Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction

Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyse the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller-Segel model for general initial data.


Introduction.
Ground states of free energies play a crucial role in the long time asymptotics of nonlinear aggregation diffusion models. These nonlocal partial differential equations (PDEs) are ubiquitous in the mathematical modeling of phenomena which involve a large number of particles. For instance, nonlocal drift-diffusion equations show up naturally in semiconductor modeling, bacterial chemotaxis, granular media, and many other areas; see [18,9,13] and the references therein. These equations are just based on two competing mechanisms, namely the attraction, modeled by a nonlocal force, and the repulsion, modeled by a nonlinear diffusion.
One of the archetypal models of nonlinear aggregation diffusion is the so-called classical parabolic-elliptic Keller-Segel model. This model was classically introduced as the simplest description for chemotactic bacteria movement in which the tendency of linear diffusion to spread repels the attraction due to the logarithmic kernel interaction in two dimensions. Although there is a large amount of literature in this field, many advances have been made in the last 10 years thanks to the combination of different ideas ranging from functional inequalities to entropy-entropy dissipation techniques passing through optimal transport. We refer the reader to [9,8,6,11] and the references therein for some aspects of this fair competition case in which there is a well-defined critical mass. In fact, here a clear dichotomy arises: if the total mass of the system is less than the critical mass, then the long time asymptotics are described by a self-similar solution, while for a mass larger than the critical one, there is finite time blow-up. For the exact critical mass case, a detailed study has also been performed in [8,6].
The existence of a well-defined critical mass can be generalized to more dimensions if one allows for degenerate diffusions. In fact, let us consider the evolution of the probability density ρ given by where d ≥ 2, and with the homogeneous kernel W(x) = |x| α /α with −d < α < 0. By choosing α = 2 − d, d ≥ 3, and m = 2 − 2/d, it was shown in [7] that there exist a critical mass and an exact dichotomy as in the classical Keller-Segel model. This is based on the fact that these equations share a common structural setting, namely, they have a well-defined free energy functional given by and that the two terms in (1.2) scale equally by dilations in that particular case. Actually, this fact is also satisfied by all the fair competition cases in which m = (d − α)/d. Therefore, there is a well-defined critical mass in all the fair competition cases by generalizing the arguments in [6]. While the analysis of the fair competition cases can be considered advanced, it is not so for both the diffusion dominated case, m > (d − α)/d, and the aggregation dominated case, m < (d − α)/d. Regarding the latter, recent results in [4] discriminate blow-up and global existence depending on the initial conditions and on the exponent m for the particular case of α = 2 − d, d ≥ 3. Other results in this direction also appear in a series of papers by Sugiyama [27,28,26]. However, in the diffusion dominated case, m > (d − α)/d, there is little information about the long time asymptotics, seemingly due to the lack of confinement by the interaction kernel; see [4,28]. It is actually proved that solutions exist globally, and that they are bounded uniformly in time without further information about their behavior at infinity. Existence of steady states in the case α = 2−d, d ≥ 3, m > 2−2/d has been analyzed in [4].
In this paper, we build upon the understanding of the long time asymptotics in the classical diffusion dominated case in two dimensions. Calvez and Carrillo proved in [10] that solutions corresponding to the classical diffusion dominated two-dimensional Keller Segel model, exist globally and are uniformly bounded in time. However, they were not able to clarify the long time asymptotics. Here we encounter once again the structural setting of a free energy functional given by Since the free energy G[ρ] is decaying in time through the evolution of the flow associated to (1.3), one may expect convergence towards the possible (global) minimizers of G[ρ] over mass densities. Due to the translational invariance of (1.3), we will consider the set of admissible functions for the functional G[ρ] as the set of zero center-of-mass densities for a given mass M > 0. This work is entirely devoted to showing the existence of a unique global minimizer of the free energy G[ρ] in Y M . Furthermore, we will show that this global minimizer is a radially decreasing compactly supported continuous density function smooth inside its support, and that it is characterized as the unique (up to translation) compactly supported stationary state of the diffusion dominated Keller-Segel model (1.3) with m > 1. This unique profile is the clear candidate to describe the long time asymptotics of the evolution model (1.3) for general initial data, which will be treated elsewhere. Finally, we point out that for the model (1.1) with d ≥ 3 and in the diffusion dominated case m > 2 − 2/d, this asymptotic regime is shown in [19,Theorem 5.6,Corollary 5.9] in the class of radially symmetric, continuous, and compactly supported initial data.
From the technical point of view, we cannot resort to classical concentrationcompactness principles as used in [21,22,7,2], which are closely related to homogeneous kernels as in (1.2). Actually, we take advantage of the logarithmic interaction kernel to show the confinement of the density in section 2. This is the basic building block in showing the existence of global minimizers that are radially decreasing due to symmetric decreasing rearrangement techniques. In section 3 we further identify them and show that they are compactly supported and smooth in their support using variational techniques. Finally, in section 4 a nonstandard application of the moving plane method shows that compactly supported stationary states of (1.3) coincide with the unique up to translation global minimizer of the previous sections.

Minimization of the free energy functional.
Our goal is to minimize the functional G[ρ] given by (1.4) defined on Y M for a given mass M > 0. Set Let us first check that W[ρ] > −∞ in this class. Notice that for each ρ ∈ Y M , Hölder's inequality implies that where ρ # is the spherical decreasing rearrangement of ρ; see, for instance, [17,3,29] for the basic definitions and related properties.
Proof. We split the proof into three parts, proving first that global minimizers must be radial, and thus we can restrict our study to Y rad M . We next show that the functional G is bounded from below in Y rad M , and finally that the infimum is achieved in Y rad M .
Step 1. The candidates to be global minimizers of G are radial. As soon as the interaction term W[ρ] is finite, the interaction functional W[ρ] decreases under rearrangement as proven in [12,Lemma 2], in the sense that This shows that Actually, all the minimizers of G in the class Y M must be radially decreasing, i.e., they lay in the class Y rad M . Indeed, if ρ is a global minimizer in Y M , by inequality (2.1) we have that ρ # is a radially decreasing global minimizer of G. Since the L m -norms of ρ and ρ # are equal, from G[ρ] = G[ρ # ] we deduce that Hence, using [12, Lemma 2] again, we find that ρ must be a translation of ρ # , that is, ρ(x) = ρ # (x − y) for some y ∈ R 2 . Moreover, we have and thus the zero center-of-mass condition holds if and only if y = 0, giving ρ = ρ # , namely, ρ is radially decreasing.
Step 2. G is bounded from below in Y rad M . Here, we follow arguments from [10]. For any ρ ∈ Y M such that ρ log ρ , ρ log(1 + |x| 2 ) ∈ L 1 (R 2 ), the logarithmic Hardy-Littlewood-Sobolev (HLS) inequality [12] implies that there exists a constant C(M ) > 0 such that Let us start by showing a bound from below in a restricted class of densities. Consider ρ in Y rad M and first assume that ρ is continuous, with compact support. Applying (2.2) to (1.4) we have Now, let us choose θ ∈ (0, 1) and a value κ θ,m > 1 such that Then, we deduce and therefore, Hence, we infer from (2.3) that This bound from below being only dependent on the L m -norm can be extended to Y rad M by a density argument that we detail next. If ρ ∈ Y rad M is less regular, let us take a nondecreasing sequence of radially decreasing, compactly supported, continuous nonnegative functions ρ n converging strongly to ρ in L 1 (R 2 ) ∩ L m (R 2 ): such a choice is always possible, since we can approximate ρ first by smooth functions; then the sequence of their rearrangements satisfies the required conditions by the L pcontraction property of the rearrangement map. If ρ n 1 = M n , let us construct the sequence Thus ρ n ∈ Y rad M . Besides, from M n M , we get ρ n ρ strongly in L 1 (R 2 )∩L m (R 2 ), ρ n 1 = M , and we can apply inequality (2.3) to deduce Hölder's inequality implies Concerning the positive part of log |x − y|, since ρ n is a nondecreasing sequence converging to ρ, we have by the monotone convergence theorem that as n → ∞, and thus H(ρ n ) → H(ρ), W(ρ n ) → W(ρ) as n → ∞. Hence, we can pass to the limit in (2.5) and get (2 and let us choose a minimizing sequence of G, i.e., a sequence {ρ n } n∈N in Y rad M such that By the control of the functional G in (2.4), we get that {ρ n } n∈N is bounded in L m (R 2 ), and hence by (2.6) it follows that {W[ρ n ]} n∈N is bounded. In order to control the behavior at infinity, we follow arguments similar to those in [24] and [23,Proposition 7.10]. For all R ≥ 1 and any ρ ∈ L 1 By Hölder's inequality, we have where m = m/(m − 1). In particular, by (2.8) Thus the fact that {W[ρ n ]} n∈N is bounded and (2.11) yield that is, the so-called confinement of the mass. In order to check that {ρ n } is locally equi-integrable, we observe that for given ε > 0, setting for all n ∈ N, that is, the sequence {ρ n } n∈N is equi-integrable. According to the Dunford-Pettis theorem using (2.12) and (2.13), there exists a function ρ 0 ∈ L 1 and ρ 0 1 = M . Furthermore, In particular, the interaction energy W[ρ 0 ] of ρ 0 is bounded from below because the functional G is. Our aim is now to show that W is lower semicontinuous with respect to the L 1 ∩ L m weak convergence, taking advantage of some arguments shown in [5]. Then fix ε ∈ (0, 1), R > 1, and write Observe that we can use the equi-integrability of the sequence {ρ n } and the fact that ρ n ρ 0 weakly in L 1 + (R 2 ) to apply Lemma 2.3 in [5] and find It remains then to get a bound from below of the last integral W R [ρ n ] for large n. In order to do this, we first point out that (2.16) implies that the sequence of densities (ρ n ⊗ ρ n )(x, y) converges to (ρ 0 ⊗ ρ 0 )(x, y) in the weak- * sense as measures. Then using the fact that the function log + |x − y| is bounded from below and obviously lower semicontinuous in the set (x, y) : In particular, combining this last inequality with (2.8) we derive that and W [ρ 0 ] is finite. Now, using (2.17) and (2.18) we get Using (2.14), this gives in turn By taking the rearrangement ρ # 0 of ρ 0 , since W[ρ 0 ] is finite, inequality (2.1) implies Let us point out that the previous proof works in any dimension since the logarithmic HLS inequality holds with a constant that depends only on the dimension and the mass. We also emphasize that the use of the logarithmic potential is crucial here, since we do not know how to prove a quantitative confinement property when the Newtonian potential for dimensions larger than two is used instead.

Identification, regularity, and uniqueness of global minimizers.
Our aim is to show a full characterization of any minimizer ρ 0 of the functional G to relate them to the steady states to the two-dimensional Keller-Segel model. We first deduce the Euler-Lagrange conditions satisfied by critical points of the functional.
Theorem 3.1. Let ρ 0 ∈ Y M be a global minimizer of the free energy functional G defined in (1.4). Then ρ 0 satisfies As a consequence, any global minimizer of G verifies Proof. The technical difficulty here is to make good variations of the minimizer under the low available regularity conditions on ρ 0 obtained from Theorem 2.1, namely, . We use some ideas from [25]. We first show (3.1). Let ρ 0 be a radially decreasing minimizer of G. Taking any ε > 0 and a test function ψ ∈ C ∞ 0 (R 2 ) such that ψ(x) = ψ(−x), let us define the function , and in addition, Due to (3.4), we have ρ 0 + εϕ ∈ Y M , and thus we can calculate the first variation (3.5) Using the first order Taylor expansion of (ρ 0 + εϕ) m at ε = 0, we have By the definition of G ε (t), it is obvious that for all t ∈ [0, 1] and ε < ε 0 , one has and then Lebesgue's dominated convergence yields In addition, as ρ 0 (x) log(1 + |x| 2 ) ∈ L 1 (R 2 ), the algebraic inequality log |x − y| ≤ 1 2 (log 2 + log(1 + |x| 2 )) + log(1 + |y| 2 ) and the estimate |ϕ(x)| ≤ 2 ψ ∞ ρ 0 (x) give W[ϕ] ≤ C. Therefore, using this last property and (3.6) to pass to the limit in (3.5) as ε → 0, we obtain that Taking −ψ instead of ψ, we finally obtain for all even functions ψ ∈ C ∞ 0 (R 2 ). Hence, we deduce Now, we turn to the proof of (3.2). Let us take an even function ψ ∈ C ∞ 0 (R 2 ) with ψ ≥ 0 such that ψ(x) ∈ [0, 1], and let us define the function In addition, denoting by | · | N the N -dimensional Lebesgue measure, we have and then ρ 0 + εϕ ≥ 0 for small ε in supp(ρ 0 ) and outside since ψ ≥ 0, and hence ρ 0 + εϕ ∈ Y M . Arguing as before, we obtain from (3.7) Remark 3.2. Let us point out that inequality (3.2) is a consequence of the positivity and mass constraints on the class of possible minimizers, i.e., due to the fact that we are working with an optimization problem with convex constraints.
Actually, we can show many properties about the regularity of global radial minimizers to the free energy functional G. Now, we give some information concerning the asymptotic behavior at infinity of the logarithmic potential of any density ρ 0 ∈ Y M , namely, the Newtonian potential With this further result in hand, we are now ready to give more information about the regularity of the radially decreasing minimizers of G. Proof. Let ρ 0 be a radially decreasing minimizer of G. Then there is a ball B R (0) such that {ρ 0 > 0} = B R (0). Let us consider the logarithmic potential of ρ 0 , namely, the function u defined in (3.9). As ρ 0 ∈ L m (R 2 ), by [16,Lemma 9.9] we have u ∈ W 2,m loc (R 2 ). By Morrey's theorem (m > 1), it follows that u ∈ L ∞ loc (R 2 ), and by (3.8) we get Thus from the monotonicity of ρ 0 we deduce ρ 0 ∈ L ∞ (R 2 ). Hence [16,Lemma 4.1] implies u ∈ C 1 (R 2 ). Now, for all r > 0 we define the mass function of ρ 0 , Take any R 1 < R and consider the following boundary value problem: The logarithmic potential (3.9) solves problem (3.12), whence u = v on B R1 (0). On the other hand, the solution of (3.12) can be written as in [29,3]: if r = |x| ∈ (0, R 1 ), where ρ * 0 is the one-dimensional decreasing rearrangement of ρ 0 . Differentiating we get u (r) = − 1 2πr that is, By identity (3.13) it follows that ρ 0 is smooth inside its support. Indeed, following some arguments of [19], first we observe that the function  Thus u = K * ρ 0 is differentiable everywhere in the positive set {ρ 0 > 0} = B R (0) of ρ 0 . This property and (3.11) imply that ρ 0 is differentiable in B R (0), so f (r) is twice differentiable. Then we can repeat this argument and conclude. Let us prove that ρ 0 has compact support. There are two different ways to prove this property: one is based on the asymptotic behavior of the log-potential, and the other relies on a pure ordinary differential equation (ODE) approach relating our global minimizers to nonlinear elliptic equations. We show both methods since they give complementary information. Concerning the first one, we simply use (3.10) to infer that u(x) ∼ M K(x) → −∞ as |x| → ∞; hence if (3.11) were satisfied for all x, for a sufficiently large R we would have ρ 0 < 0 for all |x| > R, which is a contradiction. Then ρ 0 must have compact support.
The other argument to prove that ρ 0 is compactly supported is the following. By contradiction, let us suppose that supp(ρ 0 ) = R 2 . Then (3.8) implies that the function θ := ρ m−1 0 ∈ L m m−1 solves the problem (3.14) Since θ is radial, the first equation in (3.14) (which is an Emden-Fowler type equation) can be rewritten as We obtain that in both cases m < 2, m > 2 all the proper solutions to (3.15) are oscillatory, namely, they have a sequence of zeros tending to +∞. But this contradicts the fact that θ is everywhere positive. The case m = 2 is even simpler. Indeed, in this case when θ satisfies the linear problem (recall that ρ 0 is smooth), we have −(r θ ) = r 2 θ, r > 0, and the condition θ → 0 as r → ∞ obliges θ to have the form which is clearly oscillating, leading to a contradiction. Therefore, the support of ρ 0 must be compact. Finally, the Newtonian potential being smooth, together with (3.3), implies that the density ρ 0 is Hölder continuous in R 2 with exponent 1/(m − 1). Therefore, we can write θ in terms of a scaling of the solution ζ to the same problem in the unit ball, namely, With the above regularity of global minimizers, it is easy to show that the distributional gradient in R 2 of ρ m 0 satisfies ∇ρ m 0 = m m−1 ρ 0 ∇ρ m−1 0 = mρ m−1 0 [∇ρ 0 ] + , with the last gradient being the classical gradient in its support. As a consequence, in the sense of distributions. We have deduced the following result. Now, let us show the uniqueness of stationary states among the set of radially decreasing compactly supported smooth inside their support solutions. As a consequence, we conclude the uniqueness of global minimizers taking into account Corollary 3.5 and Theorems 2.1 and 3.3. With this aim, we briefly recall some of the main results contained in [19]. We first start with the definition of mass concentration.
Definition 3.6. Let ρ 1 , ρ 2 ∈ L 1 loc (R N ), N ≥ 1, be two radially symmetric functions on R N . We say that ρ 1 is less concentrated than ρ 2 , and we write ρ 1 ≺ ρ 2 if for all r > 0 we get The partial order relationship ≺ is called comparison of mass concentrations. Of course, this definition can be suitably adapted if ρ 1 , ρ 2 are radially symmetric and locally integrable functions on a ball B R . Besides, if ρ 1 and ρ 2 are locally integrable on a general open set Ω, we say that ρ 1 is less concentrated than ρ 2 and we write again t) is a locally integrable function on R N for all times t ≥ 0, we define the time dependant mass function of ρ as If ρ(x, t) is the solution to the evolution problem (1.3) where the initial data ρ(x, 0) is a continuous, compactly supported, radially decreasing function, then it is easy to check (see [19]) that the mass function M ρ (r, t) satisfies, in the support {x : |x| < R(t)} of ρ(·, t), the PDE Let us take a function ρ(x, t) being C 1 in its positive set and ρ(·, t) ∈ L 1 (R 2 )∩L ∞ (R 2 ) for each t ≥ 0. We will say that ρ is a subsolution (resp., a supersolution) to (3.18) if the sign ≤ (resp., the sign ≥) replaces the equal sign in (3.18).
In [19] the following result concerning the mass comparison is proved, which is readily seen to hold also in dimension N = 2.
Proposition 3.7. Assume that ρ 1 , ρ 2 are, respectively, a subsolution and a supersolution to (3.18). Suppose that ρ 1 , ρ 2 preserve the mass through time, i.e., and that ρ 1 is less concentrated than ρ 2 at the initial time, namely, Then the mass functions preserve the same order for all times: It is also possible to show a two-dimensional version of [19,Theorem 5.6], showing an exponential convergence of the mass function of the solution to (1.3) with a generic radial initial data to the mass function of a steady state having the same mass. Notice that the existence of a radially decreasing steady state with given mass M is guaranteed by Corollary 3.5 and Theorems 2.1 and 3.3.
where C depends on ρ(x, 0), M , and m, and the rate λ depends only on M . Proof. We briefly provide the main arguments of the proof, since it is totally analogous to the proof of [19,Theorem 5.6], to which the interested reader should refer. We can always assume that ρ(0, 0) > 0, as otherwise ρ(0, t) will become positive in finite time (see [19,Corollary 5.5]). It is always possible to choose a small positive constant a such that (3.20) a 2 ρ 0 (ax) ≺ ρ(x, 0) and a −2 ρ 0 (a −1 x) ρ(x, 0).
For a given nonnegative function ξ(x, t) which is differentiable in its positive set, let us introduce its velocity field − → v (x, t; ξ) through the formula It is possible to prove that if we consider the velocity field − → v (x; ρ a 1 ) of ρ a 1 (x) = a 2 ρ 0 (ax), then its inward normal component satisfies, for all a ∈ (0, 1), the estimate Since in the positive set of ρ 0 we have, for suitable positive constants C 1 , C 2 , by the previous estimate we find v(r) ≥ C 1 (1 − a 2 (m − 1))a 2 r.
where we impose that the scaling factor k(t) satisfies the following ODE with initial data k(0) = a: Then one proves that φ(r, t) is a subsolution to (3.18) and that the following exponential estimate holds: Similarly, we construct a supersolution to (3.18) by taking into account the constant C 2 in (3.21) and defining the function where k(t) solves this time the following ODE with initial data k(0) = 1/a: Then η(r, t) is shown to be a supersolution to (3.18), whose mass function satisfies the estimate Now, from relations (3.20) we find φ(·, 0) ≺ ρ(·, 0) ≺ η(·, 0), and thus by Proposition 3.7 we get for all r, t > 0. Then inequalities (3.22)-(3.23)-(3.24) yield As a consequence, the uniqueness of a radially decreasing steady state of a given mass M follows: in fact, if there were two such steady states ρ 0 ,ρ 0 , inequality (3.19) would ensure that M ρ0 (r) = Mρ 0 (r) , and therefore differentiating we find ρ 0 =ρ 0 . Summarizing the results of the last two sections, we conclude with the following.

Symmetry of the steady states.
The aim of this section is to establish the symmetry of any compactly supported steady state, and not only of global minimizers, which in turn will yield the uniqueness of compactly supported steady states. Consider a nonnegative density ρ ∈ Y M and notice that, thanks to the fact that ρ log ρ and ρ log(1 + |x| 2 ) belong to L 1 (R 2 ), the logarithmic potential associated to ρ, denoted in the rest of the paper by u = K * ρ, is well defined. This is, for instance, a consequence of the logarithmic HLS inequality (2.2); see [12]. Let us specify the definition of steady state for the nonlinear diffusion Keller-Segel model (1.3) following [25,4].
Definition 4.1. A nonnegative compactly supported density ρ ∈ Y M is a stationary state for the evolution problem , and the couple (ρ, u) satisfies in the distributional sense, with u being the Newtonian potential associated to ρ as in (3.9). Let us first observe that the nonlinear term in the right-hand side of (4.1) makes sense for compactly supported steady states. Notice that the logarithmic potential u = K * ρ is an L 1 distributional solution of −Δu = ρ with ρ ∈ L m (R 2 ), m > 1. Thus, using the elliptic regularity theory [16, Lemma 9.9], we deduce that u ∈ W 2,m loc (R 2 ) and, thanks to the Sobolev embedding in dimension 2, we have that u ∈ C 1,α loc (R 2 ) for some α > 0. On the other hand, by the fact that ρ ∈ L m (R 2 ), m > 1, and ∇u is locally bounded, we see that the left-hand side of (4.1) belongs to W −1,m loc (R 2 ). Noticing that ρ m is an L 1 distributional solution of (4.1) with datum in W −1,m and that ρ m = 0 on the boundary of a sufficiently large ball by the compact support hypothesis, we see that ρ m is in fact a weak W 1,m loc solution of (4.1); cf. [20]. Sobolev embedding shows that both ρ m and ρ belong to some Hölder space C 0,α (R 2 ). Since m > 1 and ρ m−1 ∈ W 1,m loc (R 2 ), we have ∇ρ m = m m−1 ρ∇ρ m−1 . We conclude that wherever ρ is positive, (4.1) can be interpreted as for all u ∈ R and x ∈ Ω. We are now ready to state our symmetry result.
Theorem 4.2. Let ρ ∈ Y M be any nonnegative compactly supported stationary state. Then ρ is radially symmetric about the origin.
The proof of Theorem 4.2 will be achieved thanks to a nonstandard moving plane type argument for u, especially thanks to a precise decay estimate at infinity and a symmetry property for the function G introduced above. This result is known in the corresponding range of nonlinearities in larger dimensions. Here, the main technical difficulty is to deal with the logarithmic behavior of the Newtonian potential in two dimensions.
First of all, we need to prove a precise decay estimate for u: we already know, thanks to (3.10), that u(x) behaves like −M log |x| when |x| is large, but unfortunately this is not enough for our purposes. Let us assume that supp(ρ) ⊂ B ro (0), i.e., ρ(x) = 0 for any |x| > r o : we can refine the asymptotic behavior of u as given by the following.
Proposition 4.3. There are C 1 , C 2 > 0 such that for all |x| ≥ 2r o , We can proceed essentially as in the proof of [25,Lemma 1]. Notice that for |x| ≥ 2r o we have supp (ρ) ⊂ {|y| ≤ |x| 2 }. Thus, thanks to the homogeneity of the derivatives of the kernel K(x) and the zero center-of-mass condition, we have due to a simple Taylor expansion for some 0 < σ(y) < 1, leading to the desired estimate (4.3). Replacing the kernel − log |x| with its x 1 derivative −x 1 /|x| 2 (which is homogeneous of degree −1), the same proof leads to (4.4).
We will now start a moving plane type procedure in order to establish that the solution is even with respect to the first variable. Then, thanks to the rotational invariance, we will deduce that u is even with respect to any plane through the origin, which in turn means that u is in fact radial. Hence, let x λ = σ λ (x) = (2λ − x 1 , x 2 ) be the reflection of x ∈ Σ λ = {x 1 < λ} with respect to the plane T λ = {x 1 = λ}, and let u λ (x) := u(x λ ) = u(2λ − x 1 , x 2 ) be the corresponding reflection of u.
Thanks to the strict monotonicity of our kernel, we will start by showing that well away from the x 2 axis the difference of u and u λ is nonpositive as shown by the following.
For x ∈ Σ λ and λ < 0, we observe that |x| − λ ≤ |x λ | ≤ |x|. Hence, we get while u = u λ on T λ by definition. Observe that since ρ is C 0,α , by Schauder estimates, u and u λ satisfy, respectively, in the classical pointwise sense. Moreover, since for |x| > r we have that ρ(x) = 0 while ρ(x λ ) ≥ 0 always, by (4.6) we finally obtain By (4.7) and the classical comparison principle, we find u ≤ u λ on Σ λ . Next, we will show that the same property of Lemma 4.4, for a fixed negative λ, is true outside a sufficiently large ball.
Moreover, by the continuity with respect to λ and the fact that Σ λ is a decreasing set-valued function of λ, we see that Λ is in fact attained. Our aim is then to show that Λ = 0. However, since our problem is not autonomous (notice the x dependence of the nonlinearity g(x, u) given by (4.2)), we cannot proceed with a standard moving plane argument, and we need to recall from [25] an important reflection property of the function G(x) introduced above. Lemma 4.6. G(x) = G(x λ ) for any x ∈ Σ λ and λ ≤ Λ.
We are now ready to prove Theorem 4.2.

Proof of Theorem 4.2.
We will proceed with a moving plane argument for u(x) := (K * ρ)(x). Recalling the definition of Λ given in (4.12) as the maximal negative parameter λ for which the reflection u λ is larger than u, we want to prove that Λ = 0. So let us argue by contradiction and suppose that Λ < 0. Recall that u satisfies −Δu = g(x, u) with the nonlinearity g(x, u) in (4.2) being nonnegative and increasing with respect to u. By direct computation and thanks to Lemma 4.6, we also get −Δu Λ = g(x Λ , u Λ ) = g(x, u Λ ) . Now, by the continuity with respect to λ and x we have that u Λ (x) ≥ u(x) for x ∈ Σ Λ . This implies that g(x, u) ≤ g(x Λ , u Λ ) for any x ∈ Σ Λ , i.e., Δu ≥ Δu Λ . By the strong comparison principle, we infer that either u ≡ u Λ or u < u Λ on Σ Λ . But the case u ≡ u Λ can be ruled as follows. Since u satisfies (4.13) −Δu = ρ, we have that ρ(x) = ρ(x Λ ), and thus by the zero center-of-mass condition for ρ and through the change of variables (x 1 , x 2 ) = (2Λ − x 1 , x 2 ), we get therefore Λ = 0, a contradiction. Hence u < u Λ on Σ Λ . For small ε > 0 with Λ + ε < 0, let us consider the cap Σ Λ+ε with corresponding reflection u Λ+ε . We will proceed by dividing Σ Λ into four subsets as illustrated in Figure 2. Let R Λ+ε > 2r o be as in Lemma 4.5, and consider the concentric balls B(0, 2r o ) and B(0, R Λ+ε ). We divide the cap Σ Λ+ε into four subsets given by On the set A 1 , we can apply Lemma 4.5 to get that u Λ+ε (x) ≥ u(x) for all x ∈ A 1 . Observe that the set A 2 is compact; then by continuity the fact that u Λ (x) > u(x) implies that u Λ (x) ≥ u(x) + 2σ for some small σ > 0 and for all x ∈ A 2 . Thus, by the continuity with respect to λ, we see that u Λ+ε (x) ≥ u(x) + σ for all x ∈ A 2 for ε small enough, which means On the set A 3 we need to argue as follows. Since u < u Λ on Σ Λ and u = u Λ on T Λ , by Hopf's lemma we know that ∂ ν (u − u Λ ) > 0 on T Λ . But ∂ ν (u − u Λ ) = 2∂ ν u = 2u x1 > 0 on T Λ . In particular, there exists a constant σ o such that u x1 ≥ σ o > 0 on T Λ ∩ B(0, 2r o ), which implies that there exists ε > 0 small enough such that u x1 ≥ σ o /2 on A 3 by continuity. Then we can show that and thus ρ is radially decreasing. The same argument as in the proof of Theorem 3.3 shows that ρ is smooth inside its support.
Theorem 4.8. There is a unique up to translation compactly supported steady state to (1.3) with mass M . Moreover, such steady state is radially decreasing, continuous, and smooth in its support, and it coincides (up to translation) with the global minimizer of the free energy functional G in Y M .
Proof. By Theorem 4.2 and Remark 4.7 we have that any compactly supported steady state ρ is a radially symmetric decreasing continuous function smooth in its support. Then Theorem 3.9 provides its uniqueness and identifies it with the global minimizer of G in the class Y M .